2. Mathematical overview

This section introduces some quantum information concepts and defines the notation used in the chapter.

### 2.1. Density matrix and quantum channels

As the convex combination of positive matrices is also positive, then the space of positive operators forms a convex cone in Hilbert-Schmidt L C<sup>N</sup> [23]. If we restrict the matrices in the positive cone to be trace = 1, we arrive to another set of matrices, that is named the set of density matrices. This set of operators also originates a vector space, this space is denoted as D C<sup>N</sup> . Therefore, the matrices that belong to this set, or the vectors in this vector space, are named density matrices.

Definition 1. A linear positive operator ρ∈ D C<sup>N</sup> is a density matrix and represents the state of a quantum system, if it satisfies the following properties:


As the convex combination of density matrices is a density matrix, the vector space D is a convex set whose pure states are projectors onto the real numbers. A given density matrix ρ∈ D C<sup>N</sup> is a pure state if it satisfies:

$$
\rho = \rho^2,\tag{1}
$$

then the state ρ is a rank-1 matrix and it can be written as:

equivalent formulations for characterization and quantification of quantumness of correlations: quantum discord [10, 11], minimum local disturbance [12–14] and geometrical approach [15–17]. This chapter presents in detail two different ways to relate quantum entanglement and quantumness of correlations. The main purpose of this chapter is to discuss that quantumness of correlations plays an interesting role in entanglement distillation protocol. Entanglement and quantumness of correlations connect each other in two different pictures. The relation derived by Koashi and Winter [18] demonstrates the balance between quantumness of correlations and entanglement in the purification process [19]. This balance leads to a formal proof for the irreversibility of the entanglement distillation protocol, in terms of quantumness of correlations [20]. In the named activation protocol, the quantumness of correlations of a given composed system can be converted into distillable entanglement with a measurement appara-

The chapter is organized as follow. In Section 2, a mathematical overview is presented, and the notation is defined. Section 3 introduces some important concepts about the notion of quantum correlations: entanglement and quantumness of correlations. Section 4 presents the Koashi-Winter relation and its role in the irreversibility of quantum distillation process. Section 5 is intended to the description of the activation protocol, and the demonstration that

This section introduces some quantum information concepts and defines the notation used in

As the convex combination of positive matrices is also positive, then the space of positive operators forms a convex cone in Hilbert-Schmidt L C<sup>N</sup> [23]. If we restrict the matrices in the positive cone to be trace = 1, we arrive to another set of matrices, that is named the set of density matrices. This set of operators also originates a vector space, this space is denoted as D C<sup>N</sup> . Therefore, the matrices that belong to this set, or the vectors in this vector space, are named density matrices.

Definition 1. A linear positive operator ρ∈ D C<sup>N</sup> is a density matrix and represents the state of a

As the convex combination of density matrices is a density matrix, the vector space D is a convex set whose pure states are projectors onto the real numbers. A given density matrix

<sup>ρ</sup> <sup>¼</sup> <sup>ρ</sup><sup>2</sup>

; (1)

quantumness of correlation can be activated into distillable entanglement.

tus during the local measurement process [21, 22].

64 Advanced Technologies of Quantum Key Distribution

2. Mathematical overview

2.1. Density matrix and quantum channels

quantum system, if it satisfies the following properties:

the chapter.

• Hermitian: ρ = ρ†

• Trace one: Tr(ρ)=1

• Positive semi-definite: ρ ≥ 0;

ρ∈ D C<sup>N</sup> is a pure state if it satisfies:

$$
\rho = |\psi\rangle\langle\psi|.\tag{2}
$$

The set of pure states is a 2(<sup>N</sup> � 1)-dimensional subset of the (N<sup>2</sup> � 2)-dimensional boundary of D C<sup>N</sup> . Every state with at least one eigenvalue equal to zero belongs to the boundary [23]. For two-dimensional systems (it is also named qubit [24]), the boundary is just composed of pure states.

Consider a linear transformation <sup>Φ</sup> : <sup>L</sup> <sup>C</sup><sup>N</sup> ! <sup>L</sup> <sup>C</sup><sup>M</sup> . This map represents a physical process, if it satisfies some conditions, determined by the mathematical properties of the density matrices. Indeed, to represent a physical process, the transformation must map a quantum state into another quantum state, <sup>Φ</sup> : <sup>D</sup> <sup>C</sup><sup>N</sup> ! <sup>D</sup> <sup>C</sup><sup>M</sup> . It holds if <sup>Φ</sup> satisfy the following properties:

• Linearity: As a quantum state can be a convex combination of other quantum states, the map must be linear. For two arbitrary operators ρ, σ∈ D C<sup>N</sup>

$$\Phi(\rho + \sigma) = \Phi(\rho) + \Phi(\sigma);\tag{3}$$

• Trace preserving: The eigenvalues of the density matrix represent probabilities, and it sum must be one, then a quantum channel must to keep the trace of the density matrix:

$$\operatorname{Tr}[\Phi(\rho)] = 1.\tag{4}$$

• Completely positive: Consider a channel Φ : Dð Þ! C<sup>A</sup> Dð Þ C<sup>A</sup> and a quantum state ρ, σ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , then

$$\mathbb{T} \otimes \Phi(\rho) \succeq 0. \tag{5}$$

The map that satisfies this property is named completely positive map. The linear transformations mapping quantum states into quantum states are named completely positive and trace preserve (CPTP) quantum channels. The space of quantum channels that maps N � N density matrices onto <sup>M</sup> � <sup>M</sup> density matrices is denoted as <sup>C</sup> <sup>C</sup><sup>N</sup>; <sup>C</sup><sup>M</sup> .

#### 2.2. Measurement

Measurement is a classical statistical inference of quantum systems. The measurement process maps a quantum state into a classical probability distribution.

We can define a measurement as a function <sup>Π</sup> : <sup>Σ</sup> ! <sup>P</sup>ð Þ <sup>C</sup><sup>Γ</sup> <sup>1</sup> , associating an alphabet Σ to positive operators f g Π<sup>x</sup> <sup>x</sup> ⊂Pð Þ C<sup>Γ</sup> . For a given density matrix ρ∈ Dð Þ C<sup>Γ</sup> , the measurement process consists in to chose an element of Σ randomly. This random choice is represented by a

<sup>1</sup> Just to clarify the notation, when we write a subscript in the complex euclidean vector space, as CΓ, it represents a label to the space, it shall be very useful when we study composed systems. When we write a superscript on it, it represents the dimension of the complex vector space. For example, if dim(CΓ) = N, we can also represent this space as C<sup>N</sup>, the usage of the notation will depend on the context.

probability vector p ! ∈ R<sup>N</sup> <sup>þ</sup>, with <sup>N</sup> being the cardinality of the random variable described by <sup>p</sup> !. The elements of the probability vector p ! are given by:

$$p\_\mathbf{x} = \operatorname{Tr}(\Pi\_\mathbf{x} \rho),\tag{6}$$

In order to differ the set of measurement channels from a general CPTP channel, this set is represented as P. A given measurement map M∈P CΓ; C<sup>Γ</sup> ð Þ0 is a quantum channel that maps

by a diagonal density matrix as in Eq. (11). The dimension of C<sup>Γ</sup><sup>0</sup> is the number of outcomes of

For general measurements, described by positive operators valued measure (POVM), the measurement process can be described by a measurement channel Φ ∈Pð Þ CΓ; C<sup>Γ</sup> . The description performed above can be followed to describe these general measurements, indeed projective measurements are a restriction for a POVM composed by orthogonal operators. Consider a set of positive operators {Mx}x, representing a POVM, then Tr[Mxρ] = px are the elements of a

Using the Naimark's theorem, the measurement channel is described as a dephasing channel on a state in a enlarged space. In other words, for POVMs whose elements are rank-1 and linearly independent, it is possible to associate a projective measurement on an enlarged

Theorem 3 (Naimark's theorem). Given a quantum measurement M∈P CΓ; C<sup>Γ</sup> ð Þ0 , with POVM

<sup>x</sup>¼<sup>0</sup>, there exists a projective measurement <sup>Π</sup> <sup>∈</sup><sup>P</sup> <sup>C</sup><sup>Γ</sup> ð Þ0 , with elements <sup>Π</sup><sup>y</sup>

The action of the isometry on the state ρ, in the Naimark's theorem, is named as embedding operation. In this way, the isometry will be V ¼ I<sup>Γ</sup> ⊗ j i0 <sup>E</sup> and the enlarged space C<sup>Γ</sup><sup>0</sup> ¼ C<sup>Γ</sup> ⊗ CE. For this simple case, the relation between the POVM elements {Mx}<sup>x</sup> and the projective mea-

� �Π<sup>x</sup> <sup>I</sup><sup>Γ</sup> <sup>⊗</sup> j i<sup>0</sup> <sup>E</sup>

As the measurement can be described by a quantum channel, we can study how quantum

Definition 4. Given a N-partite composed system, represented by the state ρ<sup>A</sup>1,::.; AN ∈

Tr MA<sup>1</sup>

<sup>k</sup><sup>1</sup> ⊗⋯⊗ MAN

h i <sup>k</sup>

kN ρ<sup>A</sup>1,::.; AN

in the sum represents the set of indexes k1,…, kN.

k !

Mx ¼ I<sup>Γ</sup> ⊗ h j 0 <sup>E</sup>

D C<sup>A</sup><sup>1</sup> ⊗⋯⊗ CAN ð Þ, we define the measurement on each subsystem applied locally:

� � <sup>¼</sup> <sup>X</sup>

<sup>Γ</sup> , then the post-measurement state is:

Φ ρ � � <sup>¼</sup> <sup>X</sup> x

<sup>Γ</sup><sup>0</sup> . This probability vector is described

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The Role of Quantumness of Correlations in Entanglement Resource Theory

pxj i ex h j ex : (12)

Tr Mx<sup>ρ</sup> � � <sup>¼</sup> Tr <sup>Π</sup>xVρV† � �; (13)

� �: (14)

! 〉〈 k � � ! �

� �

�; (15)

� �<sup>M</sup>

<sup>y</sup>¼<sup>0</sup> such that:

a density matrix in a probability vector, M : Dð Þ! C<sup>Γ</sup> R<sup>þ</sup>

! ∈ R<sup>þ</sup>

Where {|ex〉}<sup>x</sup> is an orthonormal basis in CΓ.

where V ∈U CΓ; C<sup>Γ</sup> ð Þ0 is an isometry.

surement on the enlarged space {Πx}x:

measurements can be performed locally.

Φ<sup>A</sup><sup>1</sup> ⊗⋯⊗ ΦAN ρ<sup>A</sup>1,::.; AN

<sup>¼</sup> j i <sup>k</sup><sup>1</sup> ⊗⋯⊗ j i kN and the label k!

are the measurement operators on each subsystem.

the measurement.

probability vector p

space.

where k � �! � E

MAx kx n o

kx

elements Mf g<sup>x</sup> <sup>M</sup>

where Π<sup>x</sup> is the measurement operator associated to x ∈ Σ. The alphabet Σ is the set of measurement outcomes, and the vector p ! is the classical probability vector associated to the measurement process Π of a given density matrix ρ. As the outcomes are elements of a probability vector, these elements must be positive and sum to one. Which implies that the measurement operators must sum to identity:

$$\sum\_{\mathbf{x}} \Pi\_{\mathbf{x}} = \mathbb{I}\_{\Gamma}, \tag{7}$$

where I<sup>Γ</sup> is the identity matrix in CΓ. It is easy to check that this condition implies ∑xpx = 1:

$$\sum\_{x} p\_x = \sum\_{x} \text{Tr}(\Pi\_x \rho) = \text{Tr}\left(\sum\_{x} \Pi\_x \rho\right) = \text{Tr}(\rho) = 1. \tag{8}$$

For instance, we shall restrict the measurements to a subclass of measurement operators named projective measurements. As it is shown later, its generalization can be performed via the Naimark's theorem. For projective measurements, the cardinality of p ! is at least the dimension of ρ, and the measurement operators are projectors:

$$
\Pi\_\mathbf{x}^2 = \Pi\_\mathbf{x},
\tag{9}
$$

for any x ∈ Σ. If we consider an orthonormal basis {|ex〉}, where the vectors |ex〉 span CΓ, this set represents a projective measurement for Π<sup>x</sup> = |ex〉〈ex|. The output state is described by the expression:

$$\rho\_x = \frac{\Pi\_x \rho \Pi\_x}{\text{Tr}\left(\Pi\_x \rho\right)}.\tag{10}$$

The set of operators defines a convex hull in Pð Þ C<sup>Γ</sup> , then a measured state represents a pure state in this convex hull. In this way, the post-measurement state can be reconstructed by the convex combination of the output states ρ = ∑xpxρx.

As physical processes are described by quantum channels, it is possible to describe the classical statistical inference of the quantum measurements as a CPTP channel. A channel that maps a quantum state into a probability vector is the dephasing channel. Therefore, the postmeasurement state is the state under the action of the dephasing channel.

Theorem 2. A given map Φ ∈ C CΓ; C<sup>Γ</sup> ð Þ0 is a measurement if and only if:

$$\Phi(\rho) = \sum\_{\mathbf{x}} \text{Tr}\{M\_{\mathbf{x}}\rho\}|e\_{\mathbf{x}}\rangle\langle e\_{\mathbf{x}}|,\tag{11}$$

where ρ∈ Dð Þ C<sup>Γ</sup> , Mx ∈Pð Þ C<sup>Γ</sup> and j i ex ∈ C<sup>Γ</sup><sup>0</sup> .

In order to differ the set of measurement channels from a general CPTP channel, this set is represented as P. A given measurement map M∈P CΓ; C<sup>Γ</sup> ð Þ0 is a quantum channel that maps a density matrix in a probability vector, M : Dð Þ! C<sup>Γ</sup> R<sup>þ</sup> <sup>Γ</sup><sup>0</sup> . This probability vector is described by a diagonal density matrix as in Eq. (11). The dimension of C<sup>Γ</sup><sup>0</sup> is the number of outcomes of the measurement.

For general measurements, described by positive operators valued measure (POVM), the measurement process can be described by a measurement channel Φ ∈Pð Þ CΓ; C<sup>Γ</sup> . The description performed above can be followed to describe these general measurements, indeed projective measurements are a restriction for a POVM composed by orthogonal operators. Consider a set of positive operators {Mx}x, representing a POVM, then Tr[Mxρ] = px are the elements of a probability vector p ! ∈ R<sup>þ</sup> <sup>Γ</sup> , then the post-measurement state is:

$$\Phi(\rho) = \sum\_{\mathbf{x}} p\_{\mathbf{x}} |e\_{\mathbf{x}}\rangle\langle e\_{\mathbf{x}}|.\tag{12}$$

Where {|ex〉}<sup>x</sup> is an orthonormal basis in CΓ.

probability vector p

expression:

! ∈ R<sup>N</sup>

X x

px <sup>¼</sup> <sup>X</sup> x

dimension of ρ, and the measurement operators are projectors:

convex combination of the output states ρ = ∑xpxρx.

where ρ∈ Dð Þ C<sup>Γ</sup> , Mx ∈Pð Þ C<sup>Γ</sup> and j i ex ∈ C<sup>Γ</sup><sup>0</sup> .

The elements of the probability vector p

66 Advanced Technologies of Quantum Key Distribution

ment outcomes, and the vector p

operators must sum to identity:

<sup>þ</sup>, with <sup>N</sup> being the cardinality of the random variable described by <sup>p</sup>

px <sup>¼</sup> Tr <sup>Π</sup>x<sup>ρ</sup> � �; (6)

Π<sup>x</sup> ¼ IΓ; (7)

<sup>x</sup> ¼ Πx; (9)

Tr <sup>Π</sup>x<sup>ρ</sup> � � : (10)

Tr Mx<sup>ρ</sup> � �j i ex h j ex ; (11)

� � <sup>¼</sup> <sup>1</sup>: (8)

! is at least the

! is the classical probability vector associated to the measure-

¼ Tr ρ

! are given by:

where Π<sup>x</sup> is the measurement operator associated to x ∈ Σ. The alphabet Σ is the set of measure-

ment process Π of a given density matrix ρ. As the outcomes are elements of a probability vector, these elements must be positive and sum to one. Which implies that the measurement

> X x

where I<sup>Γ</sup> is the identity matrix in CΓ. It is easy to check that this condition implies ∑xpx = 1:

For instance, we shall restrict the measurements to a subclass of measurement operators named projective measurements. As it is shown later, its generalization can be performed via

for any x ∈ Σ. If we consider an orthonormal basis {|ex〉}, where the vectors |ex〉 span CΓ, this set represents a projective measurement for Π<sup>x</sup> = |ex〉〈ex|. The output state is described by the

<sup>ρ</sup><sup>x</sup> <sup>¼</sup> <sup>Π</sup>xρΠ<sup>x</sup>

The set of operators defines a convex hull in Pð Þ C<sup>Γ</sup> , then a measured state represents a pure state in this convex hull. In this way, the post-measurement state can be reconstructed by the

As physical processes are described by quantum channels, it is possible to describe the classical statistical inference of the quantum measurements as a CPTP channel. A channel that maps a quantum state into a probability vector is the dephasing channel. Therefore, the post-

measurement state is the state under the action of the dephasing channel.

Theorem 2. A given map Φ ∈ C CΓ; C<sup>Γ</sup> ð Þ0 is a measurement if and only if:

Φ ρ � � <sup>¼</sup> <sup>X</sup> x

x Πxρ !

Tr <sup>Π</sup>x<sup>ρ</sup> � � <sup>¼</sup> Tr <sup>X</sup>

Π2

the Naimark's theorem. For projective measurements, the cardinality of p

!.

Using the Naimark's theorem, the measurement channel is described as a dephasing channel on a state in a enlarged space. In other words, for POVMs whose elements are rank-1 and linearly independent, it is possible to associate a projective measurement on an enlarged space.

Theorem 3 (Naimark's theorem). Given a quantum measurement M∈P CΓ; C<sup>Γ</sup> ð Þ0 , with POVM elements Mf g<sup>x</sup> <sup>M</sup> <sup>x</sup>¼<sup>0</sup>, there exists a projective measurement <sup>Π</sup> <sup>∈</sup><sup>P</sup> <sup>C</sup><sup>Γ</sup> ð Þ0 , with elements <sup>Π</sup><sup>y</sup> � �<sup>M</sup> <sup>y</sup>¼<sup>0</sup> such that:

$$\operatorname{Tr}(M\_x \rho) = \operatorname{Tr}(\Pi\_x V \rho V^\dagger),\tag{13}$$

where V ∈U CΓ; C<sup>Γ</sup> ð Þ0 is an isometry.

The action of the isometry on the state ρ, in the Naimark's theorem, is named as embedding operation. In this way, the isometry will be V ¼ I<sup>Γ</sup> ⊗ j i0 <sup>E</sup> and the enlarged space C<sup>Γ</sup><sup>0</sup> ¼ C<sup>Γ</sup> ⊗ CE. For this simple case, the relation between the POVM elements {Mx}<sup>x</sup> and the projective measurement on the enlarged space {Πx}x:

$$M\_x = \left(\mathbb{I}\_\Gamma \otimes \langle \mathbf{0} \vert\_E \right) \Pi\_x \left(\mathbb{I}\_\Gamma \otimes \vert \mathbf{0} \rangle\_E \right). \tag{14}$$

As the measurement can be described by a quantum channel, we can study how quantum measurements can be performed locally.

Definition 4. Given a N-partite composed system, represented by the state ρ<sup>A</sup>1,::.; AN ∈ D C<sup>A</sup><sup>1</sup> ⊗⋯⊗ CAN ð Þ, we define the measurement on each subsystem applied locally:

$$\otimes\_{A\_{\mathbb{L}}} \otimes \dots \otimes \Phi\_{A\_{\mathbb{N}}} \left( \rho\_{A\_{\mathbb{L}}, \dots, A\_{\mathbb{N}}} \right) = \sum\_{\vec{k}} \text{Tr} \left[ M^{A\_{\mathbb{L}}}\_{\vec{k}\_{\mathbb{L}}} \otimes \dots \otimes M^{A\_{\mathbb{N}}}\_{\vec{k}\_{\mathbb{N}}} \rho\_{A\_{\mathbb{L}}, \dots, A\_{\mathbb{N}}} \right] \left| \vec{k} \right> \langle \vec{k} \rangle \,, \tag{15}$$

where k � �! � E <sup>¼</sup> j i <sup>k</sup><sup>1</sup> ⊗⋯⊗ j i kN and the label k! in the sum represents the set of indexes k1,…, kN. MAx kx n o kx are the measurement operators on each subsystem.

Suppose the measurement is performed on some subsystems, the remaining other subsystems are unmeasured. Consider a bipartite system ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> and a measurement acting on the system B, then the measurement map will be written as:

$$\mathbb{T}\_{A}\otimes\Phi\_{\mathbb{B}}\left(\rho\_{AB}\right) = \sum\_{\mathbf{x}} \mathrm{Tr}\_{\mathbb{B}}\left[\mathbb{I}\_{A}\otimes M\_{\mathbf{x}}^{\mathbb{B}}\rho\_{AB}\right]\otimes|b\_{\mathbf{x}}\rangle\langle b\_{\mathbf{x}}|.\tag{16}$$

evidencing this uncertainty, for the measurement observables of a POVM. These probability distributions are classical probability distributions extracted from the quantum systems, and

It is also possible to define a quantum analogous to the Shannon entropy. This quantum entropy is named as von Neumann entropy, and in analogy with Shannon entropy, it is

Definition 5 (von Neumann entropy). Given a density operator ρ ∈ D C<sup>N</sup> � �, the quantum version of

� � ¼ �Tr <sup>ρ</sup> log2<sup>ρ</sup> � �: (23)

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69

<sup>S</sup> <sup>ρ</sup>AB � � ¼ �Tr <sup>ρ</sup>AB log2ρAB � �: (25)

λ<sup>k</sup> log2ð Þ λ<sup>k</sup> ; (24)

� �; (26)

� � <sup>þ</sup> <sup>S</sup>ð Þ <sup>σ</sup> ; (27)

� �; (28)

the Shannon entropy quantifies the degree of surprise related to a given result.

S ρ

S ρ

maximum for the maximally mixed state I=N, where it is Sð Þ¼ I=N log2N.

joint probability. For a bipartite state ρAB, the joint von Neumann entropy is:

Follow some interesting, and useful, properties about von Neumann entropy:

S ρ<sup>A</sup>

<sup>S</sup> <sup>ρ</sup> <sup>⊗</sup> <sup>σ</sup> � � <sup>¼</sup> <sup>S</sup> <sup>ρ</sup>

S X i pi ρi !

� � ¼ �<sup>X</sup>

k

where {λk}<sup>k</sup> are the eigenvalues of ρ = ∑kλk|k〉〈k|. The von Neumann entropy has the same interpretation of the Shannon entropy for the probability distribution composed by the eigenvalues of the density matrix. The von Neumann entropy is zero of pure states, and it is

For composed systems, the von Neumann entropy is analogous to the Shannon entropy of the

1. (Pure states) For a bipartite pure state |ϕ〉AB ∈ C<sup>A</sup> ⊗ CB, the partitions have the same von

� � <sup>¼</sup> <sup>S</sup> <sup>ρ</sup><sup>B</sup>

≥ X i pi S ρ<sup>i</sup>

defined as the expectation value of the operator log2(ρ).

the Shannon entropy is defined as the function:

The von Neumann entropy can be rewritten as:

Neumann entropy:

where ρ<sup>A</sup> = TrB(|ϕ〉〈ϕ|AB).

2. (Additivity) von Neumann entropy is additive:

3. (Concavity) von Neumann entropy is a concave function:

where ρ and σ are density matrices.

for a convex combination ρ = ∑ipiρi.

As the measurement is not acting on A, the post-measured state on A will remain the same. If we write px <sup>¼</sup> TrAB <sup>I</sup><sup>A</sup> <sup>⊗</sup> MB <sup>x</sup>ρAB � � and <sup>ρ</sup><sup>A</sup> <sup>x</sup> <sup>¼</sup> Tr<sup>B</sup> <sup>I</sup><sup>A</sup> <sup>⊗</sup> MB <sup>x</sup> <sup>ρ</sup> ½ � AB TrAB I<sup>A</sup> ⊗ M<sup>B</sup> <sup>x</sup> <sup>ρ</sup>AB ½ �, the post-measured state will be:

$$\mathbb{T}\_A \otimes \Phi\_{\mathbb{B}}(\rho\_{AB}) = \sum\_{\mathbf{x}} p\_{\mathbf{x}} \rho\_{\mathbf{x}}^A \otimes |b\_{\mathbf{x}}\rangle\langle b\_{\mathbf{x}}|.\tag{17}$$

As the measurement is a classical statistical inference process, the local measurement process destroys the quantum correlations between the systems. Indeed the post-measured state is not a classical probability distribution, although it only has classical correlations.

#### 2.3. Quantum entropy

Consider that one can prepare an ensemble of quantum states ξ = {px, ρx}x, accordingly to some random variable X. Classical information can be extracted from the ensemble of quantum states, in the form of a variable Y, performing measurements on the quantum system. The conditional probability distribution to obtain a value y, given as input the state ρ<sup>x</sup> is:

$$p(y|\mathbf{x}) = \operatorname{Tr}(M\_y \rho\_x),\tag{18}$$

where {My}<sup>y</sup> is a POVM. The joint probability distribution X and Y is given by:

$$p(\mathbf{x}, y) = p\_{\mathbf{x}} \text{Tr} \{ M\_y \rho\_{\mathbf{x}} \}. \tag{19}$$

The probability distribution of Y is obtained from the marginal probability distribution:

$$p(y) = \sum\_{\mathbf{x}} p(\mathbf{x}, y) = \sum\_{\mathbf{x}} p\_{\mathbf{x}} \text{Tr}(M\_y \rho\_{\mathbf{x}}) = \text{Tr}\left(M\_y \sum\_{\mathbf{x}} p\_{\mathbf{x}} \rho\_{\mathbf{x}}\right). \tag{20}$$

Considering the Bayes rule:

$$p(\mathbf{x}, y) = p\_{\mathbf{x}} p(y|\mathbf{x}) = p(y)P(\mathbf{x}|y),\tag{21}$$

it is possible to obtain the conditional probability distribution with elements:

$$P(\mathbf{x}|y) = \frac{p\_x p(y|\mathbf{x})}{p(y)}.\tag{22}$$

Even in the case the system is always prepared in the same state, there exists an uncertainty about the measured of an observable. The probability distributions presented above are evidencing this uncertainty, for the measurement observables of a POVM. These probability distributions are classical probability distributions extracted from the quantum systems, and the Shannon entropy quantifies the degree of surprise related to a given result.

It is also possible to define a quantum analogous to the Shannon entropy. This quantum entropy is named as von Neumann entropy, and in analogy with Shannon entropy, it is defined as the expectation value of the operator log2(ρ).

Definition 5 (von Neumann entropy). Given a density operator ρ ∈ D C<sup>N</sup> � �, the quantum version of the Shannon entropy is defined as the function:

$$S(\rho) = -\text{Tr}\left[\rho \log\_2 \rho\right].\tag{23}$$

The von Neumann entropy can be rewritten as:

Suppose the measurement is performed on some subsystems, the remaining other subsystems are unmeasured. Consider a bipartite system ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> and a measurement acting on

Tr<sup>B</sup> I<sup>A</sup> ⊗ MB

<sup>x</sup> <sup>ρ</sup> ½ � AB TrAB I<sup>A</sup> ⊗ M<sup>B</sup>

As the measurement is not acting on A, the post-measured state on A will remain the same. If

x pxρ<sup>A</sup>

As the measurement is a classical statistical inference process, the local measurement process destroys the quantum correlations between the systems. Indeed the post-measured state is not

Consider that one can prepare an ensemble of quantum states ξ = {px, ρx}x, accordingly to some random variable X. Classical information can be extracted from the ensemble of quantum states, in the form of a variable Y, performing measurements on the quantum system. The

p yð Þ¼ jx Tr Myρ<sup>x</sup>

p xð Þ¼ ; y pxTr Myρ<sup>x</sup>

pxTr Myρ<sup>x</sup>

P xð Þ¼ <sup>j</sup><sup>y</sup> pxp yð Þ <sup>j</sup><sup>x</sup>

Even in the case the system is always prepared in the same state, there exists an uncertainty about the measured of an observable. The probability distributions presented above are

� � <sup>¼</sup> Tr My

The probability distribution of Y is obtained from the marginal probability distribution:

x

it is possible to obtain the conditional probability distribution with elements:

conditional probability distribution to obtain a value y, given as input the state ρ<sup>x</sup> is:

where {My}<sup>y</sup> is a POVM. The joint probability distribution X and Y is given by:

p xð Þ¼ ; <sup>y</sup> <sup>X</sup>

p yð Þ¼ <sup>X</sup> x

Considering the Bayes rule:

<sup>x</sup> <sup>¼</sup> Tr<sup>B</sup> <sup>I</sup><sup>A</sup> <sup>⊗</sup> MB

� � <sup>¼</sup> <sup>X</sup>

<sup>x</sup>ρAB

� � <sup>⊗</sup> j i bx h j bx : (16)

<sup>x</sup> <sup>ρ</sup>AB ½ �, the post-measured state will be:

<sup>x</sup> ⊗ j i bx h j bx : (17)

� �; (18)

� �: (19)

p yð Þ : (22)

: (20)

X x pxρ<sup>x</sup> !

p xð Þ¼ ; y pxp yð Þ¼ jx p yð ÞP xð Þ jy ; (21)

the system B, then the measurement map will be written as:

I<sup>A</sup> ⊗ Φ<sup>B</sup> ρAB

<sup>x</sup>ρAB � � and ρ<sup>A</sup>

we write px <sup>¼</sup> TrAB <sup>I</sup><sup>A</sup> <sup>⊗</sup> MB

68 Advanced Technologies of Quantum Key Distribution

2.3. Quantum entropy

� � <sup>¼</sup> <sup>X</sup>

I<sup>A</sup> ⊗ Φ<sup>B</sup> ρAB

a classical probability distribution, although it only has classical correlations.

x

$$S(\rho) = -\sum\_{k} \lambda\_k \log\_2(\lambda\_k),\tag{24}$$

where {λk}<sup>k</sup> are the eigenvalues of ρ = ∑kλk|k〉〈k|. The von Neumann entropy has the same interpretation of the Shannon entropy for the probability distribution composed by the eigenvalues of the density matrix. The von Neumann entropy is zero of pure states, and it is maximum for the maximally mixed state I=N, where it is Sð Þ¼ I=N log2N.

For composed systems, the von Neumann entropy is analogous to the Shannon entropy of the joint probability. For a bipartite state ρAB, the joint von Neumann entropy is:

$$S(\rho\_{AB}) = -\operatorname{Tr}(\rho\_{AB}\log\_2\rho\_{AB}).\tag{25}$$

Follow some interesting, and useful, properties about von Neumann entropy:

1. (Pure states) For a bipartite pure state |ϕ〉AB ∈ C<sup>A</sup> ⊗ CB, the partitions have the same von Neumann entropy:

$$\mathcal{S}(\rho\_A) = \mathcal{S}(\rho\_B),\tag{26}$$

where ρ<sup>A</sup> = TrB(|ϕ〉〈ϕ|AB).

2. (Additivity) von Neumann entropy is additive:

$$\mathcal{S}(\rho \otimes \sigma) = \mathcal{S}(\rho) + \mathcal{S}(\sigma),\tag{27}$$

where ρ and σ are density matrices.

3. (Concavity) von Neumann entropy is a concave function:

$$S\left(\sum\_{i} p\_{i}\rho\_{i}\right) \geq \sum\_{i} p\_{i}S(\rho\_{i}),\tag{28}$$

for a convex combination ρ = ∑ipiρi.

4. (Classical-quantum states) For bipartite state in the form ρAB = ∑xpx|x〉〈x| ⊗ ρx, the von Neumann entropy will be:

$$S\left(\sum\_{\mathbf{x}} p\_{\mathbf{x}} | \mathbf{x}\rangle\langle\mathbf{x}| \otimes \rho\_{\mathbf{x}}\right) = H(\mathbf{X}) + \sum\_{\mathbf{x}} p\_{\mathbf{x}} S(\rho\_{\mathbf{x}}),\tag{29}$$

In contrast with the von Neumman entropy, the relative entropy always decreases under the action of a quantum channel. This property has an operational meaning: two states are always

Theorem 9. Given two density matrices ρ, σ∈ Dð Þ C<sup>A</sup> and a quantum channel Γ∈Cð Þ CA; C<sup>B</sup> , the

This theorem implies into another property of the quantum mutual information: it decreases monotonically under local CPTP channels. As mutual information quantifies correlations, this

Corollary 10. Given a bipartite state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> and quantum channel Φ<sup>B</sup> ∈ C CB; C<sup>B</sup> ð Þ0 , the

<sup>≥</sup> I A : <sup>B</sup><sup>0</sup> ð Þ<sup>I</sup> <sup>⊗</sup> <sup>Φ</sup> <sup>ρ</sup>ð Þ AB

¼ S ρABjjρ<sup>A</sup> ⊗ ρ<sup>B</sup>

� � � � <sup>¼</sup> I A : <sup>B</sup><sup>0</sup> ð Þ<sup>I</sup> <sup>⊗</sup> <sup>Φ</sup> <sup>ρ</sup>ð Þ AB

� �jjΓð Þ <sup>σ</sup> � � (35)

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71

The Role of Quantumness of Correlations in Entanglement Resource Theory

� � (37)

� �: (39)

<sup>p</sup> , von Neumann entropy

2

޼�1. The negative value of the quantum conditional

I Að iB޼�S Að Þ jB : (40)

: (36)

: (38)

<sup>S</sup> <sup>ρ</sup>jj<sup>σ</sup> � � <sup>≥</sup> <sup>S</sup> <sup>Γ</sup> <sup>ρ</sup>

means that the amount of correlations reduce under local noise.

I Að Þ : B <sup>ρ</sup>AB

I Að Þ : B <sup>ρ</sup>AB

S Að Þ jB <sup>ρ</sup>AB

<sup>≥</sup> <sup>S</sup> <sup>I</sup><sup>A</sup> <sup>⊗</sup> <sup>Φ</sup><sup>B</sup> <sup>ρ</sup>AB � �jjρ<sup>A</sup> <sup>⊗</sup> <sup>Φ</sup><sup>B</sup> <sup>ρ</sup><sup>B</sup>

Analogous to the classical conditional entropy, it is possible to define a quantum version of it. For a bipartite system ρAB, the quantum conditional entropy quantifies the amount of infor-

Definition 11 (Conditional entropy). Consider a bipartite system ρAB, the quantum conditional

One interesting property of the quantum conditional entropy is that it can be negative. For

of the pure state is zero: S(|ϕ〉〈ϕ|AB) = 0. Nonetheless the reduced state is the maximally mixed state: ρ<sup>B</sup> ¼ I=2, whose von Neumann entropy is Sð Þ¼ I=2 1. Therefore, the conditional entropy

The conditional entropy has an operational meaning in the state merging protocol, where a tripartite pure state is shared by two experimentalists, one will send part of its state through a

� � �

<sup>¼</sup> <sup>S</sup> <sup>ρ</sup>AB � � � <sup>S</sup> <sup>ρ</sup><sup>B</sup>

AB <sup>¼</sup> ð Þ j i <sup>00</sup> <sup>þ</sup> j i <sup>11</sup> <sup>=</sup> ffiffiffi

less distinguishable under the action of noise.

following inequality holds:

mutual information satisfies:

using the theorem above:

entropy is defined as the function:

Proof. Given the mutual information:

I Að Þ : B <sup>ρ</sup>AB

mation of A that is available when B is known.

example, if we consider a bipartite pure state ϕ

entropy is defined as the coherent information:

of this state is negative S Að Þ <sup>j</sup><sup>B</sup> j i<sup>ϕ</sup> h j <sup>ϕ</sup> AB

where H(X) = � ∑xpx log2px

For composed system, it is possible to define a quantum analogous to the mutual information for bipartite states.

Definition 6 (Mutual information). Given a bipartite state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , the quantum mutual information is defined as:

$$I(A:B)\_{\rho\_{AB}} = \mathcal{S}(\rho\_A) + \mathcal{S}(\rho\_B) - \mathcal{S}(\rho\_{AB}).\tag{30}$$

The quantum mutual information of ρAB quantifies the correlations in quantum systems. It can be interpreted as the number of qubits that one part must send to another to destroy the correlations between the entire system. As the amount of correlations in a quantum state must be positive, it is possible to conclude that:

$$\mathcal{S}(\boldsymbol{\rho}\_A) + \mathcal{S}(\boldsymbol{\rho}\_B) \ge \mathcal{S}(\boldsymbol{\rho}\_{AB}). \tag{31}$$

From property 2, it is easy to see that mutual information is zero for product state ρAB = ρ<sup>A</sup> ⊗ ρB. The mutual information of pure states will be equal to:

$$I(A:B)\_{\psi\_{AB}} = \mathfrak{S}(\rho\_A) = \mathfrak{S}(\rho\_B),\tag{32}$$

where ψAB = |ψ〉〈ψ|AB is pure state.

The quantum version of the relative entropy quantifies the distinguishability between quantum states.

Definition 7 (Quantum relative entropy). Given two density matrices ρ, σ∈ D C<sup>N</sup> � �, the distinguishability between them can be quantified using the quantum relative entropy:

$$S(\rho||\sigma) = \operatorname{Tr}\left[\rho \log\_2 \rho - \rho \log\_2 \sigma\right].\tag{33}$$

It will be zero if ρ = σ.

The quantum relative entropy is a positive function for supp(ρ) ⊆ supp(σ), otherwise it diverges to infinity. The quantum mutual information can also be written as a quantum relative entropy.

Proposition 8. Consider a bipartite state ρAB, the following expression holds:

$$I(A:B)\_{\rho\_{AB}} = \mathcal{S}(\rho\_{AB}||\rho\_A \otimes \rho\_B),\tag{34}$$

where ρ<sup>A</sup> and ρ<sup>B</sup> are the reduced states of ρAB.

In contrast with the von Neumman entropy, the relative entropy always decreases under the action of a quantum channel. This property has an operational meaning: two states are always less distinguishable under the action of noise.

Theorem 9. Given two density matrices ρ, σ∈ Dð Þ C<sup>A</sup> and a quantum channel Γ∈Cð Þ CA; C<sup>B</sup> , the following inequality holds:

$$\mathcal{S}\left(\rho||\sigma\right) \ge \mathcal{S}\left(\Gamma\left(\rho\right)||\Gamma\left(\sigma\right)\right) \tag{35}$$

This theorem implies into another property of the quantum mutual information: it decreases monotonically under local CPTP channels. As mutual information quantifies correlations, this means that the amount of correlations reduce under local noise.

Corollary 10. Given a bipartite state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> and quantum channel Φ<sup>B</sup> ∈ C CB; C<sup>B</sup> ð Þ0 , the mutual information satisfies:

$$I(A:B)\_{\rho\_{AB}} \ge I(A:B')\_{\mathbb{L}\otimes\Phi(\rho\_{AB})}.\tag{36}$$

Proof. Given the mutual information:

$$I(A:B)\_{\rho\_{AB}} = \mathcal{S}\left(\rho\_{AB}||\rho\_A \otimes \rho\_B\right) \tag{37}$$

using the theorem above:

4. (Classical-quantum states) For bipartite state in the form ρAB = ∑xpx|x〉〈x| ⊗ ρx, the von

For composed system, it is possible to define a quantum analogous to the mutual information

Definition 6 (Mutual information). Given a bipartite state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , the quantum mutual

� � <sup>þ</sup> <sup>S</sup> <sup>ρ</sup><sup>B</sup>

� � <sup>≥</sup> <sup>S</sup> <sup>ρ</sup>AB

� � <sup>¼</sup> <sup>2</sup><sup>S</sup> <sup>ρ</sup><sup>B</sup>

<sup>S</sup> <sup>ρ</sup>jj<sup>σ</sup> � � <sup>¼</sup> Tr <sup>ρ</sup> log2<sup>ρ</sup> � <sup>ρ</sup> log2<sup>σ</sup> � �: (33)

� �; (34)

The quantum mutual information of ρAB quantifies the correlations in quantum systems. It can be interpreted as the number of qubits that one part must send to another to destroy the correlations between the entire system. As the amount of correlations in a quantum state must

From property 2, it is easy to see that mutual information is zero for product state ρAB = ρ<sup>A</sup> ⊗ ρB.

The quantum version of the relative entropy quantifies the distinguishability between quan-

Definition 7 (Quantum relative entropy). Given two density matrices ρ, σ∈ D C<sup>N</sup> � �, the distin-

The quantum relative entropy is a positive function for supp(ρ) ⊆ supp(σ), otherwise it diverges to infinity. The quantum mutual information can also be written as a quantum relative entropy.

¼ S ρABjjρ<sup>A</sup> ⊗ ρ<sup>B</sup>

¼ S ρ<sup>A</sup>

� � <sup>þ</sup> <sup>S</sup> <sup>ρ</sup><sup>B</sup>

I Að Þ : B <sup>ψ</sup>AB ¼ 2S ρ<sup>A</sup>

<sup>¼</sup> H Xð Þþ<sup>X</sup>

x

� � � <sup>S</sup> <sup>ρ</sup>AB

pxS ρ<sup>x</sup>

� �; (29)

� �: (30)

� �: (31)

� �; (32)

Neumann entropy will be:

70 Advanced Technologies of Quantum Key Distribution

where H(X) = � ∑xpx log2px

be positive, it is possible to conclude that:

where ψAB = |ψ〉〈ψ|AB is pure state.

where ρ<sup>A</sup> and ρ<sup>B</sup> are the reduced states of ρAB.

tum states.

It will be zero if ρ = σ.

The mutual information of pure states will be equal to:

for bipartite states.

information is defined as:

S X x

I Að Þ : B <sup>ρ</sup>AB

S ρ<sup>A</sup>

guishability between them can be quantified using the quantum relative entropy:

Proposition 8. Consider a bipartite state ρAB, the following expression holds:

I Að Þ : B <sup>ρ</sup>AB

pxj ix h j x ⊗ ρ<sup>x</sup> !

$$I(A:B)\_{\rho\_{AB}} \ge \mathcal{S}\left(\mathbb{I}\_A \otimes \Phi\_{\mathcal{B}}\left(\rho\_{AB}\right) || \rho\_A \otimes \Phi\_{\mathcal{B}}\left(\rho\_B\right)\right) = I(A:B')\_{\mathbb{I}\otimes\Phi\left(\rho\_{AB}\right)}.\tag{38}$$

Analogous to the classical conditional entropy, it is possible to define a quantum version of it. For a bipartite system ρAB, the quantum conditional entropy quantifies the amount of information of A that is available when B is known.

Definition 11 (Conditional entropy). Consider a bipartite system ρAB, the quantum conditional entropy is defined as the function:

$$\mathcal{S}(A|B)\_{\rho\_{AB}} = \mathcal{S}(\rho\_{AB}) - \mathcal{S}(\rho\_B). \tag{39}$$

One interesting property of the quantum conditional entropy is that it can be negative. For example, if we consider a bipartite pure state ϕ � � � AB <sup>¼</sup> ð Þ j i <sup>00</sup> <sup>þ</sup> j i <sup>11</sup> <sup>=</sup> ffiffiffi 2 <sup>p</sup> , von Neumann entropy of the pure state is zero: S(|ϕ〉〈ϕ|AB) = 0. Nonetheless the reduced state is the maximally mixed state: ρ<sup>B</sup> ¼ I=2, whose von Neumann entropy is Sð Þ¼ I=2 1. Therefore, the conditional entropy of this state is negative S Að Þ <sup>j</sup><sup>B</sup> j i<sup>ϕ</sup> h j <sup>ϕ</sup> AB ޼�1. The negative value of the quantum conditional entropy is defined as the coherent information:

$$I(A|B) = -S(A|B). \tag{40}$$

The conditional entropy has an operational meaning in the state merging protocol, where a tripartite pure state is shared by two experimentalists, one will send part of its state through a quantum channel to the other. The coherent information quantifies the amount of entanglement required to the sender be able to perform the protocol. If it is positive, they cannot use entanglement to perform the state merging, and in the end the amount of entanglement grows [25–27]. The coherent information also quantifies the capacity of a quantum channel, optimizing over all input states ρA, the output state is known to be ρB. This result is named as LSD theorem [28–31].

## 3. Quantum correlations

#### 3.1. Entanglement

This section introduces the concept of quantum entanglement, presenting its characterization and quantification.

#### 3.1.1. Separable states

Consider two systems A and B, often named the experimentalists responsible by the systems as Alice and Bob, respectively. The state of the systems A and B is described by a density matrix on a Hilbert space. In this way considering two finite Hilbert spaces C<sup>A</sup> and CB, and a basis in each one:

$$\{\left|a\_{i}\right\rangle\}\_{i=0}^{|A|-1} \in \mathbb{C}\_{A};\tag{41}$$

B. It can be checked easily via the mutual information of the state, which is clearly zero once

The concept of product state can be generalized for mixed state. Considering a composed system represented by the state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , it is called a product state if can be

where ρ<sup>A</sup> ∈ Dð Þ C<sup>A</sup> and ρ<sup>B</sup> ∈ Dð Þ C<sup>B</sup> are the states of the systems A and B, respectively. The product state for mixed states is also no correlated, as its mutual information is zero. As the space of quantum states is a convex set, the convex combination of states will also be a quantum state. The convex combination of product states generalizes the notion of product

Definition 12 (Separable states). Considering a composed system described by the state

The set of quantum channels that let separable states invariant is named local operations and classical communication (LOCC). The set of separable states form a subspace in the space of density matrices, it can be denoted as Sep(CAB). The separable state can be easily extended to multipartite systems. Considering a n-partite system, it is named m-separable if it can be

A measure of entanglement for mixed state can be obtained from the quantification of entanglement for pure states. It is possible to construct a measure of entanglement in this sense calculating the average of entanglement taken on pure states needed to form the state. The most famous measure which follow this idea is named as entanglement of formation. The entanglement of formation is interpreted as the minimal pure state entanglement required to

Definition 13. Considering a quantum state ρ ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , the entanglement of formation is

<sup>σ</sup> <sup>¼</sup> <sup>X</sup> i, j pi,j σA <sup>i</sup> ⊗ σ<sup>B</sup>

decomposed in a convex combination of product states composed by m parties.

Ef ρ

where the optimization is performed over all ensembles ξρ ¼ pi

� � <sup>¼</sup> min ξρ X i pi E ψ<sup>i</sup> � �

ρAB ¼ ρ<sup>A</sup> ⊗ ρB; (46)

The Role of Quantumness of Correlations in Entanglement Resource Theory

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73

<sup>j</sup> ; (47)

� �� ; (48)

<sup>i</sup>¼<sup>1</sup>, such that <sup>ρ</sup> <sup>=</sup> <sup>∑</sup>ipi|ψi〉

; ψ<sup>i</sup> � � � <sup>ψ</sup><sup>i</sup> � � � � �<sup>M</sup>

that the von Neumman entropy of the pure state is zero [32–34].

σ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , it is a separable state if and only if can be written as:

<sup>j</sup> ∈ Dð Þ C<sup>B</sup> .

states, that is named as separable state [35].

<sup>i</sup> <sup>∈</sup> <sup>D</sup>ð Þ <sup>C</sup><sup>A</sup> and <sup>σ</sup><sup>B</sup>

3.1.2. Entanglement quantification

build the mixed state [7].

〈ψi|, ∑ipi = 1 and pi ≥ 0.

defined as:

written as:

where σ<sup>A</sup>

$$\{\{b\_k\}\}\_{k=0}^{|\mathcal{B}|-1} \in \mathbb{C}\_B,\tag{42}$$

where |A| = dim(CA) and |B| = dim(CB). The global system, composed of A and B, can be obtained through the tensor product between the basis in the Hilbert space of each system:

$$\{\{a\_i, b\_k\}\}\_{i,j=0}^{|AB|-1} = \{ |a\_i\rangle \otimes |b\_k\rangle \}\_{i,k=0}^{|A|-1,|B|-1},\tag{43}$$

hence the dimension of the composed system is the product of the dimension: |AB| = dim (CAB) = dim(CA) � dim(CB). The Hilbert space of the composed system is denoted as CAB = C<sup>A</sup> ⊗ CB. A pure state of the composed system can be decomposed in the basis in Eq. (43):

$$|\psi\rangle\_{AB} = \sum\_{i,k} c\_{i,k} |a\_i\rangle \otimes |b\_k\rangle. \tag{44}$$

From this expression, one can realize that: in general a pure state, which describes a composed system, cannot be written as the product of the state of each system. In other words, suppose the system A and B described by the states |α〉<sup>A</sup> = ∑iai|ai〉 ∈ C<sup>A</sup> and |β〉<sup>B</sup> = ∑kbk|bk〉 ∈ CB, the composed system is described by the state:

$$
\langle |a\rangle \otimes |\beta\rangle = \sum\_{i,k} a\_i b\_k |a\_i\rangle \otimes |b\_k\rangle. \tag{45}
$$

It is the particular case where the coefficients in Eq. (44) are ci,<sup>k</sup> = ai � bk. If a composed system can be written as Eq. (45), it is called a product state, and there is no correlations between A and B. It can be checked easily via the mutual information of the state, which is clearly zero once that the von Neumman entropy of the pure state is zero [32–34].

The concept of product state can be generalized for mixed state. Considering a composed system represented by the state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , it is called a product state if can be written as:

$$
\rho\_{A\mathcal{B}} = \rho\_A \otimes \rho\_{\mathcal{B}},\tag{46}
$$

where ρ<sup>A</sup> ∈ Dð Þ C<sup>A</sup> and ρ<sup>B</sup> ∈ Dð Þ C<sup>B</sup> are the states of the systems A and B, respectively. The product state for mixed states is also no correlated, as its mutual information is zero. As the space of quantum states is a convex set, the convex combination of states will also be a quantum state. The convex combination of product states generalizes the notion of product states, that is named as separable state [35].

Definition 12 (Separable states). Considering a composed system described by the state σ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , it is a separable state if and only if can be written as:

$$
\sigma = \sum\_{i,j} p\_{i,j} \sigma\_i^A \otimes \sigma\_j^B,\tag{47}
$$

where σ<sup>A</sup> <sup>i</sup> <sup>∈</sup> <sup>D</sup>ð Þ <sup>C</sup><sup>A</sup> and <sup>σ</sup><sup>B</sup> <sup>j</sup> ∈ Dð Þ C<sup>B</sup> .

quantum channel to the other. The coherent information quantifies the amount of entanglement required to the sender be able to perform the protocol. If it is positive, they cannot use entanglement to perform the state merging, and in the end the amount of entanglement grows [25–27]. The coherent information also quantifies the capacity of a quantum channel, optimizing over all input states ρA, the output state is known to be ρB. This result is named as

This section introduces the concept of quantum entanglement, presenting its characterization

Consider two systems A and B, often named the experimentalists responsible by the systems as Alice and Bob, respectively. The state of the systems A and B is described by a density matrix on a Hilbert space. In this way considering two finite Hilbert spaces C<sup>A</sup> and CB, and a basis in each one:

<sup>i</sup>¼<sup>0</sup> <sup>∈</sup> <sup>C</sup>A; (41)

<sup>k</sup>¼<sup>0</sup> <sup>∈</sup> <sup>C</sup>B; (42)

ci, <sup>k</sup> ai j i ⊗ j i bk : (44)

aibk ai j i ⊗ j i bk : (45)

i, <sup>k</sup>¼<sup>0</sup> ; (43)

ai f g j i <sup>j</sup>Aj�<sup>1</sup>

f g j i bk <sup>j</sup>Bj�<sup>1</sup>

ai f g j i ; bk <sup>j</sup>ABj�<sup>1</sup>

composed system is described by the state:

where |A| = dim(CA) and |B| = dim(CB). The global system, composed of A and B, can be obtained through the tensor product between the basis in the Hilbert space of each system:

hence the dimension of the composed system is the product of the dimension: |AB| = dim (CAB) = dim(CA) � dim(CB). The Hilbert space of the composed system is denoted as CAB = C<sup>A</sup> ⊗ CB. A pure state of the composed system can be decomposed in the basis in Eq. (43):

j i <sup>ψ</sup> AB <sup>¼</sup> <sup>X</sup>

j i α ⊗ β � � � i, k

<sup>¼</sup> <sup>X</sup> i, k

It is the particular case where the coefficients in Eq. (44) are ci,<sup>k</sup> = ai � bk. If a composed system can be written as Eq. (45), it is called a product state, and there is no correlations between A and

From this expression, one can realize that: in general a pure state, which describes a composed system, cannot be written as the product of the state of each system. In other words, suppose the system A and B described by the states |α〉<sup>A</sup> = ∑iai|ai〉 ∈ C<sup>A</sup> and |β〉<sup>B</sup> = ∑kbk|bk〉 ∈ CB, the

i,j¼<sup>0</sup> <sup>¼</sup> ai f g j i <sup>⊗</sup> j i bk <sup>j</sup>Aj�1,jBj�<sup>1</sup>

LSD theorem [28–31].

3.1. Entanglement

and quantification.

3.1.1. Separable states

3. Quantum correlations

72 Advanced Technologies of Quantum Key Distribution

The set of quantum channels that let separable states invariant is named local operations and classical communication (LOCC). The set of separable states form a subspace in the space of density matrices, it can be denoted as Sep(CAB). The separable state can be easily extended to multipartite systems. Considering a n-partite system, it is named m-separable if it can be decomposed in a convex combination of product states composed by m parties.

#### 3.1.2. Entanglement quantification

A measure of entanglement for mixed state can be obtained from the quantification of entanglement for pure states. It is possible to construct a measure of entanglement in this sense calculating the average of entanglement taken on pure states needed to form the state. The most famous measure which follow this idea is named as entanglement of formation. The entanglement of formation is interpreted as the minimal pure state entanglement required to build the mixed state [7].

Definition 13. Considering a quantum state ρ ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , the entanglement of formation is defined as:

$$E\_f(\rho) = \min\_{\xi\_\rho} \sum\_i p\_i E(\left| \psi\_i \right>), \tag{48}$$

where the optimization is performed over all ensembles ξρ ¼ pi ; ψ<sup>i</sup> � � � <sup>ψ</sup><sup>i</sup> � � � � �<sup>M</sup> <sup>i</sup>¼<sup>1</sup>, such that <sup>ρ</sup> <sup>=</sup> <sup>∑</sup>ipi|ψi〉 〈ψi|, ∑ipi = 1 and pi ≥ 0.

The entanglement entropy E(|ψi〉) is defined as:

$$E(\left|\psi\_i\right>) = \mathcal{S}(\text{Tr}\_{\mathcal{B}}\left[\left|\psi\_i\right>\langle\psi\_i|\right]),\tag{49}$$

3.2. Quantumness of correlations

3.2.1. Classically correlated states

bility 1/2, where |ϕ〉, |ψ〉 ∈ C<sup>2</sup>

J j i ψ ; ϕ � � � �� <sup>¼</sup> <sup>1</sup>

2

existence of two distinct events:

J ϕ � � �

; j i <sup>ψ</sup> � � <sup>¼</sup> ffiffiffi

2 <sup>S</sup> <sup>ϕ</sup> � � � <sup>ϕ</sup> � �

other hand, for the quantum coin flip with states ϕ

<sup>|</sup>ψ〉 = |1〉. For this case, the overlap is <sup>ϕ</sup>j<sup>ψ</sup> � � <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffi

presented.

This section presents a revision about some basic concepts of quantumness of correlations for distinguishable systems. The notion of classically correlated states and quantum discord is

Consider a flip coin game with two distinct events described by the states {|0〉〈0|, |1〉〈1|}, each with the same probability 1/2. It is known that it is possible to distinguish the faces of the coin, with a null probability of error. The probability of error to distinguish two events, or two probability distributions, depends on the trace distance of the probability vectors of the events:

as the states are orthogonal |||0〉〈0| � |1〉〈1|||1 = 2, therefore the probability of error PE(|0〉〈0|, | 1〉〈1|) = 0, as one expected. Now suppose a quantum coin flip, which coherent superposition between the two faces of the coin, described by the events: {|ϕ〉〈ϕ|, |ψ〉〈ψ|}, with equal proba-

. As an example, consider the states ϕ

2

� � j i <sup>ψ</sup> h jjj <sup>ψ</sup> <sup>1</sup> <sup>¼</sup> ffiffiffi

þ 1 2 <sup>S</sup> <sup>ϕ</sup> � � � <sup>ϕ</sup> � �

<sup>p</sup> . The Jensen-Shannon divergence is related to the Bures distance and induces a

<sup>¼</sup> ð Þ j i<sup>0</sup> <sup>þ</sup> j i<sup>1</sup> <sup>=</sup> ffiffiffi

then the probability of error to distinguish the events is not zero. Superposition of states in quantum mechanics creates events that cannot be perfectly distinguished. The distinguishability of quantum or classical events can be quantifier by the Jensen-Shannon divergence. For two probability distributions (or events), it is defined as the symmetric and smoothed version of the Shannon relative entropy, or in the quantum case the von Neumman relative entropy [37, 38].

Definition 16. The Jensen-Shannon divergence for two arbitrary events |ψ〉, |ϕ〉 is defined as:

For the classical coin flip game, the Jensen-Shannon divergence will be just J(|0〉, |1〉) = 1. On the

metric for pure quantum states related to the Fisher-Rao metric [39], it is lager for more distinguishable events, and the largest distance characterizes complete distinguishable events. The Jensen-Shannon divergence for two arbitrary events |ψ〉, |ϕ〉 is related to the mutual information [37]:

� � �� <sup>¼</sup> I Rð Þ : <sup>E</sup> <sup>ρ</sup>RE

where R represents a register, E represents the events and ρRE ∈ Dð Þ C<sup>R</sup> ⊗ C<sup>E</sup> characterizes the

� � �

2 p ;

jjj i0 h j 0 � j i1 h jjj 1 <sup>1</sup>; (52)

The Role of Quantumness of Correlations in Entanglement Resource Theory

http://dx.doi.org/10.5772/intechopen.70396

� � �

<sup>p</sup> . The trace distance of these states is simply:

� þ j i ψ h j ψ

2

<sup>2</sup> <sup>∥</sup>j i <sup>ψ</sup> h j <sup>ψ</sup> � �: (53)

; (54)

<sup>p</sup> and |ψ〉 = |1〉, it will be

<sup>¼</sup> ð Þ j i<sup>0</sup> <sup>þ</sup> j i<sup>1</sup> <sup>=</sup> ffiffiffi

2 <sup>p</sup> and 75

PEð Þ¼ j i<sup>0</sup> h j <sup>0</sup> ; j i<sup>1</sup> h j <sup>1</sup> <sup>1</sup>

jj ϕ � � � <sup>ϕ</sup> � �

� þ j i ψ h j ψ <sup>2</sup> <sup>∥</sup> <sup>ϕ</sup> � � � <sup>ϕ</sup> � � �

� �

J j i ψ ; ϕ �

where S(TrB[|ψi〉〈ψi|]) is the von Neumann entropy of the reduced state of |ψi〉. The entanglement of formation is not easy to evaluate. Indeed the minimization process implies in to find an optimal convex hull, in function of a nonlinear function. For two qubits systems, it can be calculated analytically [36].

Quantum entanglement also enables an operational interpretation. This interpretation has two different ways: the resource required to construct a given quantum state and the resource extracted from a quantum system. The resource here refers to the amount of copies of maximally mixed state. Then, one can define the measure of this resource as a measure of entanglement in the limit of many copies.

The number of copies m of maximally entangled states required to construct n copies of a given state ρ, by means of LOCC protocols, is named entanglement cost [7]. The entanglement cost can be written as the regularized version of the entanglement of formation [4].

Definition 14 (Entanglement cost). The number of copies of the maximally entangled states required to build the state ρ is given by:

$$E\_{\mathbb{C}}(\rho) = \lim\_{n \to \infty} \frac{E\_f(\rho^{\otimes n})}{n},\tag{50}$$

where Ef(ρ⊗<sup>n</sup> ) is the entanglement of formation of the n copies of ρ.

The number of copies m of the maximally entangled state which can be extracted from n copies of a given state ρ, by LOCC, is named as distillable entanglement [7].

Definition 15 (Distillable entanglement). The distillable entanglement of a given state ρ is defined as:

$$E\_D\left(\rho\right) = \lim\_{n \to \infty} \frac{m}{n},\tag{51}$$

where m is the number of maximally entangled states that can be extracted from ρ in the limit of many copies.

The distillable entanglement is a very important operational measure of entanglement, because it quantifies how useful is a given quantum state, for the quantum information purpose.

The operational meaning of the entanglement cost and the distillable entanglement compose the research theory of quantum entanglement. The entanglement cost and the distillable entanglement of a given state are not the same. Indeed the cost of entanglement is greater than the distillable entanglement. The point is: it is more expensive to create a state ρ with copies of maximally entangled state than is possible to extract entanglement from ρ. One example is the bound entangled state, even it is entangled it is not possible to extract any maximally entangled state, although it requires an amount of maximally entangled states to build it.

#### 3.2. Quantumness of correlations

The entanglement entropy E(|ψi〉) is defined as:

74 Advanced Technologies of Quantum Key Distribution

calculated analytically [36].

ment in the limit of many copies.

to build the state ρ is given by:

where Ef(ρ⊗<sup>n</sup>

copies.

build it.

E ψ<sup>i</sup> 

be written as the regularized version of the entanglement of formation [4].

EC ρ

) is the entanglement of formation of the n copies of ρ.

of a given state ρ, by LOCC, is named as distillable entanglement [7].

<sup>¼</sup> lim<sup>n</sup>!<sup>∞</sup>

<sup>¼</sup> <sup>S</sup> Tr<sup>B</sup> <sup>ψ</sup><sup>i</sup>

where S(TrB[|ψi〉〈ψi|]) is the von Neumann entropy of the reduced state of |ψi〉. The entanglement of formation is not easy to evaluate. Indeed the minimization process implies in to find an optimal convex hull, in function of a nonlinear function. For two qubits systems, it can be

Quantum entanglement also enables an operational interpretation. This interpretation has two different ways: the resource required to construct a given quantum state and the resource extracted from a quantum system. The resource here refers to the amount of copies of maximally mixed state. Then, one can define the measure of this resource as a measure of entangle-

The number of copies m of maximally entangled states required to construct n copies of a given state ρ, by means of LOCC protocols, is named entanglement cost [7]. The entanglement cost can

Definition 14 (Entanglement cost). The number of copies of the maximally entangled states required

The number of copies m of the maximally entangled state which can be extracted from n copies

Definition 15 (Distillable entanglement). The distillable entanglement of a given state ρ is defined as:

<sup>¼</sup> lim<sup>n</sup>!<sup>∞</sup>

where m is the number of maximally entangled states that can be extracted from ρ in the limit of many

The distillable entanglement is a very important operational measure of entanglement, because it quantifies how useful is a given quantum state, for the quantum information purpose.

The operational meaning of the entanglement cost and the distillable entanglement compose the research theory of quantum entanglement. The entanglement cost and the distillable entanglement of a given state are not the same. Indeed the cost of entanglement is greater than the distillable entanglement. The point is: it is more expensive to create a state ρ with copies of maximally entangled state than is possible to extract entanglement from ρ. One example is the bound entangled state, even it is entangled it is not possible to extract any maximally entangled state, although it requires an amount of maximally entangled states to

ED ρ

Ef ρ <sup>⊗</sup> <sup>n</sup>

m

 <sup>ψ</sup><sup>i</sup> 

; (49)

<sup>n</sup> ; (50)

<sup>n</sup> ; (51)

This section presents a revision about some basic concepts of quantumness of correlations for distinguishable systems. The notion of classically correlated states and quantum discord is presented.

#### 3.2.1. Classically correlated states

Consider a flip coin game with two distinct events described by the states {|0〉〈0|, |1〉〈1|}, each with the same probability 1/2. It is known that it is possible to distinguish the faces of the coin, with a null probability of error. The probability of error to distinguish two events, or two probability distributions, depends on the trace distance of the probability vectors of the events:

$$P\_E(|0\rangle\langle 0|, |1\rangle\langle 1|) = \frac{1}{2} - \frac{1}{4} |||0\rangle\langle 0| - |1\rangle\langle 1||) \_1,\tag{52}$$

as the states are orthogonal |||0〉〈0| � |1〉〈1|||1 = 2, therefore the probability of error PE(|0〉〈0|, | 1〉〈1|) = 0, as one expected. Now suppose a quantum coin flip, which coherent superposition between the two faces of the coin, described by the events: {|ϕ〉〈ϕ|, |ψ〉〈ψ|}, with equal probability 1/2, where |ϕ〉, |ψ〉 ∈ C<sup>2</sup> . As an example, consider the states ϕ � � � <sup>¼</sup> ð Þ j i<sup>0</sup> <sup>þ</sup> j i<sup>1</sup> <sup>=</sup> ffiffiffi 2 <sup>p</sup> and <sup>|</sup>ψ〉 = |1〉. For this case, the overlap is <sup>ϕ</sup>j<sup>ψ</sup> � � <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffi 2 <sup>p</sup> . The trace distance of these states is simply:

$$\left| \left| \left| \phi \right\rangle \langle \phi \right| - \left| \psi \right\rangle \langle \psi \rangle \right| \right|\_1 = \sqrt{2},$$

then the probability of error to distinguish the events is not zero. Superposition of states in quantum mechanics creates events that cannot be perfectly distinguished. The distinguishability of quantum or classical events can be quantifier by the Jensen-Shannon divergence. For two probability distributions (or events), it is defined as the symmetric and smoothed version of the Shannon relative entropy, or in the quantum case the von Neumman relative entropy [37, 38].

Definition 16. The Jensen-Shannon divergence for two arbitrary events |ψ〉, |ϕ〉 is defined as:

$$J(|\psi\rangle, |\phi\rangle) = \frac{1}{2} S\left(\frac{|\phi\rangle\langle\phi| + |\psi\rangle\langle\psi|}{2} \| |\phi\rangle\langle\phi|\right) + \frac{1}{2} S\left(\frac{|\phi\rangle\langle\phi| + |\psi\rangle\langle\psi|}{2} \| |\psi\rangle\langle\psi|\right). \tag{53}$$

For the classical coin flip game, the Jensen-Shannon divergence will be just J(|0〉, |1〉) = 1. On the other hand, for the quantum coin flip with states ϕ � � � <sup>¼</sup> ð Þ j i<sup>0</sup> <sup>þ</sup> j i<sup>1</sup> <sup>=</sup> ffiffiffi 2 <sup>p</sup> and |ψ〉 = |1〉, it will be J ϕ � � � ; j i <sup>ψ</sup> � � <sup>¼</sup> ffiffiffi 2 <sup>p</sup> . The Jensen-Shannon divergence is related to the Bures distance and induces a metric for pure quantum states related to the Fisher-Rao metric [39], it is lager for more distinguishable events, and the largest distance characterizes complete distinguishable events. The Jensen-Shannon divergence for two arbitrary events |ψ〉, |ϕ〉 is related to the mutual information [37]:

$$J(|\psi\rangle, |\phi\rangle) = I(\mathbb{R} : E)\_{\rho\_{\mathbb{R}\mathbb{E}}},\tag{54}$$

where R represents a register, E represents the events and ρRE ∈ Dð Þ C<sup>R</sup> ⊗ C<sup>E</sup> characterizes the existence of two distinct events:

$$
\rho\_{RE} = \frac{1}{2} |0\rangle\langle 0|\_{R} \otimes |\phi\rangle\langle \phi|\_{E} + \frac{1}{2} |1\rangle\langle 1|\_{R} \otimes |\psi\rangle\langle \psi|\_{E}.\tag{55}
$$

correlations can be quantified by the amount of correlations that are not destroyed by the local

Definition 18. For a bipartite density matrix ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , the classical correlations between A and B can be quantified by the amount of correlations that can be extracted via local measurements:

where the optimization is taken over the set of local measurement maps I ⊗ B ∈P Hð Þ AB; HAX , and

Originally, Ollivier and Zurek [10] have defined this expression restricting the optimization to projective measurements. Independently, Henderson and Vedral [11] have defined the optimization of the classical correlations over general POVMs. As the mutual information quantifies the total amount of correlations in the state, it is possible to define a quantifier of quantum correlations as the difference between the total correlations in the system, quantified by mutual information, and the classical correlations, measured by Eq. (58). This measure of quantum-

¼ I Að Þ : B <sup>ρ</sup>AB

Quantum discord quantifies the amount of information, that cannot be accessed via local measurements. Therefore, it measures the quantumness shared between A and B that cannot be recovered via a classical statistical inference process. The optimization of quantum discord is a NP-hard problem [42]. A general analytical solution for quantum discord is not known or a criterion for a giving POVM to be optimal. Nonetheless, there are some analytic expressions for some specific states [43–45]. It is a natural generalization of quantum discord for the case

Definition 20. Given a bipartite state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> the quantum discord over measurements on

I Að Þ : B <sup>ρ</sup>AB

This generalization of quantum discord was first discussed in [46] in the context of the nonlocal-broadcast theorem. This definition is often named WPM-discord, because it was also studied by Wu et al. [47]. It was also studied restricting to projective measurements by some

¼ max I ⊗ B ∈ P

<sup>x</sup> ⊗ j i bx h j bx is a quantum-classical state in the space B Hð Þ <sup>A</sup> ⊗ H<sup>X</sup> .

S ρ<sup>A</sup> � � �<sup>X</sup> x

The Role of Quantumness of Correlations in Entanglement Resource Theory

of a state ρAB is defined as:

� J Að Þ : B <sup>ρ</sup>AB

� I Að Þ : <sup>B</sup> <sup>A</sup> <sup>⊗</sup> <sup>B</sup> <sup>ρ</sup>ð Þ AB n o; (60)

pxS <sup>ρ</sup><sup>A</sup> x � � ( ); (58)

http://dx.doi.org/10.5772/intechopen.70396

77

; (59)

I Að Þ : <sup>X</sup> <sup>I</sup> <sup>⊗</sup> <sup>B</sup> <sup>ρ</sup>ð Þ AB

measurement.

<sup>I</sup> <sup>⊗</sup> <sup>B</sup> <sup>ρ</sup>AB � � <sup>¼</sup> <sup>X</sup>

where I Að Þ : B <sup>ρ</sup>AB

both systems is:

authors [48, 49].

J Að Þ : B <sup>ρ</sup>AB

x pxρ<sup>A</sup> ¼ max I ⊗ B ∈P

ness of correlations is named as quantum discord:

D Að Þ : B <sup>ρ</sup>AB

the measurement is performed locally on both subsystems.

D Að Þ : B <sup>ρ</sup>AB

where A ∈Pð Þ CA; C<sup>Y</sup> and B ∈Pð Þ CB; C<sup>X</sup> .

is the von Neumann mutual information.

¼ min A ⊗ B ∈P

Definition 19. The quantum discord D Að Þ : B <sup>ρ</sup>AB

For the classical coin flip game, it is ρ<sup>c</sup> RE ¼ <sup>1</sup> <sup>2</sup> j i0 h j 0 ⊗ j i0 h j 0 þ <sup>1</sup> <sup>2</sup>j i1 h j 1 ⊗ j i1 h j 1 , with mutual information I Rð Þ : E <sup>ρ</sup><sup>c</sup> RE <sup>¼</sup> 1. For the quantum coin, the state will be <sup>ρ</sup><sup>q</sup> RE ¼ <sup>1</sup> <sup>2</sup> j i0 h j 0 ⊗ ϕ � � � <sup>ϕ</sup> � � � þ1 <sup>2</sup>j i1 h j 1 ⊗ j i ψ h j ψ , where for ϕ � � � <sup>¼</sup> ð Þ j i<sup>0</sup> <sup>þ</sup> j i<sup>1</sup> <sup>=</sup> ffiffiffi 2 <sup>p</sup> and |ψ〉 = |1〉, and the mutual information is I Rð Þ : E <sup>ρ</sup><sup>q</sup> RE <sup>¼</sup> ffiffiffi 2 <sup>p</sup> . As the mutual information is a measure of correlations between two probability distributions, one realizes that there are more correlations between the register and the events for not completely distinguishable registers, in comparison with orthogonal registers. However, two binary classical distributions cannot share more than one bit of information; in other words, their mutual information cannot be greater than one [31]. As the correlations between the quantum coin events and the register are bigger than one, it means that there are correlations beyond the classical case. A quantum state is classically correlated if there exists a local projective measurement such that the state remains the same [10–12]. The state ρ<sup>c</sup> RE is an example of classical-classical state. In general, these states are defined as:

Definition 17 (classical-classical states). Given a bipartite state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , it is strictly classically correlated (or classical-classical state) if there exists a local projective measurement ΠAB with elements Π<sup>A</sup> <sup>l</sup> ⊗ Π<sup>B</sup> k � � k,l such that the post-measured state is equal to the input state:

$$\Pi(\rho\_{AB}) = \sum\_{k,l} \Pi\_l^A \otimes \Pi\_k^B \rho\_{AB} \Pi\_l^A \otimes \Pi\_k^B = \rho\_{AB},\tag{56}$$

therefore <sup>ρ</sup>AB <sup>¼</sup> <sup>X</sup> k,l pk,l Π<sup>A</sup> <sup>l</sup> ⊗ Π<sup>B</sup> <sup>k</sup> , and Π<sup>Y</sup> <sup>x</sup> ¼ j i ex h j ex <sup>Y</sup> is a projetor in the orthonormal basis {|ex〉Y}<sup>x</sup> ∈ HY.

The state ρ<sup>q</sup> ER is an example of a classical-quantum state, because there exists a projective measurement, with elements {|0〉〈0|, |1〉〈1|}, over partition E that keep the state unchanged. On the other hand, there is not a projective measurement over partition R with this property. In general, a state ρAB is classical-quantum if there exists a projective measurement Π<sup>A</sup> with elements {Πk}<sup>k</sup> such that:

$$
\Pi\_A \otimes \mathbb{I}\_{\mathcal{B}}(\rho\_{AB}) = \rho\_{AB} = \sum\_k p\_k \Pi\_k \otimes \rho\_k. \tag{57}
$$

The set o classically correlated states is not convex, once that combination of block diagonal matrices cannot be block diagonal. As the identity matrix is block diagonal, or just diagonal, this set is connected by the maximally mixed state, and it is a thin set [40].

#### 3.2.2. Quantum discord

The amount of classical correlations in a quantum state is measured by the capacity to extract information locally [41]. As the measurement process is a classical statistical inference, classical correlations can be quantified by the amount of correlations that are not destroyed by the local measurement.

Definition 18. For a bipartite density matrix ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , the classical correlations between A and B can be quantified by the amount of correlations that can be extracted via local measurements:

$$I(A:B)\_{\rho\_{A\mathcal{B}}} = \max\_{1 \otimes \mathcal{B} \in \mathcal{P}} I(A:X)\_{1 \otimes \mathcal{B} \binom{\rho\_{A\mathcal{B}}}{\rho\_{A\mathcal{B}}}} = \max\_{1 \otimes \mathcal{B} \in \mathcal{P}} \left\{ S(\rho\_A) - \sum\_x p\_x S(\rho\_x^A) \right\},\tag{58}$$

where the optimization is taken over the set of local measurement maps I ⊗ B ∈P Hð Þ AB; HAX , and <sup>I</sup> <sup>⊗</sup> <sup>B</sup> <sup>ρ</sup>AB � � <sup>¼</sup> <sup>X</sup> x pxρ<sup>A</sup> <sup>x</sup> ⊗ j i bx h j bx is a quantum-classical state in the space B Hð Þ <sup>A</sup> ⊗ H<sup>X</sup> .

Originally, Ollivier and Zurek [10] have defined this expression restricting the optimization to projective measurements. Independently, Henderson and Vedral [11] have defined the optimization of the classical correlations over general POVMs. As the mutual information quantifies the total amount of correlations in the state, it is possible to define a quantifier of quantum correlations as the difference between the total correlations in the system, quantified by mutual information, and the classical correlations, measured by Eq. (58). This measure of quantumness of correlations is named as quantum discord:

Definition 19. The quantum discord D Að Þ : B <sup>ρ</sup>AB of a state ρAB is defined as:

$$D(A:B)\_{\rho\_{AB}} = I(A:B)\_{\rho\_{AB}} - J(A:B)\_{\rho\_{AB}},\tag{59}$$

where I Að Þ : B <sup>ρ</sup>AB is the von Neumann mutual information.

<sup>ρ</sup>RE <sup>¼</sup> <sup>1</sup> 2

> � � �

For the classical coin flip game, it is ρ<sup>c</sup>

76 Advanced Technologies of Quantum Key Distribution

RE

<sup>2</sup>j i1 h j 1 ⊗ j i ψ h j ψ , where for ϕ

mation I Rð Þ : E <sup>ρ</sup><sup>c</sup>

þ1

I Rð Þ : E <sup>ρ</sup><sup>q</sup> RE <sup>¼</sup> ffiffiffi 2

elements Π<sup>A</sup>

therefore <sup>ρ</sup>AB <sup>¼</sup> <sup>X</sup>

elements {Πk}<sup>k</sup> such that:

3.2.2. Quantum discord

{|ex〉Y}<sup>x</sup> ∈ HY. The state ρ<sup>q</sup>

<sup>l</sup> ⊗ Π<sup>B</sup> k � �

> k,l pk,l Π<sup>A</sup> <sup>l</sup> ⊗ Π<sup>B</sup>

j i0 h j 0 <sup>R</sup> ⊗ ϕ � � � <sup>ϕ</sup> � � � <sup>E</sup> þ 1 2

RE ¼ <sup>1</sup>

<sup>¼</sup> ð Þ j i<sup>0</sup> <sup>þ</sup> j i<sup>1</sup> <sup>=</sup> ffiffiffi

example of classical-classical state. In general, these states are defined as:

k, l Π<sup>A</sup> <sup>l</sup> ⊗ Π<sup>B</sup>

<sup>k</sup> , and Π<sup>Y</sup>

Π<sup>A</sup> ⊗ I<sup>B</sup> ρAB

this set is connected by the maximally mixed state, and it is a thin set [40].

Π ρAB � � <sup>¼</sup> <sup>X</sup> <sup>2</sup> j i0 h j 0 ⊗ j i0 h j 0 þ <sup>1</sup>

<sup>p</sup> . As the mutual information is a measure of correlations between two proba-

<sup>¼</sup> 1. For the quantum coin, the state will be <sup>ρ</sup><sup>q</sup>

2

bility distributions, one realizes that there are more correlations between the register and the events for not completely distinguishable registers, in comparison with orthogonal registers. However, two binary classical distributions cannot share more than one bit of information; in other words, their mutual information cannot be greater than one [31]. As the correlations between the quantum coin events and the register are bigger than one, it means that there are correlations beyond the classical case. A quantum state is classically correlated if there exists a local projective measurement such that the state remains the same [10–12]. The state ρ<sup>c</sup>

Definition 17 (classical-classical states). Given a bipartite state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , it is strictly classically correlated (or classical-classical state) if there exists a local projective measurement ΠAB with

k,l such that the post-measured state is equal to the input state:

<sup>k</sup> <sup>ρ</sup>ABΠ<sup>A</sup>

ER is an example of a classical-quantum state, because there exists a projective

k

measurement, with elements {|0〉〈0|, |1〉〈1|}, over partition E that keep the state unchanged. On the other hand, there is not a projective measurement over partition R with this property. In general, a state ρAB is classical-quantum if there exists a projective measurement Π<sup>A</sup> with

� � <sup>¼</sup> <sup>ρ</sup>AB <sup>¼</sup> <sup>X</sup>

The set o classically correlated states is not convex, once that combination of block diagonal matrices cannot be block diagonal. As the identity matrix is block diagonal, or just diagonal,

The amount of classical correlations in a quantum state is measured by the capacity to extract information locally [41]. As the measurement process is a classical statistical inference, classical

<sup>l</sup> ⊗ Π<sup>B</sup>

j i1 h j 1 <sup>R</sup> ⊗ j i ψ h j ψ <sup>E</sup>: (55)

<sup>p</sup> and |ψ〉 = |1〉, and the mutual information is

<sup>2</sup>j i1 h j 1 ⊗ j i1 h j 1 , with mutual infor-

<sup>2</sup> j i0 h j 0 ⊗ ϕ � � � <sup>ϕ</sup> � � �

RE is an

RE ¼ <sup>1</sup>

<sup>k</sup> ¼ ρAB; (56)

pkΠ<sup>k</sup> ⊗ ρk: (57)

<sup>x</sup> ¼ j i ex h j ex <sup>Y</sup> is a projetor in the orthonormal basis

Quantum discord quantifies the amount of information, that cannot be accessed via local measurements. Therefore, it measures the quantumness shared between A and B that cannot be recovered via a classical statistical inference process. The optimization of quantum discord is a NP-hard problem [42]. A general analytical solution for quantum discord is not known or a criterion for a giving POVM to be optimal. Nonetheless, there are some analytic expressions for some specific states [43–45]. It is a natural generalization of quantum discord for the case the measurement is performed locally on both subsystems.

Definition 20. Given a bipartite state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> the quantum discord over measurements on both systems is:

$$D(A:B)\_{\rho\_{\text{AB}}} = \min\_{\mathcal{A}\otimes\mathcal{B}\in\mathcal{P}} \left\{ I(A:B)\_{\rho\_{\text{AB}}} - I(A:B)\_{\mathcal{A}\otimes\mathcal{B}\left(\rho\_{\text{AB}}\right)} \right\},\tag{60}$$

where A ∈Pð Þ CA; C<sup>Y</sup> and B ∈Pð Þ CB; C<sup>X</sup> .

This generalization of quantum discord was first discussed in [46] in the context of the nonlocal-broadcast theorem. This definition is often named WPM-discord, because it was also studied by Wu et al. [47]. It was also studied restricting to projective measurements by some authors [48, 49].

#### 3.2.3. Relative entropy of quantumness and work deficit

For a given dephasing channel Π ∈P C<sup>N</sup> , acting on any state ρ∈ D C<sup>N</sup> , the support of the dephased state contains the support of the input state: supp(ρ) ⊆ supp(Π[ρ]); therefore, the measure of quantumness of correlations based on the relative entropy remains finite for every composed state [23, 31].

Suppose Alice and Bob have a common composed system described by the state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , they would like to extract work from this system. To accomplish their task, they can perform the closed set of local operations and classical communication (CLOCC). This class of operations is composed of: (i) addition of pure ancillas, (ii) local unitary operations and (iii) local dephasing channels. Classical communication is represented by a local dephasing channel. If Alice and Bob are together in the same laboratory, they can extract work globally from the total system, then the total amount of information that Alice and Bob can extract from ρAB together is defined as the total work [12].

Definition 21. The work that can be extracted from a quantum system, described by the state ρ∈ D C<sup>N</sup> , is defined as the change in the entropy:

$$\mathcal{W}\_t(\rho) = \log\_2 \mathcal{N} - \mathcal{S}(\rho),\tag{61}$$

<sup>Δ</sup> <sup>ρ</sup>AB <sup>¼</sup> Wt <sup>ρ</sup>AB � Wl <sup>ρ</sup>AB : (63)

The Role of Quantumness of Correlations in Entanglement Resource Theory

<sup>S</sup> <sup>Γ</sup> <sup>ρ</sup>AB � <sup>S</sup> <sup>ρ</sup>AB : (64)

http://dx.doi.org/10.5772/intechopen.70396

79

From the definition of the total work and the local work, we can define the work deficit as the

Even though the total and the local work depend explicitly on the dimension of the system, the work deficit should not depend on the dimension of Γ[ρAB]. Adding local pure ancillas belongs to the CLOCC cannot change the amount of work deficit. The work deficit can quantify quantum correlations, then it must not change by the simple addition of a uncorrelated system [16, 50].

In the asymptotic limit (the limit of many copies), the work deficit quantifies the amount of pure states that can be extracted locally [51, 52]. However, as a resource cannot be created freely, the addition of pure local ancillas is not allowed, then it is replaced by the addition of maximally mixed states. The set of operations that contains: (i) addition of maximally mixture states, (ii) local unitary operations and (iii) local dephasing channels, is named noise local operations and classical communication (NLOCC) [51]. The extraction of local pure states is a protocol, whose goal is to extract resource (coherence). The set of available operations are NLOCC operations, and the set of free resource states is composed only by the maximally mixture state. It is the only state without local purity [53]. It remains an open question if the

In the limit of one copy, the work deficit can quantify quantum correlations present in a given composed system [54]. The scenario where Alice and Bob can perform many steps of classical communication one each other is named two way, and the work deficit is named two-way work deficit. In this case, they can perform measurements and communicate in each step of the protocol. Mathematically, the two-way work deficit does not have a closed expression [50]. As discussed above, it is possible to activate quantum correlations performing operations on the measured system. Therefore, this many step scenario cannot quantify quantum correlations. Because if Alice and Bob can implement a sequence of non-commuting dephasing channels, the only invariante state is the maximally mixed state. In this way, it is necessary a one round description, where Alice and Bob can communicate at the end of the protocol. Following this idea, it is possible to define the one-way work deficit, which just one side can communicate. If Bob communicates to Alice, the state created at the end of the protocol is a quantum-classical

state (or a classical-quantum state if Alice communicates at the end of the protocol).

Definition 23 (one-way work deficit). Given a bipartite state ρAB, the work deficit with just one side

where Π<sup>B</sup> ∈Pð Þ C<sup>B</sup> is a local dephasing on subsystem B. The notation Δ!(ρAB) means that the

<sup>S</sup> <sup>I</sup><sup>A</sup> <sup>⊗</sup> <sup>Π</sup><sup>B</sup> <sup>ρ</sup>AB � <sup>S</sup> <sup>ρ</sup>AB ; (65)

<sup>Δ</sup> <sup>ρ</sup>AB <sup>¼</sup> inf

CLOCC class and the NLOCC class are equivalent classes [50].

communication is named one-way work deficit [12]:

<sup>Δ</sup>! <sup>ρ</sup>AB <sup>¼</sup> min

communication is from A to B and Δ (ρAB) in the opposite direction.

Π<sup>B</sup> ∈P

Γ ∈CLOCC

diference of them:

log2N is the entropy of the maximally mixed state, and S(ρ) is the von Neumann entropy of the state.

This function can be interpreted as a quantifier of information, such that if the state is a maximally mixed state no information can be extracted from it. Therefore, if the state is a pure state, we have the maximum amount of information [12, 50]. The entropy function represents the amount of information that one can get to know about the system; therefore, the function Eq. (61) represents the amount of information that one already knows. On the other hand, if Alice and Bob cannot be in the same laboratory, the information that can be extracted from the total state is restricted to be locally accessed. In the same way, it is possible to define the total information, named local work. Then, Alice and Bob should perform CLOCC operation in order to obtain the maximal amount of local information [50]:

$$\mathcal{W}\_l(\rho\_{AB}) = \log\_2 \mathcal{N} - \sup\_{\Gamma \in \text{CLOCC}} \mathcal{S}\left(\Gamma\left[\rho\_{AB}\right]\right),\tag{62}$$

where the state Γ(ρAB) is the state after the protocol. As CLOCC consist in sending one part of the state in a dephasing channel, at the end of the protocol, the whole state is with the receiver: Γ(ρAB) = ρAA.

One can be interested in the amount of information that cannot be extracted locally by Alice and Bob. This function is named work deficit and it quantifies the amount of work that is not possible to extract locally [12].

Definition 22. Given a bipartite state ρAB, the information which two parts Alice and Bob cannot access, via CLOCC, is the work deficit:

$$
\Delta(\rho\_{AB}) = \mathcal{W}\_l(\rho\_{AB}) - \mathcal{W}\_l(\rho\_{AB}).\tag{63}
$$

From the definition of the total work and the local work, we can define the work deficit as the diference of them:

3.2.3. Relative entropy of quantumness and work deficit

78 Advanced Technologies of Quantum Key Distribution

composed state [23, 31].

Γ(ρAB) = ρAA.

possible to extract locally [12].

access, via CLOCC, is the work deficit:

together is defined as the total work [12].

ρ∈ D C<sup>N</sup> , is defined as the change in the entropy:

to obtain the maximal amount of local information [50]:

Wl ρAB

<sup>¼</sup> log2<sup>N</sup> � sup

Γ ∈ CLOCC

where the state Γ(ρAB) is the state after the protocol. As CLOCC consist in sending one part of the state in a dephasing channel, at the end of the protocol, the whole state is with the receiver:

One can be interested in the amount of information that cannot be extracted locally by Alice and Bob. This function is named work deficit and it quantifies the amount of work that is not

Definition 22. Given a bipartite state ρAB, the information which two parts Alice and Bob cannot

S Γ ρAB

For a given dephasing channel Π ∈P C<sup>N</sup> , acting on any state ρ∈ D C<sup>N</sup> , the support of the dephased state contains the support of the input state: supp(ρ) ⊆ supp(Π[ρ]); therefore, the measure of quantumness of correlations based on the relative entropy remains finite for every

Suppose Alice and Bob have a common composed system described by the state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , they would like to extract work from this system. To accomplish their task, they can perform the closed set of local operations and classical communication (CLOCC). This class of operations is composed of: (i) addition of pure ancillas, (ii) local unitary operations and (iii) local dephasing channels. Classical communication is represented by a local dephasing channel. If Alice and Bob are together in the same laboratory, they can extract work globally from the total system, then the total amount of information that Alice and Bob can extract from ρAB

Definition 21. The work that can be extracted from a quantum system, described by the state

<sup>¼</sup> log2<sup>N</sup> � <sup>S</sup> <sup>ρ</sup>

log2N is the entropy of the maximally mixed state, and S(ρ) is the von Neumann entropy of the state. This function can be interpreted as a quantifier of information, such that if the state is a maximally mixed state no information can be extracted from it. Therefore, if the state is a pure state, we have the maximum amount of information [12, 50]. The entropy function represents the amount of information that one can get to know about the system; therefore, the function Eq. (61) represents the amount of information that one already knows. On the other hand, if Alice and Bob cannot be in the same laboratory, the information that can be extracted from the total state is restricted to be locally accessed. In the same way, it is possible to define the total information, named local work. Then, Alice and Bob should perform CLOCC operation in order

; (61)

; (62)

Wt ρ

$$\Delta(\rho\_{AB}) = \inf\_{\Gamma \in \text{CLCCC}} \{ \mathcal{S}(\Gamma[\rho\_{AB}]) - \mathcal{S}(\rho\_{AB}) \}. \tag{64}$$

Even though the total and the local work depend explicitly on the dimension of the system, the work deficit should not depend on the dimension of Γ[ρAB]. Adding local pure ancillas belongs to the CLOCC cannot change the amount of work deficit. The work deficit can quantify quantum correlations, then it must not change by the simple addition of a uncorrelated system [16, 50].

In the asymptotic limit (the limit of many copies), the work deficit quantifies the amount of pure states that can be extracted locally [51, 52]. However, as a resource cannot be created freely, the addition of pure local ancillas is not allowed, then it is replaced by the addition of maximally mixed states. The set of operations that contains: (i) addition of maximally mixture states, (ii) local unitary operations and (iii) local dephasing channels, is named noise local operations and classical communication (NLOCC) [51]. The extraction of local pure states is a protocol, whose goal is to extract resource (coherence). The set of available operations are NLOCC operations, and the set of free resource states is composed only by the maximally mixture state. It is the only state without local purity [53]. It remains an open question if the CLOCC class and the NLOCC class are equivalent classes [50].

In the limit of one copy, the work deficit can quantify quantum correlations present in a given composed system [54]. The scenario where Alice and Bob can perform many steps of classical communication one each other is named two way, and the work deficit is named two-way work deficit. In this case, they can perform measurements and communicate in each step of the protocol. Mathematically, the two-way work deficit does not have a closed expression [50]. As discussed above, it is possible to activate quantum correlations performing operations on the measured system. Therefore, this many step scenario cannot quantify quantum correlations. Because if Alice and Bob can implement a sequence of non-commuting dephasing channels, the only invariante state is the maximally mixed state. In this way, it is necessary a one round description, where Alice and Bob can communicate at the end of the protocol. Following this idea, it is possible to define the one-way work deficit, which just one side can communicate. If Bob communicates to Alice, the state created at the end of the protocol is a quantum-classical state (or a classical-quantum state if Alice communicates at the end of the protocol).

Definition 23 (one-way work deficit). Given a bipartite state ρAB, the work deficit with just one side communication is named one-way work deficit [12]:

$$\Delta^{\rightarrow}(\rho\_{AB}) = \min\_{\Pi\_{\mathbb{B}} \in \mathcal{P}} \left\{ \mathbb{S}(\mathbb{L}\_{A} \otimes \Pi\_{\mathbb{B}}[\rho\_{AB}]) - \mathbb{S}(\rho\_{AB}) \right\},\tag{65}$$

where Π<sup>B</sup> ∈Pð Þ C<sup>B</sup> is a local dephasing on subsystem B. The notation Δ!(ρAB) means that the communication is from A to B and Δ (ρAB) in the opposite direction.

Another definition for the work deficit is defined when both Alice and Bob communicate at the end of the protocol, this is named zero-way work deficit. The state created at the end of the protocol is a classical-classical state.

Definition 24 (zero-way work deficit). Given a bipartite state ρAB, the work deficit with no communication until the end of the protocol is named zero work deficit [12]:

$$\Delta^{\mathcal{Q}}(\rho\_{AB}) = \min\_{\Pi\_{\mathcal{A}} \otimes \Pi\_{\mathcal{B}} \in \mathcal{P}} \left\{ \mathcal{S} \left( \Pi\_{\mathcal{A}} \otimes \Pi\_{\mathcal{B}} \left[ \rho\_{AB} \right] \right) - \mathcal{S} \left( \rho\_{AB} \right) \right\}, \tag{66}$$

Δ ρ � � ≥ Er ρ

attached for bipartite pure states: |ψ〉AB ∈ C<sup>A</sup> ⊗ CB:

trace distance [56] and Hilbert-Schmidt distance [15, 57].

quantumness of correlations

understood by the Koashi-Winter relation.

where I Að Þ : X <sup>I</sup> ⊗ ⊞ρAB


ment for pure states.

where Δ(ρ) is the work deficit and Er(ρ) is the relative entropy of entanglement. The equality is

Δ Ψð Þ¼ AB Erð Þ¼ ΨAB S ρ<sup>A</sup>

where ΨAB = |ψ〉〈ψ|AB. An interesting corollary of this proposition is that the quantum discord is equal to the work deficit for pure states, because it is also equal to the entropy of entangle-

In this section, the concept of local disturbance was introduced by the definition of the work deficit. That is the smallest relative entropy between the state and its local disturbed version (obtained performing a local dephasing channel on the state). Indeed there are many other local disturbance quantumness of correlation quantifiers, which can be obtained defining a quantum state discrimination measure, for example, Bures distance [55], Schatten p-norm [17],

Given a bipartite system ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , then it is possible to purify the state in a larger space CABE of the dimension: dim(CABE) = dim(A) � dim(B) � rank(ρAB). The purification process creates quantum correlations between the system AB and the purification system E, unless the state is already pure. Intrinsically, there is a restriction in the amount of correlations that can be shared by the systems. This balance between the correlations for tripartite states can be

> ¼ max I ⊗ ∈ P

ξρ<sup>¼</sup> pi f g ;j i <sup>ψ</sup><sup>i</sup> h j <sup>ψ</sup><sup>i</sup> <sup>i</sup>

where the optimization is taken over all possible convex hull defined by the ensemble ξ = {pi, |ψi〉〈ψi|}i, such that ρAB = ∑ipi, |ψi〉〈ψi|, and E(|ψi〉) is the entropy of entanglement of

I Að Þ : <sup>X</sup> <sup>I</sup> ⊗ ⊞ρAB<sup>Þ</sup>

X i pi E ψ<sup>i</sup> � �

Þ is the mutual information of the post-measured state I ⊗ ⊞ρABÞ, and the

4. Monogamy relation: entanglement, classical correlations and

Given the definition of the classical correlations for a bipartite state ρAB:

J Að Þ : B <sup>ρ</sup>AB

optimization is taken over all local POVM measurement maps ∈Pð Þ CB, BCX .

Ef <sup>ρ</sup>AB � � <sup>¼</sup> min

Given also the definition of the entanglement of formation of a bipartite state ρAB:

� �; (71)

The Role of Quantumness of Correlations in Entanglement Resource Theory

� �; (72)

http://dx.doi.org/10.5772/intechopen.70396

81

; (73)

� �� ; (74)

where Π<sup>A</sup> ⊗ Π<sup>B</sup> ∈Pð Þ C<sup>A</sup> ⊗ C<sup>B</sup> is a local dephasing on subsystems A and B.

In analogy with the work deficit, Modi et al. proposed a measure of quantumness of correlation defined as the relative entropy of the state and the set of classical correlated states [16]. This measure is named relative entropy of quantumness.

Definition 25 (relative entropy of quantumness). The relative entropy of quantumness D(ρAB)QC for a given state ρAB is defined as the minimum relative entropy over the set of quantum-classical states [16]:

$$D\left(\rho\_{AB}\right)\_{\mathcal{QC}} = \min\_{\xi\_{AB} \in \mathcal{Q}\_{\mathcal{QC}}} S\left(\rho\_{AB} \| \left| \xi\_{AB} \right.\right), \tag{67}$$

where ΩQC is the set of quantum-classical states.

The relative entropy of quantumness for classical-classical states is denoted as D(ρAB)CC. It is analogous to Eq. (67) when the optimization is taken over the set of classical-classical states ΩCC:

$$D(\rho\_{AB})\_{\mathcal{CC}} = \min\_{\xi\_{AB} \in \Omega\_{\mathcal{CC}}} S(\rho\_{AB} \| \xi\_{AB}) \,. \tag{68}$$

As discussed previously, in the limit of one copy, the one-way and the zero-way work deficits quantify quantumness of correlations of the system. It is possible to obtain the equivalence between one-way work deficit and relative entropy of quantumness.

Theorem 26. The one-way work deficit is equal to the relative entropy of quantumness for quantumclassical states [16, 50]:

$$D\left(\rho\_{AB}\right)\_{\text{QC}} = \Delta^{\rightarrow}\left(\rho\_{AB}\right),\tag{69}$$

The same equivalence holds for zero-way work deficit and the relative entropy of quantumness of classical-classical states:

$$D(\rho\_{AB})\_{\mathcal{CC}} = \Delta^{\mathcal{O}}(\rho\_{AB}).\tag{70}$$

The one-way and zero-way work deficits quantify quantumness correlations beyond the quantum entanglement; therefore, we should be able to compare these two classes of quantum correlations. For the relative entropy, this comparison is natural of the fact that CLOCC is a subclass of LOCC operations, which naturally implies that [12]:

$$
\Delta(\rho) \ge E\_r(\rho),
\tag{71}
$$

where Δ(ρ) is the work deficit and Er(ρ) is the relative entropy of entanglement. The equality is attached for bipartite pure states: |ψ〉AB ∈ C<sup>A</sup> ⊗ CB:

Another definition for the work deficit is defined when both Alice and Bob communicate at the end of the protocol, this is named zero-way work deficit. The state created at the end of the

Definition 24 (zero-way work deficit). Given a bipartite state ρAB, the work deficit with no

In analogy with the work deficit, Modi et al. proposed a measure of quantumness of correlation defined as the relative entropy of the state and the set of classical correlated states [16].

Definition 25 (relative entropy of quantumness). The relative entropy of quantumness D(ρAB)QC for a given state ρAB is defined as the minimum relative entropy over the set of quantum-classical states [16]:

The relative entropy of quantumness for classical-classical states is denoted as D(ρAB)CC. It is analogous to Eq. (67) when the optimization is taken over the set of classical-classical states

As discussed previously, in the limit of one copy, the one-way and the zero-way work deficits quantify quantumness of correlations of the system. It is possible to obtain the equivalence

Theorem 26. The one-way work deficit is equal to the relative entropy of quantumness for quantum-

The same equivalence holds for zero-way work deficit and the relative entropy of quantum-

The one-way and zero-way work deficits quantify quantumness correlations beyond the quantum entanglement; therefore, we should be able to compare these two classes of quantum correlations. For the relative entropy, this comparison is natural of the fact that CLOCC is a

QC ¼ Δ! ρAB

CC <sup>¼</sup> <sup>Δ</sup><sup>∅</sup> <sup>ρ</sup>AB

QC ¼ min ξAB ∈ ΩQC

CC ¼ min ξAB ∈ ΩCC

S Π<sup>A</sup> ⊗ Π<sup>B</sup> ρAB

� <sup>S</sup> <sup>ρ</sup>AB

S ρAB∥ξAB

S ρAB∥ξAB

; (66)

; (67)

: (68)

; (69)

: (70)

communication until the end of the protocol is named zero work deficit [12]:

where Π<sup>A</sup> ⊗ Π<sup>B</sup> ∈Pð Þ C<sup>A</sup> ⊗ C<sup>B</sup> is a local dephasing on subsystems A and B.

D ρAB 

D ρAB 

between one-way work deficit and relative entropy of quantumness.

D ρAB 

D ρAB 

subclass of LOCC operations, which naturally implies that [12]:

Π<sup>A</sup> ⊗ Π<sup>B</sup> ∈P

<sup>¼</sup> min

protocol is a classical-classical state.

80 Advanced Technologies of Quantum Key Distribution

<sup>Δ</sup><sup>∅</sup> <sup>ρ</sup>AB

This measure is named relative entropy of quantumness.

where ΩQC is the set of quantum-classical states.

ΩCC:

classical states [16, 50]:

ness of classical-classical states:

$$
\Delta(\Psi\_{AB}) = E\_r(\Psi\_{AB}) = S(\rho\_A), \tag{72}
$$

where ΨAB = |ψ〉〈ψ|AB. An interesting corollary of this proposition is that the quantum discord is equal to the work deficit for pure states, because it is also equal to the entropy of entanglement for pure states.

In this section, the concept of local disturbance was introduced by the definition of the work deficit. That is the smallest relative entropy between the state and its local disturbed version (obtained performing a local dephasing channel on the state). Indeed there are many other local disturbance quantumness of correlation quantifiers, which can be obtained defining a quantum state discrimination measure, for example, Bures distance [55], Schatten p-norm [17], trace distance [56] and Hilbert-Schmidt distance [15, 57].
