4. Monogamy relation: entanglement, classical correlations and quantumness of correlations

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 understood by the Koashi-Winter relation.

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

$$I(A:B)\_{\rho\_{A\overline{\mathbb{B}}}} = \max\_{\mathbb{I}\otimes\mathbb{B}\in\mathcal{P}} I(A:X)\_{\mathbb{I}\otimes\mathbb{B}\in\rho\_{A\overline{\mathbb{B}}}},\tag{73}$$

where I Að Þ : X <sup>I</sup> ⊗ ⊞ρAB Þ is the mutual information of the post-measured state I ⊗ ⊞ρABÞ, and the optimization is taken over all local POVM measurement maps ∈Pð Þ CB, BCX .

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

$$E\_f(\rho\_{AB}) = \min\_{\xi\_p = \{\boldsymbol{\psi}\_i, |\boldsymbol{\psi}\_i\rangle, \{\boldsymbol{\psi}\_i\}\}\_i} \sum\_i p\_i E(|\boldsymbol{\psi}\_i\rangle),\tag{74}$$

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 |ψi〉.

Theorem 27 (Koashi-Winter relation). Considering ρABE ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> ⊗ C<sup>E</sup> a pure state then:

$$J(A:E)\_{\rho\_{A\to}} = S(\rho\_A) - E\_f(\rho\_{AB}),\tag{75}$$

TrE ρABE IAB ⊗ μ<sup>l</sup>

implies I Að Þ : E <sup>ρ</sup><sup>0</sup>

following corollary:

by the last theorem:

The equality holds for ρABC pure.

over the system E we have J Að Þ : CE <sup>ρ</sup>ACE

correlations which A can make with a third system C.

Eq. (84):

� � � <sup>μ</sup><sup>l</sup> � � � � � � � <sup>¼</sup> <sup>X</sup>

Calculating the mutual information of ρ<sup>0</sup>

AE

ij

I Að Þ : E <sup>ρ</sup><sup>0</sup>

¼ J Að Þ : E <sup>ρ</sup>AE

Given Eqs. (81) and (85), it proves the theorem.

AE

¼ S ρ<sup>A</sup>

J Að Þ : E <sup>ρ</sup>AE

Corollary 28. For any tripartite state ρABC ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> ⊗ C<sup>C</sup> , it follows:

J Að Þ : CE <sup>ρ</sup>ACE

Ef <sup>ρ</sup>AB � � <sup>þ</sup> J Að Þ : <sup>C</sup> <sup>ρ</sup>AC

Proof. If ρABC is not a pure state, there exists a purification ρABCE, then C<sup>A</sup> ⊗ C<sup>B</sup> ⊗ CCE, followed

Therefore, as the classical correlations are monotonic under local maps, then taking the trace

As the Shannon entropy of ρ<sup>A</sup> represents the effective size of A in qubits [24], this size can be approached as the capacity of the system A makes correlations with other systems B and C [18]. In other words, this means that the existence of the quantum or classical correlations between A and another system B is enough to restrict the amount of quantum or classical

<sup>≥</sup> J Að Þ : <sup>C</sup> <sup>ρ</sup>AC

<sup>þ</sup> Ef <sup>ρ</sup>AB � � <sup>¼</sup> <sup>S</sup> <sup>ρ</sup><sup>A</sup>

.

cirlicjr �

ljj i<sup>i</sup> h j<sup>j</sup> AB <sup>¼</sup> <sup>X</sup>

AE ¼ TrB ρ<sup>0</sup>

� � �<sup>X</sup> l ql

As the POVM M is the optimal measurement in the calculation of the classical correlations, it

∑lqlS(TrB[|ϕl〉〈ϕl|]) for any decomposition {pl,|ϕl〉〈ϕl|}. Substituting the mutual information in

≤ S ρ<sup>A</sup>

The Koashi-Winter equation quantifies the amount of entanglement among A and B, considering that the former is classically correlated with another system C. This property is interesting once that it is related to the monogamy of entanglement [58], where the amount of entanglement shared by three parts is limited, and this limitation is given by the amount of classical correlations among the parties. This limitation holds for any tripartite state as stated in the

i

ABE � �:

S TrB ϕ<sup>l</sup> � � � <sup>ϕ</sup><sup>l</sup> � �

≤ S ρ<sup>A</sup>

. By definition, the entanglement of formation satisfies: Ef(ρAB) ≤

cirlij i<sup>i</sup> AB ! <sup>X</sup>

j cjr � ljh jj AB

� � � � � ; (84)

� � � Ef <sup>ρ</sup>AB � �: (85)

� �: (86)

� �; (87)

1

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

A ¼ ql ϕ<sup>l</sup> � � � <sup>ϕ</sup><sup>l</sup> � � �:

(83)

83

0 @

The Role of Quantumness of Correlations in Entanglement Resource Theory

where ρ<sup>X</sup> = TrY[ρYX].

Proof. Suppose ρAB = ∑ipi|ψi〉〈ψi| is the optimum convex combination, such that Ef(ρAB) = ∑ipiS (TrB[|ψi〉〈ψi|]). The classical correlations in system AE relates this decomposition with a measurement on the subsystem E. Therefore, there exists a measurement ME j n o on system <sup>E</sup> such that ρ<sup>0</sup> ABE <sup>¼</sup> <sup>X</sup> j TrE <sup>ρ</sup>ABE <sup>I</sup>AB <sup>⊗</sup> ME j h i � � <sup>⊗</sup> ej � � � ej � � � <sup>E</sup> and TrE ρ<sup>0</sup> ABE � � <sup>¼</sup> <sup>X</sup> i pi ψ<sup>i</sup> � � � <sup>ψ</sup><sup>i</sup> � � �. Tracing over subsystem B, then the post-measurement state will be:

$$\rho'\_{AE} = \sum\_{j} p\_j Tr\_{\mathbb{B}}\left[ \left| \psi\_j \right> \left< \psi\_j \right| \right] \otimes \left| e\_i \right> \langle e\_j |, \tag{76}$$

In this way, the mutual information of the post-measurement state:

$$I(A:E)\_{\rho\_{AE}^{\prime}} = \mathbb{S}(\rho\_A) + \mathbb{S}(\rho\_E^{\prime}) - \mathbb{S}(\rho\_{AE}^{\prime}),\tag{77}$$

$$\dot{\rho} = \mathcal{S}(\rho\_A) + H(E) - H(E) - \sum\_i p\_i \mathcal{S}\left(Tr\_A\left[\left|\psi\_j\right>\left<\psi\_j\right|\right]\right),\tag{78}$$

$$\mathcal{S} = \mathcal{S}(\rho\_A) - \sum\_i p\_i \mathcal{S}\left(Tr\_A\left[\left|\psi\_j\right>\left<\psi\_j\right|\right]\right),\tag{79}$$

$$\mathcal{S} = \mathcal{S}(\rho\_A) - E\_f(\rho\_{AB}),\tag{80}$$

It was used as the property of the Shannon entropy for a block diagonal state, where

TrB[|ψj〉〈ψj|] = TrA[|ψj〉〈ψj|] and Ef(ρAB) = <sup>∑</sup>ipiS(TrB[|ψi〉〈ψi|]). By definition J Að Þ : <sup>E</sup> <sup>ρ</sup>AE ≥ I ð Þ A:E ρ<sup>0</sup> AE , then

$$J(A:E)\_{\rho\_{AE}} \ge \mathcal{S}(\rho\_A) - E\_f(\rho\_{AB}).\tag{81}$$

Now, it is proved the converse inequality. Given ρAE, there exists a POVM M∈P CE; C<sup>E</sup> ð Þ0 with rank-1 elements {Ml}, such that TrE MlρAE � � <sup>¼</sup> ql ρA <sup>l</sup> that optimizes the classical correlations <sup>J</sup> <sup>ρ</sup>AE � � <sup>¼</sup> <sup>S</sup> <sup>ρ</sup><sup>A</sup> � � �<sup>X</sup> l ql S ρ<sup>A</sup> l � �. As the elements of the POVM are rank-1, Ml = |μl〉〈μl|, and the state ρABE is pure, the state after local measurement on E will be described by an ensemble of pure states:

$$\rho'\_{ABE} = \sum\_{l} Tr\_{\mathbb{E}} \left[ \rho\_{AB\mathbb{E}} \left( \mathbb{I}\_{AB} \otimes \left| \mu\_l \right\rangle \langle \mu\_l \rangle \right) \right] \otimes |e\rangle\langle e| = \sum q\_l |\phi\_l\rangle \langle \phi\_l| \otimes |e\rangle\langle e|. \tag{82}$$

Once that ρABE = |κ〉〈κ|ABE, and the pure state can be written in the bipartite Schmidt decomposition |κ〉 = ∑ncn|n〉AB ⊗ |n〉E, if 〈μl|n〉 = rln, it is easy to see that:

The Role of Quantumness of Correlations in Entanglement Resource Theory http://dx.doi.org/10.5772/intechopen.70396 83

$$Tr\_{\mathbb{E}}\left[\rho\_{A\text{BE}}\left(\mathbb{I}\_{A\text{B}}\otimes|\mu\_{l}\rangle\langle\mu\_{l}|\right)\right] = \sum\_{\vec{\eta}}\mathbf{c}\_{l}r\_{\text{il}}\mathbf{c}\_{\vec{\eta}}^{\*}|\mathbf{i}\rangle\langle\mathbf{j}|\_{\text{AB}} = \left(\sum\_{\vec{\eta}}\mathbf{c}\_{l}r\_{\text{il}}|\mathbf{i}\rangle\_{\text{AB}}\right)\left(\sum\_{\vec{\eta}}\mathbf{c}\_{l}r\_{\vec{\eta}}^{\*}\langle\mathbf{j}|\_{\text{AB}}\right) = q\_{l}|\phi\_{l}\rangle\langle\phi\_{l}|.\tag{83}$$

Calculating the mutual information of ρ<sup>0</sup> AE ¼ TrB ρ<sup>0</sup> ABE � �:

Theorem 27 (Koashi-Winter relation). Considering ρABE ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> ⊗ C<sup>E</sup> a pure state then:

¼ S ρ<sup>A</sup>

Proof. Suppose ρAB = ∑ipi|ψi〉〈ψi| is the optimum convex combination, such that Ef(ρAB) = ∑ipiS (TrB[|ψi〉〈ψi|]). The classical correlations in system AE relates this decomposition with a mea-

AE

� � h i

i pi

� � � Ef <sup>ρ</sup>AB

<sup>E</sup> and TrE ρ<sup>0</sup>

ABE � � <sup>¼</sup> <sup>X</sup>

⊗ ej � � � ej � �

S TrA ψ<sup>j</sup> � � � E ψj D � � �

� � h i

� �; (80)

� �; (75)

j n o

�; (76)

; (78)

� ⊗ el j ih j el : (82)

≥ I

i pi ψ<sup>i</sup> � � � <sup>ψ</sup><sup>i</sup> � �

� �; (77)

; (79)

� �: (81)

<sup>l</sup> that optimizes the classical correlations

on system E such

�. Tracing over

J Að Þ : E <sup>ρ</sup>AE

surement on the subsystem E. Therefore, there exists a measurement ME

⊗ ej � � � ej � � �

j

In this way, the mutual information of the post-measurement state:

E � � � <sup>S</sup> <sup>ρ</sup><sup>0</sup>

� � <sup>þ</sup> H Eð Þ� H Eð Þ�<sup>X</sup>

S TrA ψ<sup>j</sup> � � � E ψj D � � �

It was used as the property of the Shannon entropy for a block diagonal state, where

J Að Þ : E <sup>ρ</sup>AE

� � � <sup>μ</sup><sup>l</sup> � � � � � � � <sup>⊗</sup> el j ih j el <sup>¼</sup> <sup>X</sup>ql <sup>ϕ</sup><sup>l</sup>

TrB[|ψj〉〈ψj|] = TrA[|ψj〉〈ψj|] and Ef(ρAB) = <sup>∑</sup>ipiS(TrB[|ψi〉〈ψi|]). By definition J Að Þ : <sup>E</sup> <sup>ρ</sup>AE

≥ S ρ<sup>A</sup>

Now, it is proved the converse inequality. Given ρAE, there exists a POVM M∈P CE; C<sup>E</sup> ð Þ0 with

state ρABE is pure, the state after local measurement on E will be described by an ensemble of

Once that ρABE = |κ〉〈κ|ABE, and the pure state can be written in the bipartite Schmidt decom-

� � <sup>¼</sup> ql

� � � Ef <sup>ρ</sup>AB

ρA

� �. As the elements of the POVM are rank-1, Ml = |μl〉〈μl|, and the

� � � <sup>ϕ</sup><sup>l</sup> � �

� � <sup>þ</sup> <sup>S</sup> <sup>ρ</sup><sup>0</sup>

� � �<sup>X</sup> i pi

� � � Ef <sup>ρ</sup>AB

TrE ρABE IAB ⊗ μ<sup>l</sup>

position |κ〉 = ∑ncn|n〉AB ⊗ |n〉E, if 〈μl|n〉 = rln, it is easy to see that:

where ρ<sup>X</sup> = TrY[ρYX].

ABE <sup>¼</sup> <sup>X</sup>

j

82 Advanced Technologies of Quantum Key Distribution

I Að Þ : E <sup>ρ</sup><sup>0</sup>

AE

TrE <sup>ρ</sup>ABE <sup>I</sup>AB <sup>⊗</sup> ME

¼ S ρ<sup>A</sup>

¼ S ρ<sup>A</sup>

¼ S ρ<sup>A</sup>

¼ S ρ<sup>A</sup>

rank-1 elements {Ml}, such that TrE MlρAE

l ql S ρ<sup>A</sup> l

� � �<sup>X</sup>

ABE <sup>¼</sup> <sup>X</sup> l

ρ0

subsystem B, then the post-measurement state will be:

h i � �

ρ0 AE <sup>¼</sup> <sup>X</sup> j pj TrB ψ<sup>j</sup> � � � E ψj D � � � h i

that ρ<sup>0</sup>

ð Þ A:E ρ<sup>0</sup>

J ρAE

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

pure states:

AE , then

$$I(A:E)\_{\rho\_{A\to}^{\prime}} = S(\rho\_A) - \sum\_l q\_l S(Tr\_B\left[|\phi\_l\rangle\langle\phi\_l|\right]),\tag{84}$$

As the POVM M is the optimal measurement in the calculation of the classical correlations, it implies I Að Þ : E <sup>ρ</sup><sup>0</sup> AE ¼ J Að Þ : E <sup>ρ</sup>AE . By definition, the entanglement of formation satisfies: Ef(ρAB) ≤ ∑lqlS(TrB[|ϕl〉〈ϕl|]) for any decomposition {pl,|ϕl〉〈ϕl|}. Substituting the mutual information in Eq. (84):

$$J(A:E)\_{\rho\_{A\to}} \le \mathcal{S}(\rho\_A) - E\_f(\rho\_{AB}).\tag{85}$$

Given Eqs. (81) and (85), it proves the theorem.

The Koashi-Winter equation quantifies the amount of entanglement among A and B, considering that the former is classically correlated with another system C. This property is interesting once that it is related to the monogamy of entanglement [58], where the amount of entanglement shared by three parts is limited, and this limitation is given by the amount of classical correlations among the parties. This limitation holds for any tripartite state as stated in the following corollary:

Corollary 28. For any tripartite state ρABC ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> ⊗ C<sup>C</sup> , it follows:

$$\left(E\_f\left(\rho\_{A\mathcal{B}}\right) + f(A:\mathbb{C})\_{\rho\_{A\mathcal{C}}} \leq \mathcal{S}\left(\rho\_A\right).\tag{86}$$

The equality holds for ρABC pure.

Proof. If ρABC is not a pure state, there exists a purification ρABCE, then C<sup>A</sup> ⊗ C<sup>B</sup> ⊗ CCE, followed by the last theorem:

$$J(A:\mathbb{C}E)\_{\rho\_{A\mathbb{C}\mathbb{E}}} + E\_f(\rho\_{A\mathbb{B}}) = \mathbb{S}(\rho\_A),\tag{87}$$

Therefore, as the classical correlations are monotonic under local maps, then taking the trace over the system E we have J Að Þ : CE <sup>ρ</sup>ACE <sup>≥</sup> J Að Þ : <sup>C</sup> <sup>ρ</sup>AC .

As the Shannon entropy of ρ<sup>A</sup> represents the effective size of A in qubits [24], this size can be approached as the capacity of the system A makes correlations with other systems B and C [18]. In other words, this means that the existence of the quantum or classical correlations between A and another system B is enough to restrict the amount of quantum or classical correlations which A can make with a third system C.

Summing the mutual information I Að Þ : E <sup>ρ</sup>AE on both sides of the Koashi-Winter relation, Eq. (75), it is possible to obtain a monogamy expression for the entanglement of formation of the state ρAB in function of the quantum discord [19]:

$$D(A:E)\_{\rho\_{AE}} = E\_{\sharp} \left(\rho\_{AB}\right) - S(A|E)\_{\rho\_{AE}},\tag{88}$$

<sup>D</sup><sup>∞</sup>ð Þ <sup>A</sup> : <sup>B</sup> <sup>ρ</sup>AB

D Að Þ : E <sup>ρ</sup> <sup>⊗</sup> <sup>n</sup> AE

<sup>D</sup><sup>∞</sup>ð Þ <sup>A</sup> : <sup>E</sup> <sup>ρ</sup>AE

Theorem 32 (Cornelio et al. [20]). For every mixed entangled state ρAB, if

ED <sup>ρ</sup>AB <sup>¼</sup> <sup>1</sup>

for a finite number of n and k, the entanglement is irreversible EC(ρAB) > ED(ρAB).

Taking the limit of many copies, the equation can be rewritten as:

[25, 26].

Therefore, similarly to Eq. (88) in the limit of many copies:

taking the regularization we have:

rem comes from Eq. (88).

tum discord <sup>D</sup><sup>∞</sup>ð Þ <sup>A</sup> : <sup>E</sup> <sup>ρ</sup>AE

as the conditional entropy is additive S Að Þ jE <sup>ρ</sup> <sup>⊗</sup> <sup>n</sup>

<sup>¼</sup> lim<sup>n</sup>!<sup>∞</sup> 1 n

<sup>¼</sup> Ef <sup>ρ</sup> <sup>⊗</sup> <sup>n</sup>

AE

n

k

where <sup>σ</sup>AB <sup>¼</sup> ð Þ Vk <sup>⊗</sup> <sup>I</sup> <sup>ρ</sup>AB and ED(σAB) = kED(ρAB). The quantum discord <sup>D</sup><sup>∞</sup>ð Þ <sup>A</sup> : <sup>E</sup> <sup>σ</sup>AE in this context can be viewed as the minimal amount of entanglement lost in the distillation protocol, for states belonging to the class described in the theorem [20]. This expression has an operational interpretation for quantum discord, where the quantum discord between the system and the purification system restricts the amount of e-bits lost in the distillation process. A consequence of this result is expressed by the state merging protocol [27], Alice (A), Bob (B) and the Environment (E) share a pure tripartite state ρABE, she would like to send her state to Bob, keeping the coherence with the system E. They can perform this protocol consuming an amount of entanglement in the process; the amount of entanglement is the regularized quan-

In addition to the above relations, some upper and lower bounds between quantum discord and entanglement of formation have been calculated via the Koashi-Winter relation and the properties of entropy [59–62]. Equation (88) was also used to calculate the quantum discord and the entanglement of formation analytically for systems with rank-2 and dimension 2 ⊗ n [41, 63, 64]. Experimental investigations of Eq. (88) were performed in the characterization of the information flow between system and environment of a non-Markovian process [65].

EC <sup>ρ</sup>AB <sup>¼</sup> <sup>1</sup>

D Að Þ : B <sup>ρ</sup> <sup>⊗</sup> <sup>n</sup> AB

AB � S Að Þ <sup>j</sup><sup>E</sup> <sup>ρ</sup> <sup>⊗</sup> <sup>n</sup>

<sup>¼</sup> EC <sup>ρ</sup>AB � S Að Þ <sup>j</sup><sup>E</sup> <sup>ρ</sup>AE

I A<sup>ð</sup> <sup>i</sup>BÞð Þ Vn <sup>⊗</sup> <sup>I</sup> <sup>ρ</sup> <sup>⊗</sup> <sup>n</sup>

EF ρ <sup>⊗</sup> <sup>n</sup>

AB

<sup>D</sup><sup>∞</sup>ð Þ <sup>A</sup> : <sup>E</sup> <sup>σ</sup>AE <sup>¼</sup> ECð Þ� <sup>σ</sup>AB EDð Þ <sup>σ</sup>AB ; (97)

¼ nS Að Þ jE <sup>ρ</sup>AE

AE

The Role of Quantumness of Correlations in Entanglement Resource Theory

: (92)

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

; (93)

; (94)

. Therefore, the following theo-

AB ; (96)

(95)

85

where D Að Þ : E <sup>ρ</sup>AE is the quantum discord of the state ρAE with local measurement on the subsystem E and S Að Þ jE <sup>ρ</sup>AE ¼ S AE ð Þ� S Eð Þ is the conditional entropy. As the label in the states is arbitrary, we can rewrite this expression changing the labels E ! B and vice versa to obtain D Að Þ : B <sup>ρ</sup>AB ¼ S Að Þ jB <sup>ρ</sup>AB � Ef <sup>ρ</sup>AE , taking the sum between this and Eq. (88):

$$D(A:E)\_{\rho\_{AE}} + D(A:B)\_{\rho\_{AB}} = E\_f(\rho\_{AE}) + E\_f(\rho\_{AB}),\tag{89}$$

as the total state is pure S Að Þ jE <sup>ρ</sup>AE ¼ �S Að Þ jB <sup>ρ</sup>AB . This expression means that the sum of total amount of entanglement that A shares with B and E is equal to the sum of the amount of quantum discord shared with B and E [19].

From Eq. 88, it is possible to calculate an interesting expression, which relates the irreversibility of the entanglement distillation protocol and quantum discord [20]. As discussed, the entanglement cost is larger than the distillable entanglement. Given the entanglement cost defined as the regularization of the entanglement of formation [4]:

Definition 29. For a mixed state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , the regularization of the entanglement of formation Ef(ρAB) results in the entanglement cost:

$$E\_{\mathbb{C}}(\rho\_{AB}) = \lim\_{n \to \infty} \frac{1}{n} E\_f(\rho\_{AB}^{\otimes n}). \tag{90}$$

The Hashing inequality says that the distillable entanglement of ρAB is lower bounded by the coherent information I Að iBÞ<sup>ρ</sup>AB ¼ �S Að Þ jB [3]. As the coherent information can increase under LOCC, it is possible to optimize it under LOCC attaining the distillable entanglement [3].

Definition 30. The regularized coherent information after optimization over LOCC for a mixed state ρAB gives the distillable entanglement:

$$E\_D(\rho\_{AB}) = \lim\_{n \to \infty} \frac{1}{n} I(A \backslash \mathcal{B})\_{(V\_n \otimes \mathbb{I})\rho\_{AB}^{\otimes n}},\tag{91}$$

where Vn ⊗ I acts locally on the n copies of ρAB.

It is also possible to define the regularized quantum discord:

Definition 31. The regularized quantum discord can be defined as the quantum discord of a state ρAB in the limit of many copies:

The Role of Quantumness of Correlations in Entanglement Resource Theory http://dx.doi.org/10.5772/intechopen.70396 85

$$D^{\circ \circ}(A:B)\_{\rho\_{AB}} = \lim\_{n \to \infty} \frac{1}{n} D(A:B)\_{\rho\_{AB}^{\otimes n}}.\tag{92}$$

Therefore, similarly to Eq. (88) in the limit of many copies:

$$D(A:E)\_{\rho\_{AE}^{\otimes n}} = E\_f(\rho\_{AB}^{\otimes n}) - S(A|E)\_{\rho\_{AE}^{\otimes n}},\tag{93}$$

taking the regularization we have:

Summing the mutual information I Að Þ : E <sup>ρ</sup>AE

84 Advanced Technologies of Quantum Key Distribution

where D Að Þ : E <sup>ρ</sup>AE

D Að Þ : B <sup>ρ</sup>AB

subsystem E and S Að Þ jE <sup>ρ</sup>AE

¼ S Að Þ jB <sup>ρ</sup>AB

as the total state is pure S Að Þ jE <sup>ρ</sup>AE

quantum discord shared with B and E [19].

tion Ef(ρAB) results in the entanglement cost:

coherent information I Að iBÞ<sup>ρ</sup>AB

in the limit of many copies:

ρAB gives the distillable entanglement:

where Vn ⊗ I acts locally on the n copies of ρAB.

the state ρAB in function of the quantum discord [19]:

� Ef ρAE

D Að Þ : E <sup>ρ</sup>AE

as the regularization of the entanglement of formation [4]:

D Að Þ : E <sup>ρ</sup>AE

on both sides of the Koashi-Winter relation,

. This expression means that the sum of total

: (90)

; (91)

; (88)

; (89)

Eq. (75), it is possible to obtain a monogamy expression for the entanglement of formation of

¼ Ef ρAB

is arbitrary, we can rewrite this expression changing the labels E ! B and vice versa to obtain

amount of entanglement that A shares with B and E is equal to the sum of the amount of

From Eq. 88, it is possible to calculate an interesting expression, which relates the irreversibility of the entanglement distillation protocol and quantum discord [20]. As discussed, the entanglement cost is larger than the distillable entanglement. Given the entanglement cost defined

Definition 29. For a mixed state ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , the regularization of the entanglement of forma-

The Hashing inequality says that the distillable entanglement of ρAB is lower bounded by the

Definition 30. The regularized coherent information after optimization over LOCC for a mixed state

1 n

Definition 31. The regularized quantum discord can be defined as the quantum discord of a state ρAB

LOCC, it is possible to optimize it under LOCC attaining the distillable entanglement [3].

1 n

Ef ρ <sup>⊗</sup> <sup>n</sup> AB

I A<sup>ð</sup> <sup>i</sup>BÞð Þ Vn <sup>⊗</sup> <sup>I</sup> <sup>ρ</sup> <sup>⊗</sup> <sup>n</sup>

¼ �S Að Þ jB [3]. As the coherent information can increase under

AB

þ D Að Þ : B <sup>ρ</sup>AB

¼ �S Að Þ jB <sup>ρ</sup>AB

EC ρAB

ED ρAB

It is also possible to define the regularized quantum discord:

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

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

� S Að Þ <sup>j</sup><sup>E</sup> <sup>ρ</sup>AE

, taking the sum between this and Eq. (88):

¼ Ef ρAE

is the quantum discord of the state ρAE with local measurement on the

¼ S AE ð Þ� S Eð Þ is the conditional entropy. As the label in the states

<sup>þ</sup> Ef <sup>ρ</sup>AB

$$D''(A:E)\_{\rho\_{A\mathcal{E}}} = E\_{\mathcal{C}} \left(\rho\_{A\mathcal{B}}\right) - S(A|E)\_{\rho\_{A\mathcal{E}}},\tag{94}$$

as the conditional entropy is additive S Að Þ jE <sup>ρ</sup> <sup>⊗</sup> <sup>n</sup> AE ¼ nS Að Þ jE <sup>ρ</sup>AE . Therefore, the following theorem comes from Eq. (88).

Theorem 32 (Cornelio et al. [20]). For every mixed entangled state ρAB, if

$$E\_D(\rho\_{AB}) = \frac{1}{n} I(A \backslash B)\_{(V\_n \otimes \mathbb{I})\rho\_{AB}^{\otimes n}} \tag{95}$$

$$E\_{\mathbb{C}}(\rho\_{AB}) = \frac{1}{k} E\_{\mathbb{F}}(\rho\_{AB}^{\otimes n}),\tag{96}$$

for a finite number of n and k, the entanglement is irreversible EC(ρAB) > ED(ρAB).

Taking the limit of many copies, the equation can be rewritten as:

$$D''(A:E)\_{\sigma\_{AE}} = Ec(\sigma\_{AB}) - Ec(\sigma\_{AB}),\tag{97}$$

where <sup>σ</sup>AB <sup>¼</sup> ð Þ Vk <sup>⊗</sup> <sup>I</sup> <sup>ρ</sup>AB and ED(σAB) = kED(ρAB). The quantum discord <sup>D</sup><sup>∞</sup>ð Þ <sup>A</sup> : <sup>E</sup> <sup>σ</sup>AE in this context can be viewed as the minimal amount of entanglement lost in the distillation protocol, for states belonging to the class described in the theorem [20]. This expression has an operational interpretation for quantum discord, where the quantum discord between the system and the purification system restricts the amount of e-bits lost in the distillation process. A consequence of this result is expressed by the state merging protocol [27], Alice (A), Bob (B) and the Environment (E) share a pure tripartite state ρABE, she would like to send her state to Bob, keeping the coherence with the system E. They can perform this protocol consuming an amount of entanglement in the process; the amount of entanglement is the regularized quantum discord <sup>D</sup><sup>∞</sup>ð Þ <sup>A</sup> : <sup>E</sup> <sup>ρ</sup>AE [25, 26].

In addition to the above relations, some upper and lower bounds between quantum discord and entanglement of formation have been calculated via the Koashi-Winter relation and the properties of entropy [59–62]. Equation (88) was also used to calculate the quantum discord and the entanglement of formation analytically for systems with rank-2 and dimension 2 ⊗ n [41, 63, 64]. Experimental investigations of Eq. (88) were performed in the characterization of the information flow between system and environment of a non-Markovian process [65].

### 5. Activation protocol

Physically, a measurement process can be described as an interaction between the measurement apparatus and the system, followed by a projective measurement on the apparatus. Consider a state <sup>ρ</sup><sup>S</sup> <sup>¼</sup> <sup>X</sup> k λkj ik h j k ∈ Dð Þ C<sup>S</sup> . The input state is described as ρ<sup>S</sup>:<sup>M</sup> ¼ ρ<sup>S</sup> ⊗ j i0 h j 0 <sup>M</sup>, by coupling a pure ancilla, that represents the measurement apparatus. The interaction between the system and the ancillary state is performed by a unitary evolution: <sup>U</sup><sup>S</sup>:<sup>M</sup> <sup>∈</sup>Uð Þ <sup>C</sup><sup>S</sup> <sup>⊗</sup> <sup>C</sup><sup>M</sup> , such that Tr<sup>M</sup> <sup>U</sup><sup>S</sup>:<sup>M</sup>ρ<sup>S</sup>:<sup>M</sup>U† S:M � � <sup>¼</sup> <sup>X</sup> l <sup>Π</sup>lρSΠ† <sup>l</sup> . A unitary operation satisfying this condition is given by:

$$\langle \mathcal{U}\_{\mathcal{S}:\mathcal{M}} | k \rangle\_{\mathcal{S}} | 0 \rangle\_{\mathcal{M}} = |k \rangle\_{\mathcal{S}} | k \rangle\_{\mathcal{M}},\tag{98}$$

A general bipartite state can be written as ρ = ∑i,j|i〉〈j| ⊗ Oi,j, where Oi,j is an Hermitian operator with trace different from zero. Then if the measurement is performed only on the subsystem A, the state ρ~<sup>S</sup>:<sup>M</sup> after the interaction with the measurement apparatus will be:

j i<sup>i</sup> h j<sup>j</sup> <sup>A</sup> <sup>⊗</sup> j i<sup>0</sup> h j <sup>0</sup> <sup>M</sup><sup>A</sup> <sup>⊗</sup> <sup>O</sup><sup>B</sup>

i,j

Differently of the global measurement process, for local measurements, entanglement can be

measurement apparatus creates a maximally entangle state. different from the case where the measurement is performed on the A quantum state cannot create quantum entanglement with the measurement apparatus, if it is classically correlated. As proved in the following theorem. Theorem 33 ([21, 22]). A state is classically correlated (has no quantumness of correlations), if and only if there exists an unitary operation such that the post interaction state is separable with respect to

Proof. The proof is performed for the general case, for measurements on both systems.

pk,j ak, bj �

the state after the interaction with the measurement apparatus represented by the unitary

Only if: Given a general separable state between the system and the measurement apparatus:

and the fact that the interaction is unitary, there is a convex combination such that

� � � ψα � � �

� � ak, bj � � � <sup>S</sup> ⊗ ak, bj �

� � ak, bj � � �

> � � ak, bj � � �

<sup>ρ</sup><sup>S</sup> <sup>¼</sup> <sup>X</sup> k, j

> pk,j ak, bj �

<sup>ρ</sup>~<sup>S</sup>:<sup>M</sup> <sup>¼</sup> <sup>X</sup> k, j

<sup>ρ</sup>~<sup>S</sup>:<sup>M</sup> <sup>¼</sup> <sup>X</sup>

α

p<sup>α</sup> ϕα � � � ϕα � � � <sup>S</sup> ⊗ ψα � � � ψα � � �

pαj i κα h j κα ; therefore, the interaction must act in the following way:

U<sup>S</sup>:<sup>M</sup>j i κα j i0 ¼ ϕα

<sup>S</sup>:<sup>M</sup> (102)

The Role of Quantumness of Correlations in Entanglement Resource Theory

: (104)

<sup>A</sup>:M<sup>A</sup> ⊗ I<sup>B</sup> (103)

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

87

<sup>2</sup>j ii h jj , the interaction with the

<sup>S</sup>; (105)

<sup>M</sup>; (106)

<sup>M</sup>; (107)

: (108)

i,j

1 AU†

ρ~<sup>S</sup>:<sup>M</sup> ¼ U<sup>S</sup>:<sup>M</sup> ρ<sup>S</sup>:<sup>M</sup>

<sup>¼</sup> <sup>X</sup> i, j

system and measurement apparatus.

operation UA:M<sup>A</sup> ⊗ UB:M<sup>B</sup> will be:

which is clearly separable.

<sup>ρ</sup><sup>S</sup> <sup>¼</sup> <sup>X</sup>

α

If: If the state is classically correlated:

¼ UA:M<sup>A</sup> ⊗ I<sup>B</sup>

� �U†

0 @

X i, j

j i<sup>i</sup> h j<sup>j</sup> <sup>A</sup> <sup>⊗</sup> j i<sup>i</sup> h j<sup>j</sup> <sup>M</sup><sup>A</sup> <sup>⊗</sup> <sup>O</sup><sup>B</sup>

created during the measurement process. For example, if Oij ¼ <sup>1</sup>

where {|k〉} is an orthonormal basis in CS. If the orthogonal basis {|k〉〈k|} is the canonical basis, this interaction is a Cnot gate [1]. Therefore, after the interaction, the state will be:

$$\left. \right|\_{\mathcal{S}\colon \mathcal{M}} = \mathcal{U}\_{\mathcal{S}\colon \mathcal{M}} \left( \rho\_{\mathcal{S}\colon \mathcal{M}} \right) \mathcal{U}\_{\mathcal{S}\colon \mathcal{M}}^{\dagger} = \sum\_{k} \lambda\_{k} |k\rangle \langle k|\_{\mathcal{S}} \otimes |k\rangle \langle k|\_{\mathcal{M}}.\tag{99}$$

The interaction between the system and the measurement apparatus results in a classically correlated state between the system and the apparatus. Hence performing a projective measurement on the state of the apparatus, the state of the system can be recovered.

Suppose now that the state of the system is composed, for example a bipartite system C<sup>S</sup> ¼ C<sup>A</sup> ⊗ CB. The measurements are performed locally in each system; therefore, the ancilla is also a bipartite system C<sup>M</sup> ¼ C<sup>M</sup><sup>A</sup> ⊗ C<sup>M</sup><sup>B</sup> . The unitary operator representing the interaction between the system and the measurement apparatus is U<sup>S</sup>:<sup>M</sup> ¼ UA:M<sup>A</sup> ⊗ UB:M<sup>B</sup> . Then, the post-measured state is:

$$\tilde{\rho}\_{\mathcal{S}} = \operatorname{Tr}\_{\mathcal{M}} \left[ \operatorname{LI}\_{\mathcal{S} : \mathcal{M}} \left( \rho\_{\mathcal{S}} \otimes |0\rangle\langle 0| \right) \operatorname{LI}\_{\mathcal{S} : \mathcal{M}}^{\dagger} \right] = \sum\_{k,l} \Pi\_{k}^{A} \otimes \Pi\_{l}^{B} \rho\_{AB} \Pi\_{k}^{\dagger A} \otimes \Pi\_{l}^{\dagger B}. \tag{100}$$

As aforementioned, the measurement process consists in interacting the system with an ancilla, which represents the measurement apparatus, and then perform a projective measurement over the ancilla. However, as the dimension of the ancilla is arbitrary, to represent a general measurement (POVM), it is necessary to couple another ancilla with the same size of the state: ρ<sup>S</sup><sup>0</sup> :<sup>M</sup> ¼ ρ<sup>S</sup> ⊗ j i0 h j 0 <sup>E</sup> ⊗ j i0 h j 0 <sup>M</sup>, where |0〉〈0|<sup>E</sup> is an ancillary state on space CE. Then, the interaction with the apparatus, given by a unitary evolution U<sup>S</sup><sup>0</sup> :M, results in the postmeasured state

$$\tilde{\rho}\_{\mathcal{S}} = \operatorname{Tr}\_{\mathcal{M}} \left[ \mathcal{U}\_{\mathcal{S}\colon \mathcal{M}} \rho\_{\mathcal{S}'\colon \mathcal{M}} \mathcal{U}\_{\mathcal{S}'\colon \mathcal{M}}^{\dagger} \right] = \sum\_{l} \Pi\_{l} (\rho\_{\mathcal{S}} \otimes |0\rangle\langle 0|\_{\mathcal{E}}) \Pi\_{l}. \tag{101}$$

By the Naimark's theorem Tr Π<sup>l</sup> ρ<sup>S</sup> ⊗ j i0 h j 0 <sup>E</sup> � � � � <sup>¼</sup> Tr Elρ<sup>S</sup> � �, where El <sup>¼</sup> ð Þ <sup>I</sup> <sup>⊗</sup> h j <sup>0</sup> <sup>Π</sup>lð Þ <sup>I</sup> <sup>⊗</sup> j i<sup>0</sup> is a element of a POVM.

A general bipartite state can be written as ρ = ∑i,j|i〉〈j| ⊗ Oi,j, where Oi,j is an Hermitian operator with trace different from zero. Then if the measurement is performed only on the subsystem A, the state ρ~<sup>S</sup>:<sup>M</sup> after the interaction with the measurement apparatus will be:

$$\mathfrak{p}\_{\mathcal{S}:\mathcal{M}} = \mathsf{U}\_{\mathcal{S}:\mathcal{M}}(\mathfrak{p}\_{\mathcal{S}:\mathcal{M}}) \mathsf{U}\_{\mathcal{S}:\mathcal{M}}^{\dagger} \tag{102}$$

$$\mathbb{L} = \mathsf{U}\_{A:\mathcal{M}\_A} \otimes \mathbb{I}\_B \left( \sum\_{i,j} |i\rangle\langle j|\_A \otimes |0\rangle\langle 0|\_{\mathcal{M}\_A} \otimes \mathcal{O}\_{i,j}^{\mathbb{B}} \right) \mathsf{U}\_{A:\mathcal{M}\_A}^\dagger \otimes \mathbb{I}\_B \tag{103}$$

$$0 = \sum\_{i,j} |i\rangle\langle j|\_A \otimes |i\rangle\langle j|\_{\mathcal{M}\_A} \otimes \mathcal{O}\_{i,j}^B. \tag{104}$$

Differently of the global measurement process, for local measurements, entanglement can be created during the measurement process. For example, if Oij ¼ <sup>1</sup> <sup>2</sup>j ii h jj , the interaction with the measurement apparatus creates a maximally entangle state. different from the case where the measurement is performed on the A quantum state cannot create quantum entanglement with the measurement apparatus, if it is classically correlated. As proved in the following theorem.

Theorem 33 ([21, 22]). A state is classically correlated (has no quantumness of correlations), if and only if there exists an unitary operation such that the post interaction state is separable with respect to system and measurement apparatus.

Proof. The proof is performed for the general case, for measurements on both systems.

If: If the state is classically correlated:

5. Activation protocol

86 Advanced Technologies of Quantum Key Distribution

Consider a state <sup>ρ</sup><sup>S</sup> <sup>¼</sup> <sup>X</sup>

isfying this condition is given by:

post-measured state is:

the state: ρ<sup>S</sup><sup>0</sup>

measured state

element of a POVM.

k

ρ~<sup>S</sup>:<sup>M</sup> ¼ U<sup>S</sup>:<sup>M</sup> ρ<sup>S</sup>:<sup>M</sup>

<sup>ρ</sup>~<sup>S</sup> <sup>¼</sup> Tr<sup>M</sup> <sup>U</sup><sup>S</sup>:<sup>M</sup> <sup>ρ</sup><sup>S</sup> <sup>⊗</sup> j i<sup>0</sup> h j <sup>0</sup> � �U†

the interaction with the apparatus, given by a unitary evolution U<sup>S</sup><sup>0</sup>

:<sup>M</sup>ρ<sup>S</sup><sup>0</sup>

:MU† S0 :M � � <sup>¼</sup> <sup>X</sup>

� � � � <sup>¼</sup> Tr Elρ<sup>S</sup>

ρ~<sup>S</sup> ¼ Tr<sup>M</sup> U<sup>S</sup><sup>0</sup>

By the Naimark's theorem Tr Π<sup>l</sup> ρ<sup>S</sup> ⊗ j i0 h j 0 <sup>E</sup>

<sup>U</sup><sup>S</sup>:<sup>M</sup> <sup>∈</sup>Uð Þ <sup>C</sup><sup>S</sup> <sup>⊗</sup> <sup>C</sup><sup>M</sup> , such that Tr<sup>M</sup> <sup>U</sup><sup>S</sup>:<sup>M</sup>ρ<sup>S</sup>:<sup>M</sup>U†

Physically, a measurement process can be described as an interaction between the measurement apparatus and the system, followed by a projective measurement on the apparatus.

ρ<sup>S</sup> ⊗ j i0 h j 0 <sup>M</sup>, by coupling a pure ancilla, that represents the measurement apparatus. The interaction between the system and the ancillary state is performed by a unitary evolution:

where {|k〉} is an orthonormal basis in CS. If the orthogonal basis {|k〉〈k|} is the canonical basis,

<sup>S</sup>:<sup>M</sup> <sup>¼</sup> <sup>X</sup> k

The interaction between the system and the measurement apparatus results in a classically correlated state between the system and the apparatus. Hence performing a projective mea-

Suppose now that the state of the system is composed, for example a bipartite system C<sup>S</sup> ¼ C<sup>A</sup> ⊗ CB. The measurements are performed locally in each system; therefore, the ancilla is also a bipartite system C<sup>M</sup> ¼ C<sup>M</sup><sup>A</sup> ⊗ C<sup>M</sup><sup>B</sup> . The unitary operator representing the interaction between the system and the measurement apparatus is U<sup>S</sup>:<sup>M</sup> ¼ UA:M<sup>A</sup> ⊗ UB:M<sup>B</sup> . Then, the

S:M

As aforementioned, the measurement process consists in interacting the system with an ancilla, which represents the measurement apparatus, and then perform a projective measurement over the ancilla. However, as the dimension of the ancilla is arbitrary, to represent a general measurement (POVM), it is necessary to couple another ancilla with the same size of

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

:<sup>M</sup> ¼ ρ<sup>S</sup> ⊗ j i0 h j 0 <sup>E</sup> ⊗ j i0 h j 0 <sup>M</sup>, where |0〉〈0|<sup>E</sup> is an ancillary state on space CE. Then,

l

Π<sup>l</sup> ρ<sup>S</sup> ⊗ j i0 h j 0 <sup>E</sup>

this interaction is a Cnot gate [1]. Therefore, after the interaction, the state will be:

� �U†

surement on the state of the apparatus, the state of the system can be recovered.

� � <sup>¼</sup> <sup>X</sup>

S:M � � <sup>¼</sup> <sup>X</sup>

λkj ik h j k ∈ Dð Þ C<sup>S</sup> . The input state is described as ρ<sup>S</sup>:<sup>M</sup> ¼

<sup>l</sup> . A unitary operation sat-

λkj ik h j k <sup>S</sup> ⊗ j ik h j k <sup>M</sup>: (99)

<sup>l</sup> <sup>ρ</sup>ABΠ†<sup>A</sup>

<sup>k</sup> ⊗ Π†<sup>B</sup>

� �Πl: (101)

� �, where El <sup>¼</sup> ð Þ <sup>I</sup> <sup>⊗</sup> h j <sup>0</sup> <sup>Π</sup>lð Þ <sup>I</sup> <sup>⊗</sup> j i<sup>0</sup> is a

<sup>l</sup> : (100)

:M, results in the post-

l <sup>Π</sup>lρSΠ†

U<sup>S</sup>:<sup>M</sup>j ik <sup>S</sup>j i0 <sup>M</sup> ¼ j ik <sup>S</sup>j ik <sup>M</sup>; (98)

$$\rho\_S = \sum\_{k,j} p\_{k,j} |a\_k, b\_j\rangle \langle a\_k, b\_j|\_S,\tag{105}$$

the state after the interaction with the measurement apparatus represented by the unitary operation UA:M<sup>A</sup> ⊗ UB:M<sup>B</sup> will be:

$$
\tilde{\rho}\_{\mathcal{S}:\mathcal{M}} = \sum\_{k,j} p\_{k,j} |a\_k, b\_j\rangle \langle a\_k, b\_j|\_{\mathcal{S}} \otimes |a\_k, b\_j\rangle \langle a\_k, b\_j|\_{\mathcal{M}},\tag{106}
$$

which is clearly separable.

Only if: Given a general separable state between the system and the measurement apparatus:

$$\tilde{\rho}\_{\mathcal{S}:\mathcal{M}} = \sum\_{a} p\_a |\phi\_a\rangle\langle\phi\_a|\_{\mathcal{S}} \otimes |\psi\_a\rangle\langle\psi\_a|\_{\mathcal{M}},\tag{107}$$

and the fact that the interaction is unitary, there is a convex combination such that <sup>ρ</sup><sup>S</sup> <sup>¼</sup> <sup>X</sup> α pαj i κα h j κα ; therefore, the interaction must act in the following way:

$$
\langle \mathcal{U}\_{\mathcal{S}:\mathcal{M}} | \mathbf{x}\_{\boldsymbol{\alpha}} \rangle | 0 \rangle = | \phi\_{\boldsymbol{\alpha}} \rangle | \psi\_{\boldsymbol{\alpha}} \rangle. \tag{108}
$$

On the other hand, as the state ρ<sup>S</sup> is bipartite, the pure states {|κα〉} can be written in general as: j i κα <sup>¼</sup> <sup>X</sup> l,i cα l,i a<sup>α</sup> l � � � bα i � � � , and after the interaction, the states will be:

$$\mathcal{U}\_{\mathcal{S}\colon\mathcal{M}}|\kappa\_{\boldsymbol{a}}\rangle|0\rangle = \sum\_{l,j} c\_{l,j}^{a} \left| a\_{l}^{a}, b\_{j}^{a} \right\rangle\_{\mathcal{S}} \otimes \left| a\_{l}^{a}, b\_{j}^{a} \right\rangle\_{\mathcal{M}}.\tag{109}$$

<sup>Δ</sup><sup>∅</sup> <sup>ρ</sup><sup>S</sup>

measured state of the system and <sup>ρ</sup>~<sup>S</sup>:<sup>M</sup> <sup>¼</sup> <sup>U</sup><sup>S</sup>:<sup>M</sup>ρ<sup>S</sup>:<sup>M</sup>U†

S ρ~<sup>S</sup>

where Π<sup>S</sup><sup>A</sup> ⊗ Π<sup>S</sup><sup>B</sup> ∈P C<sup>S</sup><sup>A</sup> ⊗ C<sup>S</sup><sup>B</sup>

quantumness of correlations.

where QE and EQ are related by Eq. (111).

relation to the correlations of the fermions [70].

Therefore:

<sup>¼</sup> min

� <sup>S</sup> <sup>ρ</sup>~<sup>S</sup>:<sup>M</sup>

protocol, and this conversion is ruled by the activation protocol.

Proposition 35 (Piani and Adesso [66]). For ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> :

quantumness of correlations also in the geometrical approach [17, 56].

ΠS<sup>A</sup> ⊗ ΠS<sup>B</sup> ∈ P

<sup>Δ</sup><sup>∅</sup> <sup>ρ</sup><sup>S</sup>

S Π<sup>S</sup><sup>A</sup> ⊗ Π<sup>S</sup><sup>B</sup> ρ<sup>S</sup>

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

This equation means that the activation protocol creates distillable entanglement between the system and the measurement apparatus during a local measurement. In other words, quantumness of correlations of the system can be converted resource for quantum information

From Eq. (111), it is possible to show that quantum entanglement is a lower bound for

To compare two measures of different quantities as quantumness of correlation and quantum entanglement, it is necessary a common rule. The activation protocol gives the rule to compare these two quantities and this rule says that the measures of quantumness of correlations and quantum entanglement must be related from Eq. (111). Entanglement is a lower bound for

Activation protocol determines that a composed state is classically correlated if and only if it cannot create entanglement during the measurement process, for a given unitary interaction [21, 22, 66]. This result provides an important tool for characterization of quantum correlations in identical particle systems (bosons and fermions), once that system and apparatus are distinguishable partitions, even if the particles in the system are identical. This approach have been applied to identical particles systems to prove how are the classically correlated states of bosons and fermions [69]. The activation protocol device also allows to determine the class of classically correlated states of the modes of a fermionic system and its

The entanglement generation by means of quantumness of correlations, as stated by the activation protocol, was experimentally evidenced using programmable quantum measurement [71]. In the experiment setup, the optimization on the unitary operations was performed by a set of programable quantum measurements in different local basis. As quantumness of

 <sup>¼</sup> min U<sup>M</sup>

� <sup>S</sup> <sup>ρ</sup><sup>S</sup>

is a local dephasing on subsystem <sup>A</sup> and <sup>B</sup>. As <sup>ρ</sup>~<sup>S</sup> is the

� <sup>S</sup> <sup>ρ</sup><sup>S</sup>

ED ρ~<sup>S</sup>:<sup>M</sup>

<sup>S</sup>:<sup>M</sup>, then:

; (114)

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

89

The Role of Quantumness of Correlations in Entanglement Resource Theory

:

QE <sup>ρ</sup>AB <sup>≥</sup> EQ <sup>ρ</sup>AB ; (116)

: (115)

As the state in Eq. (109) must be separable, it implies that the coefficients must satisfy:

$$c\_{i,j}^a = c\_{f(a)} \delta\_{i, \sharp f(a)} \quad \text{and} \quad |c\_{f(a)}| = 1 \tag{110}$$

where f(α) ∈ N<sup>2</sup> . As f(α) are orthogonal, it proves the theorem.

If the state of the system has quantum correlations, the local measurement process creates entanglement between the system and the measurement apparatus, for a every unitary interaction. Then, it is possible to fix the base of the ancilla and change the base of the system. Then, rewriting the evolution as U<sup>S</sup>:<sup>M</sup> ¼ C<sup>S</sup>:<sup>M</sup>ð Þ U<sup>S</sup> ⊗ I<sup>M</sup> , where for bipartite systems U<sup>M</sup> = UA ⊗ UB is a local unitary operation and C<sup>S</sup>:<sup>M</sup> ¼ CA:M<sup>A</sup> ⊗ CB:M<sup>B</sup> is a Cnot gate acting on the system as the control, and the apparatus as the target. It is possible to quantify the amount of quantum correlation in a given system starting on the amount of entanglement created with the measurement apparatus.

Definition 34 ([21, 22]). Each measure of entanglement used to quantify the entanglement between the system and the apparatus will result in a measure of quantumness of correlations.

$$Q\_{\mathcal{E}}\left(\rho\_{\mathcal{S}}\right) = \min\_{\mathcal{U}\_{\mathcal{S}}} E\_{\mathcal{Q}}\left(\rho\_{\mathcal{S}:\mathcal{M}}\right). \tag{111}$$

Different entanglement measures will lead, in principle, to different quantifiers for the quantumness of correlations. The only requirement is that the entanglement measure must be an entanglement monotone [21, 22, 66]. Some quantifiers of quantumness of correlations can be recovered with the activation protocol: the quantum discord [22], one-way work deficit [22], zero-way work deficit [21] and the geometrical measure of discord via trace norm [66], are some examples. Taking the distillable entanglement in Eq. (111) is quite simple to see that it results in zero-way work deficit. As shown in Eq. (106), the interaction with the measurement apparatus results in the state

$$\tilde{\rho}\_{\mathcal{S}\therefore\mathcal{M}} = \sum\_{k,j} p\_{k,j} |a\_k, b\_j\rangle\langle a\_k, b\_j|\_{\mathcal{S}} \otimes |a\_k, b\_j\rangle\langle a\_k, b\_j|\_{\mathcal{M}}.\tag{112}$$

That is named maximally correlated state, and as showed in Ref.[67], the distillable entanglement of this state attach the Hashing inequality [68]:

$$E\_D(\tilde{\rho}\_{\mathcal{S}:\mathcal{M}}) = -\mathcal{S}(\mathcal{S}|\mathcal{M}),\tag{113}$$

where Sð Þ¼ SjM S ρ~<sup>S</sup> � � � <sup>S</sup> <sup>ρ</sup>~<sup>S</sup>:<sup>M</sup> � � is conditional entropy of <sup>ρ</sup>~<sup>S</sup>:<sup>M</sup>. On the other hand, the zeroway work deficit for ρ<sup>S</sup> is:

The Role of Quantumness of Correlations in Entanglement Resource Theory http://dx.doi.org/10.5772/intechopen.70396 89

$$\Delta^{\mathcal{B}}(\rho\_{\mathcal{S}}) = \min\_{\Pi\_{\mathcal{S}\_A} \otimes \Pi\_{\mathcal{S}\_{\mathcal{B}}}} \{ \mathcal{S}(\Pi\_{\mathcal{S}\_A} \otimes \Pi\_{\mathcal{S}\_{\mathcal{S}}}[\rho\_{\mathcal{S}}]) - \mathcal{S}(\rho\_{\mathcal{S}}) \}, \tag{114}$$

where Π<sup>S</sup><sup>A</sup> ⊗ Π<sup>S</sup><sup>B</sup> ∈P C<sup>S</sup><sup>A</sup> ⊗ C<sup>S</sup><sup>B</sup> is a local dephasing on subsystem <sup>A</sup> and <sup>B</sup>. As <sup>ρ</sup>~<sup>S</sup> is the measured state of the system and <sup>ρ</sup>~<sup>S</sup>:<sup>M</sup> <sup>¼</sup> <sup>U</sup><sup>S</sup>:<sup>M</sup>ρ<sup>S</sup>:<sup>M</sup>U† <sup>S</sup>:<sup>M</sup>, then:

$$\mathcal{S}(\tilde{\rho}\_{\mathcal{S}}) - \mathcal{S}(\tilde{\rho}\_{\mathcal{S}:\mathcal{M}}) = \mathcal{S}(\Pi\_{\mathcal{S}\_{\mathcal{A}}} \otimes \Pi\_{\mathcal{S}\_{\mathcal{S}}}[\rho\_{\mathcal{S}}]) - \mathcal{S}(\rho\_{\mathcal{S}}) .$$

Therefore:

On the other hand, as the state ρ<sup>S</sup> is bipartite, the pure states {|κα〉} can be written in general

, and after the interaction, the states will be:

� � �

If the state of the system has quantum correlations, the local measurement process creates entanglement between the system and the measurement apparatus, for a every unitary interaction. Then, it is possible to fix the base of the ancilla and change the base of the system. Then, rewriting the evolution as U<sup>S</sup>:<sup>M</sup> ¼ C<sup>S</sup>:<sup>M</sup>ð Þ U<sup>S</sup> ⊗ I<sup>M</sup> , where for bipartite systems U<sup>M</sup> = UA ⊗ UB is a local unitary operation and C<sup>S</sup>:<sup>M</sup> ¼ CA:M<sup>A</sup> ⊗ CB:M<sup>B</sup> is a Cnot gate acting on the system as the control, and the apparatus as the target. It is possible to quantify the amount of quantum correlation in a given system starting on the amount of entanglement created with the mea-

Definition 34 ([21, 22]). Each measure of entanglement used to quantify the entanglement between the

Different entanglement measures will lead, in principle, to different quantifiers for the quantumness of correlations. The only requirement is that the entanglement measure must be an entanglement monotone [21, 22, 66]. Some quantifiers of quantumness of correlations can be recovered with the activation protocol: the quantum discord [22], one-way work deficit [22], zero-way work deficit [21] and the geometrical measure of discord via trace norm [66], are some examples. Taking the distillable entanglement in Eq. (111) is quite simple to see that it results in zero-way work deficit. As shown in Eq. (106), the interaction with the measurement

EQ ρ<sup>S</sup>:<sup>M</sup>

� � ak, bj � � �

� � is conditional entropy of <sup>ρ</sup>~<sup>S</sup>:<sup>M</sup>. On the other hand, the zero-

� � ¼ �Sð Þ <sup>S</sup>j<sup>M</sup> ; (113)

� � <sup>¼</sup> min U<sup>S</sup>

E S ⊗ a<sup>α</sup> <sup>l</sup> ; b<sup>α</sup> j

� � � E

i,j ¼ cfð Þ <sup>α</sup> δi,j;fð Þ <sup>α</sup> and jcfð Þ <sup>α</sup> j ¼ 1 (110)

M: (109)

� �: (111)

<sup>M</sup>: (112)

l, j c α l,j a<sup>α</sup> <sup>l</sup> ; b<sup>α</sup> j

As the state in Eq. (109) must be separable, it implies that the coefficients must satisfy:

. As f(α) are orthogonal, it proves the theorem.

system and the apparatus will result in a measure of quantumness of correlations.

QE ρ<sup>S</sup>

pk,j ak, bj �

ED ρ~<sup>S</sup>:<sup>M</sup>

� � ak, bj � � � <sup>S</sup> ⊗ ak, bj �

That is named maximally correlated state, and as showed in Ref.[67], the distillable entanglement

<sup>ρ</sup>~<sup>S</sup>:<sup>M</sup> <sup>¼</sup> <sup>X</sup> k, j

of this state attach the Hashing inequality [68]:

� � � <sup>S</sup> <sup>ρ</sup>~<sup>S</sup>:<sup>M</sup>

<sup>U</sup><sup>S</sup>:<sup>M</sup>j i κα j i<sup>0</sup> <sup>¼</sup> <sup>X</sup>

c α

as: j i κα <sup>¼</sup> <sup>X</sup>

where f(α) ∈ N<sup>2</sup>

surement apparatus.

apparatus results in the state

where Sð Þ¼ SjM S ρ~<sup>S</sup>

way work deficit for ρ<sup>S</sup> is:

l,i cα l,i a<sup>α</sup> l � � � bα i � � �

88 Advanced Technologies of Quantum Key Distribution

$$\Delta^{\mathcal{D}}\left(\rho\_{\mathcal{S}}\right) = \min\_{\mathcal{U}\_{\mathcal{M}}} E\_{\mathcal{D}}\left(\tilde{\rho}\_{\mathcal{S}:\mathcal{M}}\right). \tag{115}$$

This equation means that the activation protocol creates distillable entanglement between the system and the measurement apparatus during a local measurement. In other words, quantumness of correlations of the system can be converted resource for quantum information protocol, and this conversion is ruled by the activation protocol.

From Eq. (111), it is possible to show that quantum entanglement is a lower bound for quantumness of correlations.

Proposition 35 (Piani and Adesso [66]). For ρAB ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> :

$$\mathbb{Q}\_{\mathbb{E}}(\rho\_{AB}) \ge \mathbb{E}\_{\mathbb{Q}}(\rho\_{AB}),\tag{116}$$

where QE and EQ are related by Eq. (111).

To compare two measures of different quantities as quantumness of correlation and quantum entanglement, it is necessary a common rule. The activation protocol gives the rule to compare these two quantities and this rule says that the measures of quantumness of correlations and quantum entanglement must be related from Eq. (111). Entanglement is a lower bound for quantumness of correlations also in the geometrical approach [17, 56].

Activation protocol determines that a composed state is classically correlated if and only if it cannot create entanglement during the measurement process, for a given unitary interaction [21, 22, 66]. This result provides an important tool for characterization of quantum correlations in identical particle systems (bosons and fermions), once that system and apparatus are distinguishable partitions, even if the particles in the system are identical. This approach have been applied to identical particles systems to prove how are the classically correlated states of bosons and fermions [69]. The activation protocol device also allows to determine the class of classically correlated states of the modes of a fermionic system and its relation to the correlations of the fermions [70].

The entanglement generation by means of quantumness of correlations, as stated by the activation protocol, was experimentally evidenced using programmable quantum measurement [71]. In the experiment setup, the optimization on the unitary operations was performed by a set of programable quantum measurements in different local basis. As quantumness of correlation can be generated by local operations [10], activation protocol was explored experimentally in the generation of distillable entanglement via local operations on the measured partition of the system [72].

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