1. Introduction

Quantum entanglement plays the fundamental role in quantum information and computation [1, 2]. The resource theory of quantum entanglement, entanglement distillation [3] and entanglement cost [4] revealed one of the most fundamental aspects of quantum mechanics. Entanglement distillation protocol consists in converting a number of copies of an entangled state into few copies of maximally entangled states, by means of local operations and classical communication (LOCC) [5]. As maximally entangled states are the main resource of the quantum information, entanglement distillation protocol has many applications in this scenario, as quantum teleport [6], quantum error correction [7] and quantum cryptography [8]. A family of quantum information protocol arises from distillation of quantum entanglement and secret keys [3, 9]

However independently Ollivier and Zurek [10], and Henderson and Vedral [11] found a new quantum property, without counterpart in classical systems. They named it as the quantumness of correlations. This new kind of correlation reveals the amount of information destroyed during the local measurement process and goes beyond the quantum entanglement. There are many

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

equivalent formulations for characterization and quantification of quantumness of correlations: quantum discord [10, 11], minimum local disturbance [12–14] and geometrical approach [15–17].

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

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

• 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:

• Completely positive: Consider a channel Φ : Dð Þ! C<sup>A</sup> Dð Þ C<sup>A</sup> and a quantum state

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

Measurement is a classical statistical inference of quantum systems. The measurement process

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

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

I ⊗ Φ ρ

<sup>Φ</sup> <sup>ρ</sup> <sup>þ</sup> <sup>σ</sup> <sup>¼</sup> <sup>Φ</sup> <sup>ρ</sup>

map must be linear. For two arbitrary operators ρ, σ∈ D C<sup>N</sup>

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> .

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>

ρ, σ∈ Dð Þ C<sup>A</sup> ⊗ C<sup>B</sup> , then

2.2. Measurement

the notation will depend on the context.

1

ρ ¼ j i ψ h j ψ : (2)

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

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The Role of Quantumness of Correlations in Entanglement Resource Theory

<sup>þ</sup> <sup>Φ</sup>ð Þ <sup>σ</sup> ; (3)

Tr <sup>Φ</sup> <sup>ρ</sup> <sup>¼</sup> <sup>1</sup>: (4)

≥ 0: (5)

, associating an alphabet Σ to

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

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 quantumness of correlation can be activated into distillable entanglement.
