Quantumness and the role of locality on quantum correlations

G. Bellomo, A. Plastino, and A. R. Plastino

Phys. Rev. A 93, 062322 – Published 20 June 2016

Abstract

Quantum correlations in a physical system are usually studied with respect to a unique and fixed decomposition of the system into subsystems, without fully exploiting the rich structure of the state space. Here, we show several examples in which the consideration of different ways to decompose a physical system enhances the quantum resources and accounts for a more flexible definition of quantumness measures. Furthermore, we give a different perspective regarding how to reassess the fact that local operations play a key role in general quantumness measures that go beyond entanglement—as discordlike ones. We propose a family of measures to quantify the maximum quantumness of a given state. For the discord-based case, we present some analytical results for $2 \times d$ -dimensional states. Applying our definition to low-dimensional bipartite states, we show that different behaviors can be reported for separable and entangled states vis-à-vis those corresponding to the usual measures of quantum correlations. We show that there is a close link between our proposal and the criterion to witness quantum correlations based on the rank of the correlation matrix, proposed by Dakić, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)].

Received 11 January 2016
Revised 14 April 2016

DOI:https://doi.org/10.1103/PhysRevA.93.062322

Physics Subject Headings (PhySH)

Entanglement measures

Quantum Information, Science & Technology

Authors & Affiliations

G. Bellomo^* and A. Plastino

Instituto de Física La Plata, UNLP, CONICET, and Departamento de Física, Facultad de Ciencias Exactas, La Plata, Argentina

A. R. Plastino

CeBio and Secretaría de Investigaciones, UNNOBA, CONICET, Junín, Argentina

^*gbellomo@fisica.unlp.edu.ar

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 93, Iss. 6 — June 2016

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
For a composite system $A + B$ , CC with respect to that bipartition, coupling auxiliary uncorrelated local systems ( $\bar{A}$ and $\bar{B}$ ), and performing LU operations generally produces a QC state (see Proposition t1 for details). Here, the local unitaries have the form $U^{\bar{A} A} \otimes U^{B \bar{B}}$ and act by rearranging the local degrees of freedom, thus yielding “new” subsystems $A^{'}, {\bar{A}}^{'}, B^{'}, {\bar{B}}^{'}$ . In other words, the local unitaries induce new decompositions of $\bar{A} A$ and $B \bar{B}$ into different “primed” subsystems. (See text for details.)
Reuse & Permissions
Figure 2
Classically correlated states can exhibit some quantumness revealed by PD. That is the case for the $ρ_{η}^{C}$ family of fully CC states (orange circles), which attains maximal PD when $η = 1 / 2$ . Isotropic (green dotted line) and Werner (red dashed line) families have a PD that equals their QD, as no LO can increase quantum correlations for them (see Sec. 4a for a detailed discussion). PD for states of the $ρ_{η}^{C}$ family fits very well with the function $P^{δ} (η) \approx 0.2018 - 0.6979 {(η - 0.5)}^{2} - 0.4204 {(η - 0.5)}^{4}$ .
Reuse & Permissions
Figure 3
Any separable state with QD $\leq 0.2018$ can be reached from a CC state of two qubits, performing local operations. Nonetheless, there are separable states with QD above 0.2018. The comparison between QD and PD displays this behavior. The upper bound is given by the $ρ_{γ}^{M}$ family (orange circles; see text for details). Green data points correspond to $\sim 10^{5}$ randomly generated states. The (red shaded) region at the left of the vertical dashed line corresponds to the values of QD achievable by separable states. Inset: detail of the bottom leftmost region.
Reuse & Permissions
Figure 4
Different measures of quantum correlations for the $ρ_{γ}^{M}$ family. Entanglement of formation (red dashed line) is finite for any $γ > 0$ . QD (green dotted line) behaves in the same qualitative way as entanglement, while PD (orange circles) exhibits a transition: below $γ = 1 / 2$ PD takes a constant minimum value and grows monotonically with $γ$ for $γ \geq 1 / 2$ .
Reuse & Permissions
Figure 5
Under an amplitude damping channel, QD (green dotted line) and PD (orange dashed-dotted line) exhibit different behaviors, indicating the fact that both measures reflect different aspects of the quantumness of a given state. Inset: QD and PD as a function of time (in units of the inverse of the relaxation rate $Γ$ ).
Reuse & Permissions
Figure 6
Under global unitary operations, QD of a fixed state can be increased by an amount depending on the mixedness of the state. In particular, pseudopure states can be transformed into isotropic states (black dots), maximizing the QD for a certain value of entropy. Werner states (grey line) provide the upper bound for almost every value of entropy. Orange data points correspond to $\sim 10^{5}$ random states. Inset: original QD for the same states. The region between isotropic and Werner states is upper bounded by families of two parameters (see Ref. [41] for details).
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information