Quantum Correlations and Bell Inequality Violation under Decoherence

Quantum Correlations are studied extensively in quantum information domain. Entanglement Measures and Quantum Discord are good examples of these actively studied correlations. Detection of violation in Bell inequalities is also a widely active area in quantum information theory world. In this work, we revisit the problem of analyzing the behavior of quantum correlations and violation of Bell inequalities in noisy channels. We extend the problem defined in [1] by observing the changes in negativity measure, quantum discord and a modified version of Horodecki measure for violation of Bell inequalities under amplitude damping, phase damping and depolarizing channels. We report different interesting results for each of these correlations and measures. All these correlations and measures decrease under decoherence channels, but some changes are very dramatical comparing to others. We investigate also separability conditions of example studied states.

hot topic. We can define classical correlations in the context of quantum information as those arising from the use of LOCC. If we look at a quantum system and can not simulate them classically, we generally have quantum correlations. Suppose we have a noisy quantum system and we are working on it on LOCC. In this process we can obtain such a system state that we can do some things we can not achieve with classical correlations, such as violating Bell inequality. In this case, we can obtain these effects by quantum correlations (like quantum discord) in the initial system state that are already present at the source location (even if it is a very noisy system state), not after the LOCC operations. This is the most important point of the entanglement studies.
Until now there has been few studies focused on the relation between quantum discord and Bell inequalities violation [1]. In this work, we extend the problem defined in [1] for two other noisy channels: phase damping (PDC) and depolarizing (DPC). Especially, for these correlations and measures we proved the following: • Under DPC, the studied state becomes separable very rapidly. Negativity measure for this type states is more robust to ADC and PDC than the DPC.
• Under PDC and DPC quantum discord values decreases more rapidly then the ones under ADC. Quantum discord is more robust to ADC than the other noisy channels for this type of quantum states.

Materials and Methods
In this section we make the definitions of our scientific materials and methods. First, we define the studied state. Secondly, we definition one of most actively studied entanglement measure Negativity. In third order, we define quantum discord which is a special quantum correlation. Next, we define violation of Bell inequalities and finally we define the decoherence channels which we study here: amplitude damping, phase damping and depolarizing.

Definitions about Studied State
We consider a two-level quantum system whose computational bases are and , and then assume that two observer, Alice and Bob which are sensitive only to mode and , are respectively sharing an entangled initial state , (1) where the parameter α ϵ (0,1).

Negativity
Negativity is a quantitative version of Peres-Horodecki criterion. It is defined for two particle two level general quantum systems as follows [25,26]: Here µmin value is the minimum eigenvalue of ρ's partial transpose. ρ is the density matrix of the quantum state. Negativity, which is defined by the equation above is a value between 0 and 1 like Concurrence. Similarly like for concurrence, 1 means maximal entanglement. 0 means that the state is a separable state.
Vidal and Werner shown that Negativity is a monotone function for entanglement [26].

Quantum Discord
The definitions used in this subsection are well reviewed and organized in a review by Streltsov [33]. Quantum discord is considered as the first measure of quantum correlations beyond entanglement [27]. The definition of quantum discord is made on the basis of mutual information between two random variables X and Y and it is possible this definition with the following equations Here, is the random variable X's Shannon entropy where px is the probability when X takes the value x. H(X,Y) is the joint entropy of both variables X and Y. The conditional entropy H(X|Y) can be written as (5) where py is the probability that the random variable Y takes the value y, and H(X|y) is the entropy of the variable X conditioned on the variable Y taking the value y: and px|y is the probability of x given y.
The following equality of I and J is coming from Bayes rule px|y = pxy/py, which can be used to show that H(X|Y)=H(X,Y)-H(Y). However, I and J are no longer equal if quantum theory is applied [27,33]. More precisely, for a quantum state ρ AB the mutual information between A and B can be defined as (6) with the von Neumann entropy S, and the reduced density operators and This expression is the generalization of the classical mutual information I(X:Y) to the quantum theory.
On the other hand, the generalization of J(X:Y) is not completely obvious. The following way to generalize J to the quantum theory was proposed [27]: for a bipartite quantum state the conditional entropy of A conditioned on a measurement on B was defined as: (7) where are measurement operators corresponding to a von Neumann measurement on the subsystem B, i.e, orthogonal projectors with rank one. The probability pi for obtaining the outcome i can be given by , and the corresponding post-measurement state of the subsystem A is defined by . The equality J can now be extended to quantum states as follows [27]: (8) where the index clarifies that the value depends on the choice of the measurement operators . The quantity J represents the amount of information gained about the subsystem A by measuring the subsystem B [27].
Quantum discord is defined as the difference of these two inequivalent expressions for the mutual information, minimized over all von Neumann measurements: (9) where the minimum over all von Neumann measurements is taken in order to have a measurement-independent expression [27,33]. As was also shown in [27], quantum discord is nonnegative, and is equal to zero on quantum-classical states only. These are states of the form .

Violation of Bell inequality: Measure of CHSH violation
The CHSH inequality for a two-qubit state ρ ≡ ρAB can be written as [29,30]: in terms of the CHSH operator   are unit vectors describing the measurements on side A (Alice) and B (Bob), respectively. A shown by Horodecki et al. [29,30] by optimizing the vectors a  ,  , the maximum possible average value of the Bell operator for the state ρ is given by and hj (j=1,2,3) are the eigenvalues of the matrix U=T T T constructed from the correlation matrix T and its transpose T T . [28] In other words, M(ρ) = τ1 + τ2 where τ1 and τ2 are maximum eigenvalues of symmetric U matrix.
It can be showed that Now our question become calculating τ1 + τ2 for any random states / density matrix.
In order to quantify the violation of CHSH inequality one can use M(ρ) or equivalently where B=0 means CHSH inequality is not violated and B=1 means is maximally violated. [28]

Decoherence Channels
In this section, we give the definitions of the three decoherence channels (ADC, PDC and DPC). In general, the decoherence channels are given in the Kraus representation (15) where Eµ are the Kraus operators that satisfy (16) where 1 is an identity matrix. In the following subsections, we give the definitions about the three channels consecutively [31]

Amplitude Damping Channel (ADC)
The ADC describes the dissipation process. For a single qubit, the Kraus operators of the ADC are (17) where the decoherence strength represents the probability of decay from the upper level to the lower level , with and γ1 is the damping rate [31].

Phase Damping Channel (PDC)
The Kraus operators for the PDC are given by (18) The PDC is a prototype model of dephasing or pure decoherence. The decoherence strength is comparable with a concrete dephasing model by replacing p with , where γ2 is associated with the relaxation in spin resonance [31].

Depolarizing Channel (DPC)
The Kraus operators of the DPC are given by (19) where . For the DPC, the spin is depolarized to the maximally mixed state /2 with probability p or is unchanged with probability .

Results
In this section we give the numerical results that we achieved during our study. This section is organized as follows: First we give Negativity values (a good example of Entanglement measures) changes in ADC, PDC and DPC decoherence channels. Secondly, we show changes in Quantum Discord (correlations beyond entanglement) under mentioned decoherence channels. And finally, we give the changes in B which is a modified Horodecki measure for Violation of Bell inequalities under the same noisy channels.

Negativity under Decoherence Channels
In this subsection we show changes in Negativity values under three noisy channels respectively. The changes in Figure 1  We can view that under DPC, the studied state becomes separable very rapidly. Negativity measure for this type states is more robust to ADC and PDC than the DPC.

Quantum Discord under Decoherence Channels
In this subsection we show changes in Quantum Discord values under three noisy channels respectively. The changes in Figure 5 and 6 which are values under PDC and DPC are quite similar. In Figure 4 we can see that value changes under ADC quite different. Under PDC and DPC quantum discord values decreases more rapidly then the ones under ADC. Quantum discord is more robust to ADC than the other noisy channels for this type of quantum states.

Violation of Bell inequalities (B) under Decoherence Channels
In this subsection, we give the changes in B which is a modified Horodecki measure for Violation of Bell inequalities under the same noisy channels. In Figure 7, our finding are quite interesting. After the value of p=0.2928 none of the states are violating Bell inequalities under ADC. The changes in B can be viewed in the 3D plot shown in Figure 8.    In Figure 13, we show in 3D plot the changes in Negativity and B values as functions of p and α.

Conclusions
In this work, we extend the problem defined in [1] for two other noisy channels: phase damping (PDC) and depolarizing (DPC). We can summarize the following observations achieved during this study: For the Negativity case, especially when the value decoherence strength p increase, the studied state becomes separable. Under DPC, the studied state becomes separable more rapidly and Negativity measure for this type states is more robust to ADC and PDC than the DPC. For example, under DPC when p = 0.2, the state is separable for values α< 0.15 and α > ~0.98. When p=0.3, the state is separable for values α< 0.27 and α > ~0.96. when p becomes 0.4 the changes are more dramatical. The state is separable for values α< ~0.5 and α > ~0.85. For larger p values all states are separable.
For the Quantum Discord case; under PDC and DPC quantum discord values decreases more rapidly then the ones under ADC. Quantum discord is more robust to ADC than the other noisy channels for this type of quantum states.