Correlations, Nonlocality and Usefulness of an Efficient Class of Two-Qubit Mixed Entangled States

2.1 Amplitude-Damping Channel

We first proceed to analyse the effect of decoherence and weak measurements by establishing a relation between the maximum expectation value of the Bell-CHSH operator, noise parameter and weak measurement strengths. For this, we start with a scenario where Charlie prepares a two-qubit pure state |Ψ〉=α|00〉+β|11〉 (|α|²+|β|²=1) and sends one qubit each to Alice and Bob through an amplitude-damping channel. The single-qubit Kraus operators for an amplitude-damping channel can be given as

where γ is the magnitude of decoherence, and i=1 or i=2 represents the qubit index. Therefore, the two-qubit pure state, after passing through the amplitude-damping channel, evolves as

where γ′i=1−γi. In order to find whether the above state violates the Bell-CHSH inequality or not, we need to evaluate the maximum expectation value of the Bell-CHSH operator given in (1). In terms of expectation values, (1) can be re-expressed as

Considering a pair of mutually orthogonal unit vectors ĉ, ĉ′ such that b^+b^′=2cosθc^, and b^−b^′=2sinθc^′, (4) can be rewritten as

where unit vectors a and c are defined as

Similar definitions stand for a′ and c′ with prime on angles. The first term representing the expectation value 〈AC〉 gives

The expectation value of 〈AC〉 can be maximised with respect to θ_a, such that

where we have used the fact that the maximum value of psinθ₁+qcosθ₁ is p2+q2 and cos²ϕ_ac=cos²(ϕ_a+ϕ_c)=1. Similarly, the second term representing the expectation value 〈A′C′〉 gives

Equation (5) is maximized with respect to θ, and therefore, we have

To optimize the expectation value for the operator B(ρAγ), we use the orthogonality relation between ĉ and ĉ′ such as cos²θ_c=sin²θ_c′, and hence,

If Charlie sends both the qubits through perfect channels such that γ_i=0, then, the optimised expectation value of the Bell-CHSH operator will be 2[1+4α2β2]12 as it should be for transmission through an ideal quantum channel [55].

Figure 1 clearly demonstrates that the Bell-CHSH inequality is violated for a small region only where the value of noise parameters γ_i are very small; even for the violation region where the values of noise parameters are small, the violation decreases very fast. The analytical result obtained, here, is in complete agreement with the numerical optimisation of the Bell-CHSH operator for ρAγ. The effect of noise on nonlocality is depicted in Figure 2, which describes the degradation of nonlocal correlations due to decoherence for different initial states, i.e. for different α values. If we consider the noise parameters to be the same, then, the Bell-CHSH inequality is violated by the finally shared states iff (α²+β²(1−2γ)²)²+4α²β²(γ−1)²>1. For example, if we start with a maximally entangled initial state, then, the range of decoherence parameter for the Bell inequality violation is 0≤γ<0.2334. Therefore, if we start with a maximally entangled state, then, the finally shared state does not violate the Bell inequality for 0.2334≤γ≤1. Interestingly, the nonviolation region for a state with α=0.95 is 0.3203≤γ≤1, i.e. a partially entangled two-qubit state is more robust towards decoherence in comparison to a maximally entangled two-qubit state.

Figure 1:

Estimation of B(ρAγ)opt with respect to decoherence parameters γ₁ and γ₂ for a maximally entangled two-qubit state.

Figure 2:

Effect of noise parameter γ, considering (γ₁=γ₂=γ) on B(ρAγ)opt for different two-qubit entangled states.

We now move forward to analyse the effect of weak measurements on the existence of nonlocal correlation in noisy conditions. For this, we assume that Charlie prepares a two-qubit entangled state |Ψ〉=α|00〉+β|11〉, (|α|²+|β|²=1) and performs weak measurements on both qubits before sending them through amplitude-damping channels. Similarly, after receiving the qubits, both Alice and Bob carry out the reversal of the weak measurement on their qubits. The weak measurement Λiwk and the reverse weak measurement Λiwkr operators performed at both ends can be given by

where η_i and η_ri are the strengths of the weak measurement and the weak measurement reversal operations, respectively. The optimal weak measurement strength is defined by η_ri=η_i+γ_i(1−η_i), where i=1, 2 [21], [22], [23]. Assuming that the strength of the weak measurement reversal is optimal, the finally shared state between Alice and Bob evolves as

where NA=1+{γ1η′1(1+γ2η′2)+γ2η′2}|β|2 and η′i=(1−ηi) and (i=1, 2). For analytical optimisation of the Bell-CHSH operator, we first consider the first term representing the expectation value 〈AC〉 in (5), such that

Similar to the way we evaluated the optimum value of the Bell-CHSH operator for ρAγ, one can show that the optimum expectation value of the Bell-CHSH operator for ρAwk is

For η_i=1, the expression in (15) will be the same as for a pure state. This is possible as the state ρAwk becomes a pure state free from any decoherence for η_i=1.

Figure 3 demonstrates the effect of the weak measurement strengths on the Bell-CHSH operator for a decoherence parameter value of γ₁=γ₂=0.5 and α2=12. The state clearly violates the Bell inequality for γ_i=0.5 when the values of η_i’s exceed a certain minimum, and the amount of violation increases with the increase in weak measurement strength. From Figure 2, one can conclude that if one starts with a maximally entangled two-qubit state, then, for γ_i=0.5, the Bell-CHSH inequality is not violated. However, performing weak measurement and weak measurement reversal operations allows for the violation of the Bell-CHSH operator confirming the existence of nonlocal correlations in the finally shared entangled state. For simplicity, we consider a scenario where both channels have the same decoherence, i.e. γ₁=γ₂=γ, and both qubits are subjected to identical weak measurement strengths, i.e. η₁=η₂=η. In such a case, the optimal expectation value of the Bell-CHSH operator is

Figure 3:

Estimation of B(ρAwk)opt with respect to weak measurement strengths η₁ and η₂ for a maximally entangled initial state, considering γ=0.5.

where N′A=(α2+(1+γ(1−η))2β2). In order to compare the effects of the weak measurement vs. amplitude damping, in Figure 4, we show plots between the optimal expectation value of the Bell-CHSH operator and the weak measurement strength for different initial states considering that γ=0.6. We consider a higher value of the noise parameter as in the absence of weak measurement, the Bell-CHSH inequality is not violated by all states. We again observe that a partially entangled initial state with a higher α is a better and robust resource in comparison to a maximally entangled initial state. Moreover, depending on the initial state used, the Bell-CHSH inequality is violated by the finally shared state only after a certain value of the weak measurement strength η. In general, the nonviolation regime increases with a decrease in the value of α. For a given initial state, we further deduce a condition ((α²+(1+γ(η−1))²β²)²+4α²β²)≥(α²+(1+γ(1−η))²β²)² for the violation of the Bell-CHSH inequality by a finally shared state. For example, if γ=0.6, then, for an initial state with α=12, the strength of the weak measurement required for the violation of the Bell-CHSH inequality is 0.5953≤η≤1.

Figure 4:

Effect of weak measurement strength η on B(ρAnew)opt, for different two-qubit entangled states, considering the noise parameter γ=0.6.

For a maximally entangled initial state, Figures 5 and 6 describe the effects of the noise parameter γ for different values of the weak measurement strength η and the effects of the weak measurement strength η for the different values of the noise parameter γ. Clearly, the weak measurement and its reversal is a win-win situation for enhancing the correlations between the qubits. Similarly, Figures 7 and 8 illustrate the relation between the violation of the Bell-CHSH operator and the value of α for different values of the noise parameter γ for η=0 and the different values of the weak measurement strength for γ=0.6, respectively.

Figure 5:

Effect of decoherence on B(ρAnew)opt for a maximally entangled input state at different values of weak measurement strength η.

Figure 6:

Comparison of the Bell inequality violation vs. weak measurement strength η for a maximally entangled input state at different values of decoherence parameter.

Figure 7:

Comparison of B(ρAnew)opt vs. state parameter α at different values of noise parameter γ, considering η=0.

Figure 8:

Comparison of B(ρAnew)opt vs. state parameter α for different values of weak measurement strength considering γ=0.6.

2.2 Alternative Method to Estimate Violation of the Bell-CHSH Operator

Horodecki et al. [56] have shown a necessary and sufficient condition for the violation of the CHSH inequality by an arbitrary spin−12 state. According to Horodecki’s theorem, the maximum possible violation of the CHSH inequality for any arbitrary 2-qubit state ρ is given by

where M(ρ)=max_i<j (u_i+u_j), and u_i (i=1, 2, 3) are the eigenvalues of U=T^TT. Here, T^T denotes the transposition of T, and inequality (1) is violated by an arbitrary two-qubit state iff M(ρ)>1. Therefore, we calculate M(ρAwk) considering the identical noise parameters γ₁=γ₂=γ and identical weak measurement strengths η₁=η₂=η, such that

and hence,

One can clearly see that (19) is the same as (16) obtained in this article analytically.

2.3 Phase-Damping Channel

In the case of a phase-damping channel, the Kraus operators can be represented as

where γ represents the phase-damping noise parameter, and i=1, 2 represents the qubit index. Again, for simplicity, we consider that both qubits comprising the initial input state are transmitted through an identical decoherence channel, i.e. γ₁=γ₂=γ. Similar to the case of the amplitude-damping channel, the input state shared between Alice and Bob now evolves as

Therefore, using the Horodecki’s theorem, the optimum expectation value of the Bell-CHSH operator is given as

Figure 9 describes the effect of decoherence on the correlations after both the qubits pass through phase-damping channels. Unlike the case of the amplitude-damping channel, where a nonmaximally entangled state seems to be more robust than a maximally entangled state for a particular range of decoherence parameter, here, the maximally entangled state is always robust in comparison to the nonmaximally entangled states. For a given two-qubit initial state, the finally shared state always violates the Bell inequality for the whole range of decoherence parameters.

Figure 9:

Effect of decoherence γ on B(ρPγ)opt for different two-qubit entangled states.

In order to analyse the effect of the weak measurement and its reversal, we again assume that before sending the qubits through phase-damping channels, Charlie first performs weak measurements on both the qubits as given in (12). After receiving the qubits, Alice and Bob perform weak measurement reversal operations on their respective qubits. Therefore, the finally shared state between Alice and Bob evolves as

where η′=(1−η), η′r=(1−ηr),γ′=(1−γ), and N_P=(|α|² (1−η_r)²+|β|²(1−η)²). The optimised value of the Bell-CHSH operator for the state ρPwk can be calculated in a similar fashion as in the case of ρPγ, and can be given as

The optimal weak measurement reversal strength leading to maximum correlations between the qubits is evaluated to be ηr=1−(1−η)βα. Hence, assuming the strength of the measurement reversal operation to be optimal, the expectation value of the Bell-CHSH operator is given as

Equation (25) interestingly shows that the maximum expectation value of the Bell-CHSH operator of a shared bipartite state is independent of the parameter α and weak measurement strength η. Moreover, B(ρPwk)opt≥B(ρPγ)opt provided α2β2≤12, which is always true for a two-qubit entangled state. Hence, the application of weak measurement can, indeed, be useful in upgrading the nonlocal correlations against the phase-damping decoherence. For optimal reversing weak measurement strength, Figure 10 clearly indicates that the maximum expectation value of the Bell-CHSH operator for the finally shared does not depend on the initial input state and is always the same as for a maximally entangled initial state. The use of weak measurement and its reversal protocol, therefore, provides a flexibility to communication protocols such that one can choose to start with any initial two-qubit pure state. However, for non-optimal weak measurement reversal, i.e. assuming η_r=0.2 and a given input state considering α=0.42, the effect of the weak measurement strengths on the maximum expectation value of the Bell-CHSH operator for different values of the noise parameter is depicted in Figure 11.

Figure 10:

Effect of decoherence on B(ρPwk)opt for a maximally entangled state and any partially entangled input state under the application of weak measurement.

Figure 11:

Effect of weak measurement on B(ρPwk)opt for non-optimal weak measurement reversal strength η_r=0.2, for different values of decoherence parameter γ considering α=0.42.

2.4 Depolarising Channel

Finally, we consider another important decoherence channel characterised by depolarising noise such that the single-qubit Kraus operators are described as

where γ is a decoherence parameter. Here, we again consider the identical decoherence channel, i.e. γ₁=γ₂=γ. In this case, the initial state after passing through the depolarising channel can be given as

where

Thus, the optimum expectation value of B(ρDγ) of the two-qubit state ρDγ shared between Alice and Bob is

Figure 12 demonstrates the effect of the noise parameter γ on the expectation value of the Bell-CHSH operator for three different initial states, i.e. for α=12,α=0.95 and α=0.42. One can observe that the violation of the Bell-CHSH inequality decreases very fast even for small values of noise parameters.

Figure 12:

Effect of decoherence γ on B(ρDγ)opt for different two-qubit entangled states.

We now consider to analyse the effect of the weak measurement and quantum measurement reversal on nonlocal correlations of the finally shared state. For the depolarising channel, we replace 1−η with μ and 1−ηr with μ_r in (12) such that the expressions of the weak measurement and the weak measurement reversal operations are now given as Δwk=(100μ) and Δwkr=(μr001), respectively [57]. Similar to the previous cases, the finally shared state between Alice and Bob evolves as

where

The optimised value of the Bell-CHSH operator for the state ρDwk can be obtained in a similar fashion as discussed above, and can be given as

The optimal reversing weak measurement strength, in the case of a depolarising channel, for maximising the amount of entanglement in the finally shared state is evaluated as

Using (33), the maximum expectation value of the Bell-CHSH operator can be achieved for μ²=|α|/|β|, and can be expressed as

From (29) and (34), one can further deduce that B(ρDwk)opt is always greater than B(ρDγ)opt. Furthermore, for the optimal weak measurement reversal strength and a maximally entangled initial input state, Figure 13 describes the effect of the weak measurement strength on the maximum expectation value of the Bell-CHSH operator considering that γ=0.1.

Figure 13:

Effect of weak measurement on B(ρDwk)opt for a maximal entangled initial input state considering γ=0.1.

4.1 Quantum Teleportation

Quantum teleportation allows a sender to communicate quantum information using an entangled resource without sending the information through any medium. Horodecki et al. [24] described a measure of usefulness of the two-qubit mixed entangled states in terms of fidelity of quantum teleportation, namely,

where H(ρ)=Σi=13ui and u_i (i=1, 2, 3) are the eigenvalues of the real symmetric matrix U_ρ=T^TT. They further deduced that a given state is useful as a resource for quantum teleportation iff H(ρ)>1. In this subsection, we show that the class of states proposed in the previous section can always be used as a resource for quantum teleportation irrespective of the strength of decoherence and weak measurements. Surprisingly, when it comes to the fidelity of quantum teleportation, our states outperform many other mixed entangled two-qubit states. For this, we first calculate H(ϱ) for ϱ, such that

and hence,

Figure 17 clearly indicates that the teleportation fidelity of the proposed state is always greater than 2/3. Therefore, this class of states are useful resources for quantum teleportation irrespective of the values of the noise parameter for the whole range of weak measurement parameter η.

Figure 17:

Teleportation fidelity of the proposed class of states ϱ as a function of the weak measurement strength η at different values of decoherence parameter γ.

Further, we compare the efficiency of our states as resources for teleportation fidelity with other existing bipartite mixed states. We first consider the two-qubit mixed Werner state [59], i.e.

where p stands for the probability, I stands for the identity matrix representing a white noise and |ψ⁺〉 represents a maximally entangled Bell state, given by

The teleportation fidelity of Werner state using (39) is

Similarly, for Horodecki’s state [26], namely,

where a stands for the state parameter, teleportation fidelity can be evaluated as

We further consider another important class of two-qubit mixed states [60], termed as maximally entangled mixed states (MEMS), given by

ρMEMS=(z(δ)00δ201−2z(δ)000000δ200z(δ))

where

with δ, a state parameter, denoting the concurrence of ρ_MEMS.

One can calculate the optimal teleportation fidelity for ρ_MEMS, such that

A more general class of MEMS was proposed by Wei et al. [61] as a mixture of maximally entangled Bell state |Φ₊〉 and mixed diagonal state. Therefore, the general form of MEMS is given by

ρMEMSG=(q+λ200λ20s0000t0λ200r+λ2)

where q, r, s, t and λ are non-negative real state parameters such that (q+r+λ+s+t)=1. The teleportation fidelity for ρMEMSG is given by

From (49), the optimal teleportation fidelity of ρMEMSG can be obtained by considering s=0, t=0 such that we get

Figure 18 compares the efficiency of the proposed state in this article with the two-qubit Werner state, Horodecki state, ρ_MEMS and ρMEMSG for quantum teleportation. It clearly shows that the state proposed here is always a better resource in comparison to the Werner, Horodecki and ρ_MEMS states. In the case of ρMEMSG, our state proves to be a better resource for teleportation under weak decoherence; however, for strong decoherence, either our state or ρMEMSG can be considered as a preferred resource depending on the values of the state parameters. Moreover, for s≠0 or t≠0, the proposed state will always be a better resource for teleportation compared to ρMEMSG.

Figure 18:

Comparison of the usefulness of proposed class with other existing bipartite entangled mixed states.

4.2 Fully Entangled Fraction

Our analysis in the last subsection suggests that the proposed state is always a better resource in comparison to other established two-qubit mixed state for quantum teleportation protocol. In this subsection, we extend our analysis to the FEF f_ent, which is an entanglement witness and can be defined as

where the maximum is taken over all maximally entangled two-qubit states |ϕ〉 [25], [62]. Entanglement witnesses facilitates the experimental detection of entanglement and exist as a result of the Hahn-Banach theorem [26], [27], [63]. Moreover, FEF is considered as an emerging tool in describing many practical quantum information processing protocols [64], [65], [66], [67], [68], [69], [70], [71], [72]. For quantum teleportation, Horodecki et al. [25] have shown that a shared bipartite entangled state is useful for teleportation iff fent>12, where the relation between FEF and the optimal teleportation fidelity F_max is given by

As teleportation fidelity of the proposed class of states is always greater than 2/3, the FEF of the proposed class of states is always greater than 1/2. Figure 19 indicates the same by showing the effect of the weak measurement strength on the FEF of the proposed state at three different values of the amplitude-damping parameter.

Figure 19:

FEF as a function of weak measurement strength η at different values of decoherence parameter γ.

We further consider three different witnesses to measure entanglement and correlations in the proposed class of states, namely, the modified or rescaled version of FEF [73], nonlinear entropic measure [74] and Horodecki’s measure M(ρ) [26]. The modified FEF detects a larger set of entangled states in comparison to the other two measures. For this, Bartkiewicz et al. proposed an efficient and realistic experimental procedure based on entanglement swapping to detect entanglement using the modified FEF defined as

where F^W<0 corresponds to separable states, and the maximum value of F^W=1 corresponds to the maximally entangled states.

The form of the nonlinear entropic entanglement witness [74] is given by

where the value of E^W varies from 0 for separable states to 1 for the maximally entangled two-qubit states. In addition, for quantifying nonlocal correlations in two-qubit states, one may define Horodecki’s measure M(ρ) [26] as

which is greater than 0 if a two-qubit state violates the Bell-CHSH inequality and attains the maximum value 1 for the maximally entangled state. M(ρ) is directly related to the degree of Bell-CHSH inequality violation, such that B′(ρ)=max[0, MW] [75], [76].

Figure 20 shows that the proposed class of states are always entangled for the complete range of weak measurement strength η for γ=0.6, thereby, highlighting the importance of the proposed class of states for quantum information processing protocols. However, the linear entropic witness E^W and Bell nonlocality measure M^W can detect the entanglement and nonlocality in the proposed class of states for ηEW≥max{0, (1−0.4534γ)}, and ηMW≥max{0, (1−0.2428γ)}, respectively.

Figure 20:

Estimating the entanglement of proposed class of states using the Bell nonlocality measure M^W, nonlinear entanglement witness E^W and the rescaled FEF F^W as a function of weak measurement strength η, considering γ=0.6.

We further compare the FEF of the proposed class of states for a given degree of nonlocality B′(ϱ) with the randomly generated two-qubit states [76].

For example, Figure 21 numerically estimates the FEF of the proposed class of states, pure states and 10⁶ randomly generated two-qubit states. It shows that the proposed class of states have higher FEF than the pure states and a large set of mixed two-qubit states for a given nonlocality. Here, we have only considered the states lying between our states and the pure states. In Table 1, we present characteristic points ϱ^k (where k=1, …, 6) for different values of η and γ.

Figure 21:

Estimation of FEF for a given nonlocality for proposed class of states, pure states and for randomly generated two-qubit states (green area) and the coordinates of points ϱ^k (where k=1, …, 6).

4.3 Dense Coding

Superdense coding is one of the simplest application of quantum information processing [1]. The usefulness of any shared entangled resource in dense coding is measured in terms of channel capacity, i.e. the maximum number of classical bits transmitted from a sender to a receiver using the shared resource [28] where the channel capacity using a bipartite entangled state ρ^AB shared between Alice and Bob is given by

where D_A is the dimension of Alice’s subsystem, S(ρ^B) is the von-Neumann entropy of Bob’s subsystem ρ^B and S(ρ^AB) is the von-Neumann entropy of the entangled state ρ^AB.

Figure 22 suggests that the channel capacity of the proposed class is always greater than 1 for weak decoherence and increases with an increase in the value of the weak measurement strength. However, for strong decoherence, the channel capacity only exceeds the classical channel capacity for large values of the weak measurement strength. Therefore, our states are useful resources for superdense coding even at high decoherence for a certain range of η. Furthermore, Figure 23 compares the efficiency of the proposed class with the Werner states, Horodecki states and ρ_MEMS for superdense coding in terms of channel capacity. It clearly shows that our states are better resources for superdense coding in comparison to the Werner, Horodecki and ρ_MEMS states.

Figure 22:

Channel capacity of superdense coding of the proposed class ϱ as a function of the weak measurement strength η at a different value of decoherence parameter γ.

Figure 23:

Comparison of the channel capacity of the proposed class with other bipartite entangled mixed states.