Optimal GHZ Paradox for Three Qubits

Quatum nonlocality as a valuable resource is of vital importance in quantum information processing. The characterization of the resource has been extensively investigated mainly for pure states, while relatively less is know for mixed states. Here we prove the existence of the optimal GHZ paradox by using a novel and simple method to extract an optimal state that can saturate the tradeoff relation between quantum nonlocality and the state purity. In this paradox, the logical inequality which is formulated by the GHZ-typed event probabilities can be violated maximally by the optimal state for any fixed amount of purity (or mixedness). Moreover, the optimal state can be described as a standard GHZ state suffering flipped color noise. The maximal amount of noise that the optimal state can resist is 50%. We suggest our result to be a step toward deeper understanding of the role played by the AVN proof of quantum nonlocality as a useful physical resource.

In this work, we present the optimal three-qubit GHZ paradox in the sense that an optimal mixed state can be found such that the logical inequality, formulated by the GHZ-typed event probabilities in Ref. 22, can be maximally violated for any fixed state mixedness.

Results
The optimal GHZ paradox. To investigate the optimal GHZ paradox of a three-qubit system, we start with the original case for the standard GHZ state where |0〉 and |1〉 are the eigenstates of the Pauli matrix σ z associated to the eigenvalues + 1 and − 1, respectively. Consider a set of four mutually commutative observables x , and x y y where σ 1x is defined as the Pauli matrix σ x measured on the 1-st qubit (similarly for the others), and state (1) is the common eigenstate of these four operators, with the eigenvalues being + 1, − 1, − 1, − 1, respectively.
However, as shown in Ref. 4, a contradiction arises if one tries to interpret the quantum result with local hidden variable (LHV) models, in which each local observable has two definite values, + 1 and − 1, even before the measurements. Specifically, we denote the supposedly definite values of σ 1x , σ 2y , ··· as v 1x , v 2y , ···, then a product of the last three observables, according to LHV models, yields From the point of view of experiments, environment-induced noise is generally unavoidable, and the efficiency of generating three-qubit entangled states in the laboratory is usually below 90%. Hence it is important and highly nontrivial to take account of the case of mixed states when studying the GHZ paradox. Ghirardi and Marinatto 23 demonstrated that the GHZ proof of nonlocality exists for a mixed state ρ if the following inequality is violated, where q i is defined as the event probability for each observable mentioned above that happens with certainty: The degree of violation of (2) exhibits the degree of nonlocality. Thus one may ask: What is the optimal state that violates (2) maximally so as to show the largest degree of nonlocality? To address the problem we use the notion of linear entropy 26 , which is a measure of state mixedness and computed as where the fraction is for normalization, with d = 2 N and N being the number of qubits. Put more precisely, the problem now becomes: Which state, for a fixed value of its linear entropy, can achieve the maximal violation of (2)? We usually need to solve a optimization problem by evaluating all states in the Hilbert space. The computation complexity increases very rapidly as the dimension of Hilbert space increases. Here we focus on three qubits and present a simple and rigorous method to obtain the optimal state, which reads , and subscript DM indicates that it is the form of physical density matrix giving maximal entropy for a given amount of violation.
The rest of the paper will be contributed to a proof of (8). But before the proof, we would like to give some discussions on the physical aspect of the state. Let us rewrite the optimal state (8) as where p rst 's are real coefficients, σ 0 denotes the identity matrix, and p 000 = 1 for consistency. Of course, p rst should be subject to some constraints: (i) ρ should be positive semi-definite, and (ii) the trace of ρ should be unity. The linear entropy of state (11) equals the sum of square of each coefficient, i.e., It is evident that the state that gives the maximal violation of (2) for a fixed linear entropy is equivalent to the state that gives the maximal linear entropy for a fixed quantum violation. For the given state of the form (11), the four event probabilities defined by Eq. (3-6) can be obtained as except the fact that it is not a density matrix. As we know, a real density matrix is positive semi-definite and its trace is unity. According to the sufficient and necessary condition of positive semi-definite matrix and similar discussion above, it is easily and rigorously to obtain the form of optimal positive semi-definite matrix that has the maximal entropy for a given amount of violation f 1 , which can be expressed as where one has to set ρ 11 = ρ 88 = f 1 . Compared with positive semi-definite matrix, density matrix has one more restrict, that is 1 Obviously, only when f 1 = 1/2, Eq. (21) represents a real physical density matrix, which means the optimal density matrix can reach that of positive semi-definite matrix Eq. (21). In this case it is nothing but the maximally entangled pure state which given the maximal violation of Eq. (18). However, when f 1 < 1/2, the optimal positive semi-definite matrix is not a density matrix. To obtain the form of the optimal density matrix, the diagonal matrix elements should have nonzero terms and the sum of them must be 1, hence, in computing the linear entropy we note that That is, due to the geometric-mean inequality, the maximal linear entropy can be obtained when ρ 11 = ρ 88 = f 1 and ρ 22 , ρ 33 , ··· ,ρ 77 are equal to each other, as is precisely the case of (8). Hence state (8) is indeed the optimal state that has the largest degree of nonlocality.

Discussion
In conclusion, we have proved the optimal GHZ paradox by finding the hull of quantum states that saturate the trade-off relation between the linear entropy and the quantum violation of (2). The optimal state can be considered as the standard GHZ state suffering flipped color noise, and we have shown that the exhibition of the GHZ paradox for the optimal state depends on the amount of the noise: the stronger the noise is, the less nonlocality the optimal state has. When the amount of noise is over 50%, the optimal state does not bear the nonlocality that violates inequality (2). The method we use in the present paper provides a particularly new perspective to understand the GHZ paradox for mixed states, and our results may have potential applications in quantum information processing.