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Clauser-Horne-Shimony-Holt關係式的研究

梁凱翔

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

In this thesis, a short review on the Clauser-Horne-Shimony-Holt (CHSH) Inequality is given. The maximization of the expectation value of the CHSH operator for pure bipartite qubit systems is derived explicitly. The derivation reduces the maximization of hCHSHi to an eigenvalue problem. We show that hCHSHimax = 2p 1 + 2 where 1and 2 are the greatest two eigenvalues of a certain matrix with eigenvalues {1,D,D} where D 4 | det c |2 1 and c is the coefficient of the expansion of the states, i.e. | i = Pi,j cij |iji. We discover the relation between the maximum of hCHSHi and the directions for spin measurement ˆr and ˆq, and these values can be continuously deformed even when the state is not a Bell State. We then specialize to pure classically correlated density matrices and also Bell State and arrive at results consistent with previous conclusions, i.e. For classically correlated density matrices hCHSHi 2 , For pure quantum states 2 hCHSHi 2p2. We also briefly discuss the case of Werner density matrix.

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In this thesis, a short review on the Clauser-Horne-Shimony-Holt (CHSH) Inequality is given.
The maximization of the expectation value of the CHSH operator for pure bipartite qubit
systems is derived explicitly.
The derivation reduces the maximization of hCHSHi to an eigenvalue problem. We show that
hCHSHimax = 2p 1 + 2
where 1and 2 are the greatest two eigenvalues of a certain matrix with eigenvalues {1,D,D}
where D 4 | det c |2 1 and c is the coefficient of the expansion of the states, i.e. | i =
Pi,j cij |iji. We discover the relation between the maximum of hCHSHi and the directions for spin measurement ˆr and ˆq, and these values can be continuously deformed even when the state is not a Bell State.
We then specialize to pure classically correlated density matrices and also Bell State and arrive at results consistent with previous conclusions, i.e.
For classically correlated density matrices hCHSHi 2 ,
For pure quantum states 2 hCHSHi 2p2.
We also briefly discuss the case of Werner density matrix.
Abstract Violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality by a bipartite quantum state is now used in many quantum applications. However, the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-s measurements is still known only for s = 1 / 2 . In the present article, for an arbitrary state of two spin-s qudits, each of dimension d = 2 s + 1 ⩾ 2 , we introduce the notion of the spin-s correlation matrix, which has dimension 3 × 3 for all s ⩾ 1 2 ; establish its relation to the general correlation ( d 2 − 1 ) × ( d 2 − 1 ) matrix of this state within the generalized Pauli representation and derive in terms of the spin-s correlation matrix the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-s measurements in this state. Specifying this general expression for the two-qudit Greenberger–Horne–Zeilinger state, the nonlocal two-qudit Werner state, and some nonseparable pure two-qudit states, we find that, under local Alice and Bob high-spin ( s ⩾ 1 ) measurements in each of these nonseparable states, including the maximally entangled one, the CHSH inequality is not violated. Moreover, unlike the case of spin- 1 / 2 measurements, where each pure nonseparable two-qubit state violates the CHSH inequality and the maximal value of its CHSH expectation increases monotonically with a growth of its entanglement, the situation under high-spin measurements is quite different—for a pure two-qudit state with a higher degree of entanglement, the maximal value of the CHSH expectation turns out to be less than for a pure two-qudit state with lower entanglement and even for a separable one.
The correlation of projections of the momentum operators of two particles is used to derive a quantum inequality for continuous variables, which must be satisfied by any bipartite system in a pure state. This inequality resembles a Clauser–Horne–Shimony–Holt (CHSH)-type inequality except for additional terms related to the imaginary component of the weak value of the momentum, which normally remains concealed in the usual quantum description but turns out to be of relevance for entangled states. Our results shed new light on the link between noncommutativity, entanglement and nonlocality of the quantum description.
The Bell--Clauser-Horne-Shimony-Holt (BCHSH) inequality (in terms of correlation functions) of two qutrits is studied in detail by employing tritter measurements. A uniform formula for the maximum value of this inequality for tritter measurements is obtained. Based on this formula, we show that nonmaximally entangled states violate the BCHSH inequality more strongly than the maximally entangled one. This result is consistent with what was obtained by Ac\'{\i}n et al. [Phys. Rev. A 65,65,65, 052325 (2002)] using the Bell--Clauser-Horne inequality (in terms of probabilities).
We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to arbitrarily high-dimensional systems. Based on this generalization, we construct the general CHSH inequality for bipartite quantum systems of arbitrarily high dimensionality, which takes the same simple form as CHSH inequality for two dimensions. This inequality is optimal in the same sense as the CHSH inequality for two-dimensional systems, namely, the maximal amount by which the inequality is violated consists of the maximal resistance to noise. We also discuss the physical meaning and general definition of the correlation functions. Furthermore, by giving another specific set of the correlation functions with the same physical meaning, we realize the inequality presented by Collins et al. [Phys. Rev. Lett. 88, 040404 (2002)]].
We evaluate the maximal Clauser-Horne-Shimony-Holt (CHSH) violation for a generic (typically mixed) qubit-qudit state, obtaining easily computable expressions in arbitrary qudit dimension. This represents the optimal (2-2-2) Bell nonlocality for this kind of systems. The work generalizes the well-known Horodeckis result for a qubit-qubit setup. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues, namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. In addition, we show the impossibility of improving the amount of CHSH violation by embedding the qudit in a Hilbert space of larger dimension. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.
The Clauser-Horne-Shimony-Holt inequality (CHSH) is one of the most popular and well-studied witnesses of Bell's theorem, separating classical from quantum correlations. In this work, for every $d\ensuremath{\ge}2$, we present a generalization of the CHSH inequality with the property that maximal violation is achieved uniquely by the maximally entangled state of local dimension ddd. This provides an avenue for device-independent certification of maximally entangled states of arbitrary local dimension.