Linearity of quantum probability measure and Hardy’s model

We re-examine d = 4 hidden-variables model for a system of two spin-1/2 particles in view of the concrete model of Hardy, who analyzed the criterion of entanglement without referring to inequality. The basis of our analysis is the linearity of the probability measure related to the Born probability interpretation, which excludes noncontextual hidden-variables model in d≥3. To be specific, we note the inconsistency of the noncontextual hidden-variables model in d = 4 with the linearity of the quantum mechanical probability measure in the sense 〈ψ|a⋅σ ⊗b ⋅σ|ψ〉+ 〈ψ|a ⋅σ ⊗b′ ⋅σ|ψ〉 = 〈ψ|a⋅σ⊗(b + b′)⋅σ|ψ〉 for noncollinear b and b′. It is then shown that Hardy's model in d = 4 does not lead to a unique mathematical expression in the demonstration of the discrepancy of local realism (hidden-variables model) with entanglement and thus his proof is incomplete. We identify the origin of this nonuniqueness with the nonuniqueness of translating quantum mechanical expressions into expressions in hidden-variables model, which results from the failure of the above linearity of the probability measure. In contrast, if the linearity of the probability measure is strictly imposed, which tantamounts to asking that the noncontextual hidden-variables model in d = 4 gives the Clauser–Horne–Shimony–Holt (CHSH) inequality |〈B〉|≤2 uniquely, it is shown that the hidden-variables model can describe only separable quantum mechanical states; this conclusion is in perfect agreement with the so-called Gisin's theorem which states that |〈B〉|≤2 implies separable states.We re-examine d = 4 hidden-variables model for a system of two spin-1/2 particles in view of the concrete model of Hardy, who analyzed the criterion of entanglement without referring to inequality. The basis of our analysis is the linearity of the probability measure related to the Born probability interpretation, which excludes noncontextual hidden-variables model in d≥3. To be specific, we note the inconsistency of the noncontextual hidden-variables model in d = 4 with the linearity of the quantum mechanical probability measure in the sense 〈ψ|a⋅σ ⊗b ⋅σ|ψ〉+ 〈ψ|a ⋅σ ⊗b′ ⋅σ|ψ〉 = 〈ψ|a⋅σ⊗(b + b′)⋅σ|ψ〉 for noncollinear b and b′. It is then shown that Hardy's model in d = 4 does not lead to a unique mathematical expression in the demonstration of the discrepancy of local realism (hidden-variables model) with entanglement and thus his proof is incomplete. We identify the origin of this nonuniqueness with the nonuniqueness of translating quantum mechanical expressions into expressions in hidden-variables model, which results from the failure of the above linearity of the probability measure. In contrast, if the linearity of the probability measure is strictly imposed, which tantamounts to asking that the noncontextual hidden-variables model in d = 4 gives the Clauser–Horne–Shimony–Holt (CHSH) inequality |〈B〉|≤2 uniquely, it is shown that the hidden-variables model can describe only separable quantum mechanical states; this conclusion is in perfect agreement with the so-called Gisin's theorem which states that |〈B〉|≤2 implies separable states.

arXiv:1401.0064International Journal of Modern Physics AJan 20, 20142 citations

Does CHSH Inequality Test the Model of Local Hidden Variables?

K. Fujikawa

It is pointed out that the local hidden-variable model of Bell and Clauser-Horne-Shimony-Holt (CHSH) gives |≪B >| ≤2√2 or |≪B >| ≤2 for the quantum CHSH operator B = a·σ ⊗(b + b′) ·σ + a′ ·σ ⊗ (b - b′) ·σ depending on two different ways of evaluation, when it is applied to a d = 4 system of two spin-1/2 particles. This is due to the failure of linearity, and it shows that the conventional CHSH inequality |≪B >| ≤2 does not provide a reliable test of the d = 4 local non-contextual hidden-variable model. To achieve |≪B >| ≤2 uniquely, one needs to impose a linearity requirement on the hidden-variable model, which in turn adds a von Neumann-type stricture. It is then shown that the local model is converted to a factored product of two non-contextual d = 2 hidden-variable models. This factored product implies pure separable quantum states and satisfies |≪B >| ≤2, but no more a proper hidden-variable model in d = 4. The conventional CHSH inequality |≪B >| ≤2 thus characterizes the pure separable quantum mechanical states but does not test the model of local hidden variables in d = 4, to be consistent with Gleason's theorem which excludes non-contextual models in d = 4. This observation is also consistent with an application of the CHSH inequality to quantum cryptography by Ekert, which is based on mixed separable states without referring to hidden variables.It is pointed out that the local hidden-variable model of Bell and Clauser-Horne-Shimony-Holt (CHSH) gives |≪B >| ≤2√2 or |≪B >| ≤2 for the quantum CHSH operator B = a·σ ⊗(b + b′) ·σ + a′ ·σ ⊗ (b - b′) ·σ depending on two different ways of evaluation, when it is applied to a d = 4 system of two spin-1/2 particles. This is due to the failure of linearity, and it shows that the conventional CHSH inequality |≪B >| ≤2 does not provide a reliable test of the d = 4 local non-contextual hidden-variable model. To achieve |≪B >| ≤2 uniquely, one needs to impose a linearity requirement on the hidden-variable model, which in turn adds a von Neumann-type stricture. It is then shown that the local model is converted to a factored product of two non-contextual d = 2 hidden-variable models. This factored product implies pure separable quantum states and satisfies |≪B >| ≤2, but no more a proper hidden-variable model in d = 4. The conventional CHSH inequality |≪B >| ≤2 thus characterizes the pure separable quantum mechanical states but does not test the model of local hidden variables in d = 4, to be consistent with Gleason's theorem which excludes non-contextual models in d = 4. This observation is also consistent with an application of the CHSH inequality to quantum cryptography by Ekert, which is based on mixed separable states without referring to hidden variables.

arXiv:1204.3437Progress of Theoretical PhysicsJun 1, 20126 citations

Classical-hidden-variable description for entanglement dynamics of two-qubit pure states

L. S. Silveira, R. M. Angelo

A hidden-variable model is explicitly constructed by use of a Liouvillian description for the dynamics of two coupled spin-1/2 particles. In this model, the underlying Hamiltonian trajectories play the role of deterministic hidden variables, whereas the shape of the initial probability distribution figures as a hidden variable that regulates the capacity of the model in producing correlations. We show that even though the model can very well describe the short-time entanglement dynamics of initially separated pure states, it is incapable of violating the Clauser-Horne-Shimony-Holt inequality. Our work suggests that, if one takes the reluctance of a given quantum resource to be emulated by a local-hidden-variable model as a signature of its nonclassicality degree, then one can conclude that entanglement and nonlocality are nonequivalent even in the context of two-qubit pure states.A hidden-variable model is explicitly constructed by use of a Liouvillian description for the dynamics of two coupled spin-1/2 particles. In this model, the underlying Hamiltonian trajectories play the role of deterministic hidden variables, whereas the shape of the initial probability distribution figures as a hidden variable that regulates the capacity of the model in producing correlations. We show that even though the model can very well describe the short-time entanglement dynamics of initially separated pure states, it is incapable of violating the Clauser-Horne-Shimony-Holt inequality. Our work suggests that, if one takes the reluctance of a given quantum resource to be emulated by a local-hidden-variable model as a signature of its nonclassicality degree, then one can conclude that entanglement and nonlocality are nonequivalent even in the context of two-qubit pure states.

arXiv:1704.02000Physical Review AJun 5, 20177 citations

A hidden-variables version of Gisin's theorem

Kazuo Fujikawa, Koichiro Umetsu

It is generally assumed that {\em local realism} represented by a noncontextual and local hidden-variables model in $d=4$ such as the one used by Bell always gives rise to CHSH inequality $|\langle B\rangle|\leq 2$ . On the other hand, the contraposition of Gisin's theorem states that the inequality $|\langle B\rangle|\leq 2$ for arbitrary parameters implies (pure) separable quantum states. The fact that local realism can describe only pure separable quantum states is naturally established in hidden-variables models, and it is quantified by $G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0$ for any two projection operators $P({\bf a})$ and $P({\bf b})$ . The test of local realism by the deviation of $G({\bf a},{\bf b})$ from $G({\bf a},{\bf b})=0$ is shown to be very efficient using the past experimental setup of Aspect and his collaborators in 1981.It is generally assumed that {\em local realism} represented by a noncontextual and local hidden-variables model in $d=4$ such as the one used by Bell always gives rise to CHSH inequality $|\langle B\rangle|\leq 2$. On the other hand, the contraposition of Gisin's theorem states that the inequality $|\langle B\rangle|\leq 2$ for arbitrary parameters implies (pure) separable quantum states. The fact that local realism can describe only pure separable quantum states is naturally established in hidden-variables models, and it is quantified by $G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0$ for any two projection operators $P({\bf a})$ and $P({\bf b})$. The test of local realism by the deviation of $G({\bf a},{\bf b})$ from $G({\bf a},{\bf b})=0$ is shown to be very efficient using the past experimental setup of Aspect and his collaborators in 1981.

arXiv:1410.1702Oct 7, 20140 citations

Quantum contextuality in the Mermin-Peres square: A hidden-variable perspective

Brian R. La Cour

The question of a hidden variable interpretation of quantum contextuality in the Mermin-Peres square is considered. The Kochen-Specker theorem implies that quantum mechanics may be interpreted as a contextual hidden-variable theory. It is shown that such a hidden-variable description can be viewed as either contextual in the random variables mapping hidden states to observable outcomes or in the probability measure on the hidden state space. The latter view suggests that this apparent contextuality may be interpreted as a simple consequence of measurement disturbance, wherein the initial hidden state is altered through interaction with the measuring device, thereby giving rise to a possibly different final hidden-variable state from which the measurement outcome is obtained. In light of this observation, a less restrictive and, arguably, more reasonable definition of noncontextuality is suggested. To prove that such a description is possible, an explicit and, in this sense, noncontextual hidden-variable model is constructed, which reproduces all quantum theoretic predictions for the Mermin-Peres square. A critical analysis of some recent and proposed experimental tests of contextuality is also provided. Although the discussion is restricted to a four-dimensional Hilbert space, the approach and conclusions are expected to generalize to any Hilbert space.The question of a hidden variable interpretation of quantum contextuality in the Mermin-Peres square is considered. The Kochen-Specker theorem implies that quantum mechanics may be interpreted as a contextual hidden-variable theory. It is shown that such a hidden-variable description can be viewed as either contextual in the random variables mapping hidden states to observable outcomes or in the probability measure on the hidden state space. The latter view suggests that this apparent contextuality may be interpreted as a simple consequence of measurement disturbance, wherein the initial hidden state is altered through interaction with the measuring device, thereby giving rise to a possibly different final hidden-variable state from which the measurement outcome is obtained. In light of this observation, a less restrictive and, arguably, more reasonable definition of noncontextuality is suggested. To prove that such a description is possible, an explicit and, in this sense, noncontextual hidden-variable model is constructed, which reproduces all quantum theoretic predictions for the Mermin-Peres square. A critical analysis of some recent and proposed experimental tests of contextuality is also provided. Although the discussion is restricted to a four-dimensional Hilbert space, the approach and conclusions are expected to generalize to any Hilbert space.

arXiv:2105.00940Physical Review AJan 5, 200923 citations

Nonlocality without inequalities for almost all entangled states for two particles

Sheldon Goldstein

We provide a streamlined proof of Hardy's theorem, that almost every entangled state for a pair of quantum particles admits a ``proof of nonlocality'' without inequalities. Moreover, our analysis covers a larger class of observables. Thus we also strengthen Hardy's assertion that the argument fails for maximally entangled states, such as the singlet state. At the same time, we formulate the argument in such a manner that the relations which must be satisfied for local hidden variables are entirely deterministic, making no reference whatsoever to probability, let alone probablistic inequalities.We provide a streamlined proof of Hardy's theorem, that almost every entangled state for a pair of quantum particles admits a ``proof of nonlocality'' without inequalities. Moreover, our analysis covers a larger class of observables. Thus we also strengthen Hardy's assertion that the argument fails for maximally entangled states, such as the singlet state. At the same time, we formulate the argument in such a manner that the relations which must be satisfied for local hidden variables are entirely deterministic, making no reference whatsoever to probability, let alone probablistic inequalities.

Physical Review LettersMar 28, 1994129 citations

It is generally assumed that {\em local realism} represented by a noncontextual and local hidden-variables model in

d=4

such as the one used by Bell always gives rise to CHSH inequality

|\langle B\rangle|\leq 2

. On the other hand, the contraposition of Gisin's theorem states that the inequality

|\langle B\rangle|\leq 2

for arbitrary parameters implies (pure) separable quantum states. The fact that local realism can describe only pure separable quantum states is naturally established in hidden-variables models, and it is quantified by

G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0

for any two projection operators

P({\bf a})

and

P({\bf b})

. The test of local realism by the deviation of

G({\bf a},{\bf b})

from

G({\bf a},{\bf b})=0

is shown to be very efficient using the past experimental setup of Aspect and his collaborators in 1981.