Bell's theorem states that quantum mechanics is not a locally causal theory. This state is often interpreted as nonlocality in quantum mechanics. Toner and Bacon [Phys. Rev. Lett. 91, 187904 (2003)] have shown that a shared random-variable theory augmented by one bit of classical communication exactly simulates the Bell correlation in a singlet state. In this paper, we show that in Toner and Bacon protocol, one of the parties (Bob) can deduce another party's (Alice) measurement outputs, if she only informs Bob of one of her own outputs. Afterwards, we suggest a nonlocal version of Toner and Bacon protocol wherein classical communications is replaced by nonlocal effects, so that Alice's measurements cause instantaneous effects on Bob's outputs. In the nonlocal version of Toner and Bacon's protocol, we get the same result again. We also demonstrate that the same approach is applicable to Svozil's protocol.
Bell's theorem states that quantum mechanics is not a locally causal theory. This state is often interpreted as nonlocality in quantum mechanics. Toner and Bacon [Phys. Rev. Lett. 91, 187904 (2003)] have shown that a shared random-variable theory augmented by one bit of classical communication exactly simulates the Bell correlation in a singlet state. In this paper, we show that in Toner and Bacon protocol, one of the parties (Bob) can deduce another party's (Alice) measurement outputs, if she only informs Bob of one of her own outputs. Afterwards, we suggest a nonlocal version of Toner and Bacon protocol wherein classical communications is replaced by nonlocal effects, so that Alice's measurements cause instantaneous effects on Bob's outputs. In the nonlocal version of Toner and Bacon's protocol, we get the same result again. We also demonstrate that the same approach is applicable to Svozil's protocol.
In all local realistic theories worked out till now, locality is considered as a basic assumption. Most people in the field consider the inconsistency between local realistic theories and quantum mechanics to be a result of non-local nature of quantum mechanics. In this paper, we derive Bell's inequality for particles with instantaneous interactions, and show that the aforementioned contradiction still exists between quantum mechanics and non-local hidden variable models. Then, we use this non-locality to obtain the GHZ theorem. In what follows, we show that Bacon and Toner's protocol, for the simulation of Bell correlation, by using local hidden variables augmented by classical communication, have some inconsistency with quantum mechanics. Our approach can answer to Brassard questions from another viewpoint, we show that if we accept that our nature obeys quantum mechanical laws, then all of quantum mechanic results cannot be simulated by realistic theories augmented by classical communication or a single instance use of non-local box.
The Bell theorem expresses that quantum mechanics is not a local-realistic theory, which is often interpreted as nonlocality of the nature. This result has led to this belief that nonlocality and entanglement are the same resources. However, this belief has been critically challenged in the literature. Here, we reexamine the relation between nonlocality and entanglement in light of the Brassard-Cleve-Tapp (BCT) model, which was originally proposed for simulating quantum correlation of Bell's states by using shared random variables augmented by classical communications. We derive a new criterion for distinguishing quantum mechanics from the BCT model through suggesting an observable event based on the perfect correlations (anti-correlations) relation. In particular, we show that in the BCT model one can obtain equal outputs for two opposite input settings with the nonzero probability 0.284. Hence, in this sense we argue that the BCT model can give rise to an unphysical result. We also show the same problem with a nonlocal version of the BCT model.
Bell's theorem implies that the outcomes of local measurements on two maximally entangled systems cannot be simulated without classical communication between the parties. The communication cost is finite for n Bell states, but it grows exponentially in n. Three simple protocols are presented that provide approximate simulations for low-dimensional entangled systems and require a linearly growing amount of communication. We have tested them by performing some simulations for a family of measurements. The maximal error is less than 1 in three dimensions and grows sublinearly with the number of entangled bits in the range numerically tested. One protocol is the multidimensional generalization of the exact Toner-Bacon [Phys. Rev. Lett. 91, 187904 (2003)] model for a single Bell state. The other two protocols are generalizations of an alternative exact model, which we derive from the Kochen-Specker [J. Math. Mech. 17, 59 (1967)] scheme for simulating single-qubit measurements. These protocols can give some indication for finding optimal one-way communication protocols that classically simulate entanglement and quantum channels. Furthermore they can be useful for deciding if a quantum communication protocol provides an advantage on classical protocols.
We consider general prepare-and-measure scenarios in which Alice can transmit qubit states to Bob, who can perform general measurements in the form of positive operator-valued measures (POVMs). We show that the statistics obtained in any such quantum protocol can be simulated by the purely classical means of shared randomness and two bits of communication. Furthermore, we prove that two bits of communication is the minimal cost of a perfect classical simulation. In addition, we apply our methods to Bell scenarios, which extends the well-known Toner and Bacon protocol. In particular, two bits of communication are enough to simulate all quantum correlations associated to arbitrary local POVMs applied to any entangled two-qubit state.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
If nonlocality is to be inferred from a violation of Bell's inequality, an important assumption is that the measurement settings are freely chosen by the observers, or alternatively, that they are random and uncorrelated with the hypothetical local variables. We demonstrate a connection between models that weaken this assumption, allowing partial correlation, and (i) models that allow classical communication between the distant parties, (ii) models that exploit the detection loophole. Even if Bob's choices are completely independent, all correlations from projective measurements on a singlet can be reproduced, with mutual information between Alice's choice and local variables less than or equal to one bit.