Md. Rajjak Gazi

Degree of complementarity determines the nonlocality in quantum mechanics

Complementarity principle is one of the central concepts in quantum mechanics which restricts joint measurement for certain observables. Of course, later development shows that joint measurement could be possible for such observables with the introduction of a certain degree of unsharpness or fuzziness in the measurement. In this paper, we show that the optimal degree of unsharpness, which guarantees the joint measurement of all possible pairs of dichotomic observables, determines the degree of nonlocality in quantum mechanics as well as in more general no-signaling theories.

Optimal free will on one side in reproducing the singlet correlation

Bell’s theorem teaches us that there are quantum correlations that cannot be simulated by just shared randomness (local hidden variable). There are some recent results which simulate the singlet correlation by using either 1 bit or a binary (no-signaling) correlation which violates Bell’s inequality maximally. But there is one more possible way to simulate quantum correlation by relaxing the condition of independency of measurement on shared randomness. Recently, Hall showed that the statistics of a singlet state can be generated by sacrificing measurement independence where underlying distribution of hidden variables depends on measurement directions of both parties (Hall 2010 Phys. Rev. Lett. 105 250404). He also proved that for any model of singlet correlation, 86% measurement independence is optimal. In this paper, we show that 59% measurement independence is optimal for simulating the singlet correlation when the underlying distribution of hidden variables depends only on the measurements of one party. We also show that a distribution corresponding to this optimal lack of free will already exists in the literature which first appeared in the context of detection efficiency loophole (Gisin and Gisin 1999 Phys. Lett. A 323–7).

A complementary relation between classical bits and randomness in local part in the simulating singlet state

Recently, simulating the statistics of the singlet state with non-quantum resources has generated much interest. The singlet state statistics can be simulated by 1 bit of classical communication without using any further non-local correlation. But, interestingly, the singlet state statistics can also be simulated with no classical cost if a non-local box is used. In the first case, the output is completely deterministic whereas in the second case, outputs are completely random. We suggest a (possibly) signaling correlation resource which successfully simulates the singlet statistics, and subsequently leads to a complementary relation between the required classical bits and randomness in the local output involved in the simulation. Our result reproduces the above two models of simulation as extreme cases. We also discuss some important features in Leggetts non-local model and the model presented by Groblacher et al.

Local simulation of singlet statistics for a restricted set of measurements

The essence of Bell’s theorem is that, in general, quantum statistics cannot be reproduced by a local hidden variable (LHV) model. This impossibility is strongly manifested when statistics collected by measuring certain local observables on a singlet state, violates the Bell inequality. In this work, we search for local POVMs with binary outcomes for which an LHV model can be constructed for a singlet state. We provide various subsets of observables for which an LHV model can be provided for singlet statistics.

Local orthogonality provides a tight upper bound for Hardy's nonlocality

The amount of nonlocality in quantum theory is limited compared to that allowed in generalized no-signaling theory [Found. Phys. 24, 379 (1994)]. This feature, for example, gets manifested in the amount of Bell inequality violation as well as in the degree of success probability of Hardys (Cabellos) nonlocality argument. Physical principles like information causality and macroscopic locality have been proposed for analyzing restricted nonlocality in quantum mechanics---viz. explaining the Cirelson bound. However, these principles are not that much successful in explaining the maximum success probability of Hardys as well as Cabellos argument in quantum theory. Here we show that, a newly proposed physical principle namely Local Orthogonality does better by providing a tighter upper bound on the success probability for Hardys nonlocality. This bound is relatively closer to the corresponding quantum value compared to the bounds achieved from other principles.

Local randomness in Hardy's correlations: implications from the information causality principle

Study of non-local correlations in terms of Hardys argument has been quite popular in quantum mechanics. Hardys non-locality argument depends on some kind of asymmetry, but a two-qubit maximally entangled state, being symmetric, does not exhibit this kind of non-locality. Here we ask the following question: can this feature be explained by some principle outside quantum mechanics? The no-signaling condition does not provide a solution. But, interestingly, the information causality principle (Pawlowski et al 2009 Nature 461 1101) offers an explanation. It shows that any generalized probability theory which gives completely random results for local dichotomic observable, cannot provide Hardys non-local correlation if it is restricted by a necessary condition for respecting the information causality principle. In fact, the applied necessary condition imposes even more restrictions on the local randomness of measured observable. Still, there are some restrictions imposed by quantum mechanics that are not reproduced from the considered information causality condition.

Classical communication and non-classical fidelity of quantum teleportation

In quantum teleportation, the role of entanglement has been much discussed. It is known that entanglement is necessary for achieving non-classical teleportation fidelity. Here we focus on the amount of classical communication that is necessary to obtain non-classical fidelity in teleportation. We quantify the amount of classical communication that is sufficient for achieving non-classical fidelity for two independent 1-bit and single 2-bits noisy classical channels. It is shown that on average 0.208 bits of classical communication is sufficient to get non-classical fidelity. We also find the necessary amount of classical communication in case of isotropic transformation. Finally we study how the amount of sufficient classical communication increases with weakening of entanglement used in the teleportation process.

Macroscopic locality with equal bias reproduces with high fidelity a quantum distribution achieving the Tsirelson's bound

Two physical principles, Macroscopic Locality (ML) and Information Causality (IC), so far, have been most successful in distinguishing quantum correlations from post-quantum correlations. However, there are also some post-quantum probability distributions which cannot be distinguished with the help of these principles. Thus, it is interesting to see whether consideration of these two principles, separately, along with some additional physically plausible constraints, can explain some interesting quantum features which are otherwise hard to reproduce. In this paper we show that, in a Bell-CHSH scenario, ML along with constraint of equal-biasness for the concerned observables, almost reproduces the quantum joint probability distribution corresponding to maximal quantum Bell violation, which is unique up to relabeling. From this example and earlier work of Cavalcanti, Salles and Scarani, we conclude that IC and ML are in-equivalent physical principles; satisfying one does not imply that the other is satisfied.

Bound on tri-partite Hardy’s nonlocality respecting all bi-partite principles

Study of certain class of tripartite correlations under a number of recently proposed bi-partite physical principles has produced important insight regarding such principles. We find a lower bound on success probability of tri-partite Hardy’s correlation respecting all bi-partite physical principles. The no-signaling principle does not reveal any gap in Hardy’s maximum success probability for bi-partite and tri-partite system, whereas quantum mechanically there is such a gap. Interestingly, we show that, Information causality principle is successful in qualitatively revealing this quantum feature.