Measurement incompatibility and Schr\"odinger-EPR steering in a class of probabilistic theories

Steering is one of the most counter intuitive non classical features of bipartite quantum system, first noticed by Schr\"odinger at the early days of quantum theory. On the other hand measurement incompatibility is another non classical feature of quantum theory, initially pointed out by N. Bohr. Recently the authors of Refs. [\href{http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.160402}{Phys. Rev. Lett. {\bf 113}, 160402 (2014)}] and [\href{http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.160403}{Phys. Rev. Lett. {\bf 113}, 160403 (2014)}] have investigated the relation between these two distinct non classical features. They have shown that a set of measurements is not jointly measurable (i.e. incompatible) if and only if they can be used for demonstrating Schr\"odinger-Einstein-Podolsky-Rosen steering. The concept of steering has been generalized for more general abstract tensor product theories rather than just Hilbert space quantum mechanics. In this article we discuss that the notion of measurement incompatibility can be extended for general probability theories. Further we show that the connection between steering and measurement incompatibility holds in a border class of tensor product theories rather than just quantum theory.Comment: Accepted in Journal of Mathematical Physics (close to accepted version). arXiv admin note: text overlap with arXiv:1402.6562, arXiv:1406.6976, arXiv:1311.1462, arXiv:0707.0620 by other author

Several foundational and information theoretic implications of Bell's theorem

In 1935, Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics. EPR also postulated the existence of more fundamental theory where physical reality of any system would be completely describe by the variables/states of that fundamental theory. This variable is commonly called hidden variable and the theory is called hidden variable theory (HVT). In 1964, John Bell proposed an empirically verifiable criterion to test for the existence of these HVTs. He derived an inequality, which must be satisfied by any theory that fulfill the conditions of locality and reality} He also showed that quantum mechanics, as it violates this inequality, is incompatible with any local-realistic theory. Later it has been shown that Bell's inequality can be derived from different set of assumptions and it also find applications in useful information theoretic protocols. In this review we will discuss various foundational as well as information theoretic implications of Bell's inequality. We will also discuss about some restricted nonlocal feature of quantum nonlocality and elaborate the role of Uncertainty principle and Complementarity principle in explaining this feature.Comment: 27 pages; This article is a modified version of the tutorial lecture delivered by G. Kar at IPQI, February 17-28, 2014, Institute of Physics, Bhubaneswar, Indi

Biased Non-Causal Game

The standard formulation of quantum theory assumes that events are ordered is a background global causal structure. Recently in Ref.[$\href{http://www.nature.com/ncomms/journal/v3/n10/full/ncomms2076.html}{Nat. Commun. {\bf3}, 1092 (2012)}$], the authors have developed a new formalism, namely, the \emph{process matrix} formalism, which is locally in agreement with quantum physics but assumes no global causal order. They have further shown that there exist \emph{non-causal} correlations originating from \emph{inseparable} process matrices that violate a \emph{causal inequality} (CI) derived under the assumption that events are ordered with respect to some global causal relation. This CI can be understood as a guessing game, where two separate parties, say Alice and Bob, generate random bits (say input bit) in their respective local laboratories. Bob generates another random bit (say decision bit) which determines their goal: whether Alice has to guess Bob's bit or vice-verse. Here we study this causal game but with biased bits and derive a biased causal inequality (BCI). We then study the possibility of violation of this BCI by inseparable process matrices. Interestingly, we show that there exist \emph{inseparable} qubit process matrices that can be used to violate the BCI for an arbitrary bias in the decision bit. In such scenario, we also derive the maximal violation of the BCI under local operations involving traceless binary observables. However, for biased input bits, we find that there is a threshold bias beyond which no valid qubit process matrix can be used to violate the causal inequality under \emph{measurement-repreparation} type operation.Comment: Revised manuscript; comments are welcom

Lack of measurement independence can simulate quantum correlation even when signaling cannot

In Bell scenario, any nonlocal correlation, shared between two spatially separated parties, can be modeled deterministically either by allowing communications between the two parties or by restricting their free will in choosing the measurement settings. Recently, Bell scenario has been generalized into `semi-quantum' scenario where external quantum inputs are provided to the parties. We show that in `semi-quantum' scenario, entangled states produce correlations whose deterministic explanation is possible only if measurement independence is reduced. Thus in simulating quantum correlation `semi-quantum' scenario reveals a qualitative distinction between signaling and measurement dependence which is absent in Bell scenario. We further show that such distinction is not observed in `steering game' scenario, a special case of `semi-quantum' scenario.Comment: 5 pages (double column); accepted in Phys. Rev.

Wigner-Yanase skew information and entanglement generation in quantum measurement

The first step of quantum measurement procedure is known as \emph{premeasurement}, during when correlation between measuring system and measurement apparatus is established. One compelling non-classical correlation is entanglement, a useful resource for various quantum information theoretic protocols. Quantifying the amount of entanglement in the premeasurement state, therefore, seeks importance from practical ground and this is the central issue of the present paper. Interestingly, for a two-label quantum system we obtain that the amount of entanglement, measured in term of \emph{negativity}, generated in premeasurement process is actually quantified by two factors: \emph{skew information} between system's initial state and the measurement direction, which quantifies the amount of information on the values of observables not commuting with the conserved quantity of the system, and \emph{mixedness parameter} of the system's initial state.Comment: 6 pages, 3 figure

Study of nonlocal correlations in macroscopic measurement scenario

Nonlocality is one of the main characteristic features of quantum systems involving more than one spatially separated subsystems. It is manifested theoretically as well as experimentally through violation of some local realistic inequality. On the other hand, classical behavior of all physical phenomena in the macroscopic limit gives a general intuition that any physical theory for describing microscopic phenomena should resemble classical physics in the macroscopic regime-- the so-called macro-realism. In the 2-2-2 scenario (two parties, each performing two measurements, each measurement with two outcomes), contemplating all the no-signaling correlations, we characterize which of them would exhibit classical (local-realistic) behavior in the macroscopic limit. Interestingly, we find correlations which at single copy level violate the Bell-Clauser-Horne-Shimony-Holt inequality by an amount less than optimal quantum violation (i.e. the Cirel'son bound $2\sqrt{2}$), but in the macroscopic limit gives rise to a value which is higher than $2\sqrt{2}$. Such correlations are therefore not considered as physical. Our study thus provides a sufficient criterion to identify some of unphysical correlations.Comment: Accepted in Physical Review A, Close to accepted versio

Implications of Coupling in Quantum Thermodynamic Machines

We study coupled quantum systems as the working media of thermodynamic machines. Under a suitable phase-space transformation, the coupled systems can be expressed as a composition of independent subsystems. We find that for the coupled systems, the figures of merit, that is the efficiency for engine and the coefficient of performance for refrigerator, are bounded (both from above and from below) by the corresponding figures of merit of the independent subsystems. We also show that the optimum work extractable from a coupled system is upper bounded by the optimum work obtained from the uncoupled system, thereby showing that the quantum correlations do not help in optimal work extraction. Further, we study two explicit examples, coupled spin-$1/2$ systems and coupled quantum oscillators with analogous interactions. Interestingly, for particular kind of interactions, the efficiency of the coupled oscillators outperforms that of the coupled spin-$1/2$ systems when they work as heat engines. However, for the same interaction, the coefficient of performance behaves in a reverse manner, while the systems work as the refrigerator. Thus the same coupling can cause opposite effects in the figures of merit of heat engine and refrigerator.Comment: 19 pages, 8 figure

Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition

In a multipartite scenario quantum entanglement manifests its most dramatic form when the state is genuinely entangled. Such a state is more beneficial for information theoretic applications if it contains distillable entanglement in every bipartition. It is, therefore, of significant operational interest to identify subspaces of multipartite quantum systems that contain such properties apriori. In this letter, we introduce the notion of unextendible biseparable bases (UBB) that provides an adequate method to construct genuinely entangled subspaces (GES). We provide an explicit construction of two types of UBBs -- party symmetric and party asymmetric -- for every $3$-{\it qudit} quantum system, with local dimension d\ge 3. Further, we show that the GES resulting from the symmetric construction is indeed a {\it bidistillable} subspace, i.e., all the states supported on it contain distillable entanglement across every bipartition.Comment: Close to published versio

Uncertainty principle certifies genuine source of intrinsic randomness

The Born's rule introduces intrinsic randomness to the outcomes of a measurement performed on a quantum mechanical system. But, if the system is prepared in the eigenstate of an observable then the measurement outcome of that observable is completely predictable and hence there is no intrinsic randomness. On the other hand, if two incompatible observables are measured (either sequentially on a particle or simultaneously on two identical copies of the particle) then uncertainty principle guarantees intrinsic randomness in the subsequent outcomes independent of the preparation state of the system. In this article we show that this is true not only in quantum mechanics but for any no-signaling probabilistic theory. Also the minimum amount of intrinsic randomness that can be guaranteed for arbitrarily prepared state of the system is quantified by the amount of (un)certainty.Comment: Accepted in Quantum Info. Proces

Exclusivity principle and unphysicality of Garg-Mermin correlation

The question concerning the physical realizability of a probability distribution is of quite importance in Quantum foundations. Specker first pointed out that this question cannot be answered from Kolmogorov's axioms alone. Lately, this observation of Specker has motivated simple principles (exclusivity principle/ local orthogonality principle) that can explain quantum limit regarding the possible sets of experimental probabilities in various nonlocality and contextuality experiments. We study Specker's observation in the simplest scenario involving three inputs each with two outputs. Then using only linear constraints imposed on joint probabilities by this principle, we reveal unphysical nature of Garg-Mermin (GM) correlation. Interestingly, GM correlation was proposed to falsify the following suggestion by Fine: if the inequalities of Clauser and Horne (CH) holds, then there exists a deterministic local hidden-variable model for a spin-1/2 correlation experiment of the Einstein-Podolsky-Rosen type, even when more than two observables are involved on each side. Our result establishes that, unlike in the CH scenario, the local orthogonality principle at single copy level is not equivalent to the no-signaling condition in the GM scenario.Comment: New references are adde