In this chapter we introduce the principle of local discriminability, which stipulates the possibility of discriminating states of composite systems via local measurements on the component systems. The phenomenon of entanglement has relevant consequences on the accessibility of information in composite systems and on the complexity of characterization of transformations: local discriminability brings dramatic simplifications to the structure of the sets of states and transformations, reconciling the holism of entangled states with the reductionist scientific approach. In order to illustrate the structural relevance of the principle, we will briefly review some OPTs that do not satisfy it, contrasting the features of such theories with those of quantum and classical probabilistic theories, where the principle holds. Entanglement and Holism Holism is a widely known feature of quantum theory, related to the existence of entangled states. In quantum theory one has pure entangled states, whose marginal states are completely mixed. This situation corresponds to having maximal knowledge of the whole with minimal knowledge of the parts. The main manifestation of the holism of entanglement is the impossibility of reconciling the statistics of joint measurements on entangled states with local realism . Local realism is the assumption that any measurement on a system is just the reading of a pre-existing quantity encoded on the system. In other words, we cannot explain the statistics of measurements on the parts of a composite system as a manifestation of our ignorance of locally pre-existing values, as in classical OPT. Does entanglement make information about composite systems locally unaccessible? This would pose severe practical issues on the possibility of acquiring information about physical systems. For example, suppose that two distant telescopes observe a source of entangled photons from a stellar emission. Holism would imply that some features of the emission process could never be detected, unless we are able to make the entangled photons interact in a joint measurement. On the other hand, if a reductionist approach is possible, local observations of the two photons at the sites of the two telescopes would be enough to characterize the emitting source. Thanks to the local discriminability principle, this is actually the case with entangled quantum sources. Another consequence of holism is the fact that transformations generally cannot be characterized by local experiments, namely by applying the unknown transformation only to its input system.
In this chapter we introduce the principle of local discriminability, which stipulates the possibility of discriminating states of composite systems via local measurements on the component systems. The phenomenon of entanglement has relevant consequences on the accessibility of information in composite systems and on the complexity of characterization of transformations: local discriminability brings dramatic simplifications to the structure of the sets of states and transformations, reconciling the holism of entangled states with the reductionist scientific approach. In order to illustrate the structural relevance of the principle, we will briefly review some OPTs that do not satisfy it, contrasting the features of such theories with those of quantum and classical probabilistic theories, where the principle holds. Entanglement and Holism Holism is a widely known feature of quantum theory, related to the existence of entangled states. In quantum theory one has pure entangled states, whose marginal states are completely mixed. This situation corresponds to having maximal knowledge of the whole with minimal knowledge of the parts. The main manifestation of the holism of entanglement is the impossibility of reconciling the statistics of joint measurements on entangled states with local realism . Local realism is the assumption that any measurement on a system is just the reading of a pre-existing quantity encoded on the system. In other words, we cannot explain the statistics of measurements on the parts of a composite system as a manifestation of our ignorance of locally pre-existing values, as in classical OPT. Does entanglement make information about composite systems locally unaccessible? This would pose severe practical issues on the possibility of acquiring information about physical systems. For example, suppose that two distant telescopes observe a source of entangled photons from a stellar emission. Holism would imply that some features of the emission process could never be detected, unless we are able to make the entangled photons interact in a joint measurement. On the other hand, if a reductionist approach is possible, local observations of the two photons at the sites of the two telescopes would be enough to characterize the emitting source. Thanks to the local discriminability principle, this is actually the case with entangled quantum sources. Another consequence of holism is the fact that transformations generally cannot be characterized by local experiments, namely by applying the unknown transformation only to its input system.
An operational probabilistic theory where all systems are classical, and all pure states of composite systems are entangled, is constructed. The theory is endowed with a rule for composing an arbitrary number of systems, and with a nontrivial set of transformations. Hence, we demonstrate that the presence of entanglement is independent of the existence of incompatible measurements. We then study a variety of phenomena occurring in the theory, some of them contradicting both classical and quantum theories, including cloning, entanglement swapping, dense coding, additivity of classical capacities, nonmonogamous entanglement, hypersignaling. We also prove the existence, in the theory, of a universal processor. The theory is causal and satisfies the no-restriction hypothesis. At the same time, it violates a number of information-theoretic principles enjoyed by quantum theory, most notably, local discriminability, purity of parallel composition of states, and purification. Moreover, we introduce an exhaustive procedure to construct generic operational probabilistic theories, and a sufficient set of conditions to verify their consistency. In addition, we prove a characterization theorem for the parallel composition rules of arbitrary theories, and specialize it to the case of bilocal-tomographic theories. We conclude pointing out some open problems. In particular, on the basis of the fact that every separable state of the theory is a statistical mixture of entangled states, we formulate a no-go conjecture for the existence of a local-realistic ontological model.
Quantum nonlocality has different manifestations that, in general, are revealed by local measurements of the parts of a composite system. In this paper, we study nonlocality arising from a set of orthogonal states that cannot be perfectly distinguished by local operations and classical communication (LOCC). Such a set is deemed nonlocal, for a joint measurement on the whole system is necessary for perfect discrimination of the states with certainty. On the other hand, a set of orthogonal states that can be perfectly distinguished by LOCC is believed to be devoid of nonlocal properties. Here, we show that there exist orthogonal sets that are locally distinguishable but without local redundancy (i.e., they become nonorthogonal on discarding one or more subsystems) whose nonlocality can be activated by local measurements. In particular, a state chosen from such a set can be locally converted, with certainty, into another state, the identity of which can now only be ascertained by global measurement and no longer by LOCC. In other words, a locally distinguishable set without local redundancy may be locally converted into a locally indistinguishable set with certainty. We also suggest an application, namely, local hiding of information, that allows us to locally hide locally available information without losing any part. Once hidden, the information in its entirety can only be retrieved using entanglement.
We investigate operational probabilistic theories where the pure states of every system are the vertices of a simplex. A special case of such theories is that of classical theories, i.e., simplicial theories whose pure states are jointly perfectly discriminable. The usual classical theory satisfies also local discriminability. However, simplicial theories---including the classical ones---can violate local discriminability, thus admitting entangled states. First, we prove sufficient conditions for the presence of entangled states in arbitrary probabilistic theories. Then we prove that simplicial theories are necessarily causal, and this represents a no-go theorem for conceiving noncausal classical theories. We then provide necessary and sufficient conditions for simplicial theories to exhibit entanglement and classify their system-composition rules. We conclude by proving that, in simplicial theories, an operational formulation of the superposition principle cannot be satisfied, and that---under the hypothesis of n-local discriminability---no mixed state admits a purification. Our results hold also in the general case where the sets of states fail to be convex.
A local measurement on one member of an entangled composite system is said to change the distant member's state instantaneously. Taken without qualification, this is misleading. A selective measurement changes the conditional state assigned to the remote subsystem once a local outcome is specified; a non-selective measurement, where that outcome is ignored, leaves the remote reduced density operator unchanged. This distinction reconciles quantum state update with operational no-signalling and provides the setting for Einstein–Podolsky–Rosen steering. This review develops measurement using density operators, completely positive instruments, Kraus representations, partial traces, conditional states, and steering assemblages. It derives the invariance of remote marginal statistics under local trace-preserving operations and contrasts this with Bell-local factorization. It examines the hierarchy among entanglement, steering, and Bell nonlocality, joint measurability, remote state preparation, and one-sided device-independent protocols. The article considers spacelike-separated measurements, microcausality in relativistic quantum theory, and ontological readings of selective state update offered by Everettian, Bohmian, collapse, and relational approaches. Decoherence is treated as the process through which local outcomes become records, while quantum Darwinism accounts for their redundant local accessibility. It highlights gaps in literature, and concludes that entanglement permits nonclassical conditional correlations, but local measurement supplies no controllable superluminal signal.
Nonlocality is the most characteristic feature of quantum mechanics. John Bell, in his seminal 1964 work, proved that local-realism imposes a bound on the correlations among the measurement statistics of distant observers. Surpassing this bound rules out local-realistic description of microscopic phenomena, establishing the presence of nonlocal correlation. To manifest nonlocality, it requires, in the simplest scenario, two measurements performed randomly by each of two distant observers. In this work, we propose a novel framework where three measurements, two on Alice's side and one on Bob's side, suffice to reveal quantum nonlocality and hence does not require all-out randomness in measurement choice. Our method relies on a very naive operational task in quantum information theory, namely, the minimal error state discrimination. As a practical implication this method constitutes an economical entanglement detection scheme, which uses a less number of entangled states compared to all such existing schemes. Moreover, the method applies to class of generalized probability theories containing quantum theory as a special example.
Abstract These two accompanying papers are concerned with two mode entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. Entanglement is a key quantum feature of composite systems in which the probabilities for joint measurements on the composite sub-systems are no longer determined from measurement probabilities on the separate sub-systems. There are many aspects of entanglement that can be studied. This two-part review focuses on the meaning of entanglement, the quantum paradoxes associated with entangled states, and the important tests that allow an experimentalist to determine whether a quantum state—in particular, one for massive bosons is entangled. An overall outcome of the review is to distinguish criteria (and hence experiments) for entanglement that fully utilize the symmetrization principle and the super-selection rules that can be applied to bosonic massive particles. In the first paper (I), the background is given for the meaning of entanglement in the context of systems of identical particles. For such systems, the requirement is that the relevant quantum density operators must satisfy the symmetrization principle and that global and local super-selection rules prohibit states in which there are coherences between differing particle numbers. The justification for these requirements is fully discussed. In the second quantization approach that is used, both the system and the sub-systems are modes (or sets of modes) rather than particles, particles being associated with different occupancies of the modes. The definition of entangled states is based on first defining the non-entangled states—after specifying which modes constitute the sub-systems. This work mainly focuses on the two mode entanglement for massive bosons, but is put in the context of tests of local hidden variable theories, where one may not be able to make the above restrictions. The review provides the detailed arguments necessary for the conclusions of a recent paper, where the question of how to rigorously demonstrate the entanglement of a two-mode Bose–Einstein condensate (BEC) has been examined. In the accompanying review paper (II), we consider spin squeezing and other tests for entanglement that have been proposed for two-mode bosonic systems. We apply the approach of review (I) to determine which tests, and which modifications of the tests, are useful for detecting entanglement in massive bosonic (BEC), as opposed to photonic, systems. Several new inequalities are derived, a theory for the required two-mode interferometry is presented, and key experiments to date are analyzed.