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arXiv:1405.1109

Indistinguishability of states for composite quantum system

Chengjun Wu, Junhui Li, Bin Luo, Hong Mei Guo

Published May 6, 2014• 0 citations
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Abstract

We consider the states of composite system from quantum probabilistic amplitude superposition, and define the local indistinguishability (LID) and nonlocal indistinguishability (NLID). For pure states, the NLID is the same as nonlocal coherence. NLID can be considered as the measure of entanglement. LID can be considered as the measure of quantum correlation. From this respect, entanglement, which must have probabilistic amplitude superposition between product basis, is explicitly different from other quantum correlation. LID and NLID are also generalized to multipartite cases. These results are useful in the exploration of the distinctive quantum feature of quantum systems.

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We consider the states of composite system from quantum probabilistic amplitude superposition, and define the local indistinguishability (LID) and nonlocal indistinguishability (NLID). For pure states, the NLID is the same as nonlocal coherence. NLID can be considered as the measure of entanglement. LID can be considered as the measure of quantum correlation. From this respect, entanglement, which must have probabilistic amplitude superposition between product basis, is explicitly different from other quantum correlation. LID and NLID are also generalized to multipartite cases. These results are useful in the exploration of the distinctive quantum feature of quantum systems.
We study different notions of quantum correlations in multipartite systems of distinguishable and indistinguishable particles. Based on the definition of quantum coherence for a single particle, we consider two possible extensions of this concept to the many-particle scenario and determine the influence of the exchange symmetry. Moreover, we characterize the relation of multiparticle coherence to the entanglement of the compound quantum system. To support our general treatment with examples, we consider the quantum correlations of a collection of qudits. The impact of local and global quantum superpositions on the different forms of quantum correlations is discussed. For differently correlated states in the bipartite and multipartite scenarios, we provide a comprehensive characterization of the various forms and origins of quantum correlations.
We analyze a family of measures of general quantum correlations for composite systems, defined in terms of the bipartite entanglement necessarily created between systems and apparatuses during local measurements. For every entanglement monotone EEE, this operational correspondence provides a different measure QE{Q}_{E}QE​ of quantum correlations. Examples of such measures are the relative entropy of quantumness, the quantum deficit, and the negativity of quantumness. In general, we prove that any so-defined quantum correlation measure is always greater than (or equal to) the corresponding entanglement between the subsystems, ${Q}_{E}\ensuremath{\ge}E$, for arbitrary states of composite quantum systems. We analyze qualitatively and quantitatively the flow of correlations in iterated measurements, showing that general quantum correlations and entanglement can never decrease along von Neumann chains, and that genuine multipartite entanglement in the initial state of the observed system always gives rise to genuine multipartite entanglement among all subsystems and all measurement apparatuses at any level in the chain. Our results provide a comprehensive framework to understand and quantify general quantum correlations in multipartite states.
It has been suggested that there may exist quantum correlations that go beyond entanglement. The existence of such correlations can be revealed by information theoretic quantities such as quantum discord, but not by the conventional measures of entanglement. We argue that a state displays quantumness, that can be of local and nonlocal origin. Information theoretic measures not only characterize the nonlocal quantumness, but also the local quantumness, such as the “local superposition”. This can be a reason, why such measures are nonzero, when there is no entanglement. We consider a generalized version of the Werner state to demonstrate the interplay of local quantumness, nonlocal quantumness and classical mixedness of a state.
Recently, based on some quantum states which are locally irreducible in all bipartitions, strong quantum nonlocality has been presented by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. However, the remaining questions are whether a set of quantum states can reveal strong quantum nonlocality when these states are locally irreducible in all tripartitions or more multipartitions and how to construct strong nonlocality of orthogonal product states in more than tripartite quantum systems. Here we present a general definition of strong quantum nonlocality based on the local irreducibility. Then, using a $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$ quantum system as an example, we show three sets of locally indistinguishable orthogonal product states to explain the general definition of strong quantum nonlocality. Furthermore, in $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$, we construct a set of strong nonlocality of orthogonal product states; these states do not form a complete basis. Our results demonstrate the phenomenon of strong nonlocality without entanglement for the multipartite quantum systems.
In this paper, we mainly study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum systems. In the Hilbert space of $3\ensuremath{\bigotimes}3$, Walgate and Hardy [Phys. Rev. Lett. 89, 147901 (2002)] presented a very simple proof for nonlocality of nine orthogonal product basis quantum states which are given by Bennett et al. [Phys. Rev. A 59, 1070 (1999)]. In the quantum system of $d\ensuremath{\bigotimes}d$, where ddd is odd, we construct d2{d}^{2}d2 orthogonal product basis quantum states and prove these states are locally indistinguishable. Then we are able to construct some locally indistinguishable product basis quantum states in the multipartite systems. All these results reveal the phenomenon of ``nonlocality without entanglement.''
Quantum nonlocality is usually associated with entangled states by their violations of Bell-type inequalities. However, even unentangled systems, whose parts may have been prepared separately, can show nonlocal properties. In particular, a set of product states is said to exhibit "quantum nonlocality without entanglement" if the states are locally indistinguishable; i.e., it is not possible to optimally distinguish the states by any sequence of local operations and classical communication. Here, we present a stronger manifestation of this kind of nonlocality in multiparty systems through the notion of local irreducibility. A set of multiparty orthogonal quantum states is defined to be locally irreducible if it is not possible to locally eliminate one or more states from the set while preserving orthogonality of the postmeasurement states. Such a set, by definition, is locally indistinguishable, but we show that the converse does not always hold. We provide the first examples of orthogonal product bases on C^{d}⊗C^{d}⊗C^{d} for d=3, 4 that are locally irreducible in all bipartitions, where the construction for d=3 achieves the minimum dimension necessary for such product states to exist. The existence of such product bases implies that local implementation of a multiparty separable measurement may require entangled resources across all bipartitions.