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arXiv:1107.5071 · International Journal of Quantum Information

ENTANGLED IDENTICAL PARTICLES AND NOISE

Giuseppe Argentieri, Fabio Benatti, Roberto Floreanini B, Ugo Marzolino

Published Oct 1, 2011• 25 citations
arXivHTMLPDF

Abstract

For systems of identical Bosons, it is necessary to reformulate the notions of separability and entanglement in algebraic terms shifting the emphasis from the particle aspect of first quantization to the mode description typical of second quantization. Within this new framework, we show that, unlike for systems consisting of distinguishable qubits, negativity is an exhaustive bipartite entanglement witness for systems with fixed number of Bosons; further, we investigate the impact of dephasing noise in relation to the use of such many-body Bosonic systems in metrological applications.

DOI
10.1142/s0219749911008210

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For systems of identical Bosons, it is necessary to reformulate the notions of separability and entanglement in algebraic terms shifting the emphasis from the particle aspect of first quantization to the mode description typical of second quantization. Within this new framework, we show that, unlike for systems consisting of distinguishable qubits, negativity is an exhaustive bipartite entanglement witness for systems with fixed number of Bosons; further, we investigate the impact of dephasing noise in relation to the use of such many-body Bosonic systems in metrological applications.
Unlike for bipartite states consisting of distinguishable particles, in the case of identical parties the notion of entanglement is still under debate. In the following, we review two different approaches to the entanglement of systems consisting of two bosons or fermions; the first approach is based on the particle aspect typical of first quantization and identifies separable pure states as those that allow to assign two orthogonal single particle vector states to both parties. The second approach makes full use of the mode aspect of second quantization whereby separability can be formulated as absence of nonlocal correlation among two different sets of modes. While the first approach applies to pure states only, the second one is more general and characterizes generic entangled states. In the following, we shall show that the mode-based approach indeed contains the particle-based one.
We study the role of entanglement and non-locality in quantum protocols that make use of systems of identical particles. Unlike in the case of distinguishable particles, the notions of entanglement and non-locality for systems whose constituents cannot be distinguished and singly addressed are still debated. We clarify why the only approach that avoids incongruities and paradoxes is the one based on the second quantization formalism, whereby it is the entanglement of the modes that can be populated by the particles that really matters and not the particles themselves. Indeed, by means of a metrological and of a teleportation protocol, we show that inconsistencies arise in formulations that force entanglement and non-locality to be properties of the identical particles rather than of the modes they can occupy. The reason resides in the fact that orthogonal modes can always be addressed while identical particles cannot.
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.
The robustness properties of bipartite entanglement in systems of NNN bosons distributed in MMM different modes are analyzed using a definition of separability based on commuting algebras of observables, a natural choice when dealing with identical particles. Within this framework, expressions for the robustness and generalized robustness of entanglement can be explicitly given for large classes of boson states: Their entanglement content turns out to be, in general, much more stable than that of distinguishable particles states. Using these results, the geometrical structure of the space of NNN-boson states can be explicitly addressed.
The existence of fundamentally identical particles represents a foundational distinction between classical and quantum mechanics.Because of their exchange symmetry, identical particles can appear to be entangled-another uniquely quantum phenomenon with far-reaching practical implications.However, a long-standing debate has questioned whether identical particle entanglement is physical or merely a mathematical artifact.In this work, we provide such particle entanglement with a consistent theoretical description as a quantum resource in processes frequently encountered in optical and cold atomic systems.This leads to a plethora of applications of immediate practical impact.On the one hand, we show that the metrological advantage for estimating phase shifts in systems of identical bosons amounts to a measure of their particle entanglement, with a clear-cut operational meaning.On the other hand, we demonstrate in general terms that particle entanglement is the property resulting in directly usable mode entanglement when distributed to separated parties, with particle conservation laws in play.Application of our tools to an experimental implementation with Bose-Einstein condensates leads to the first quantitative estimation of identical particle entanglement.Further connections are revealed between particle entanglement and other resources such as optical nonclassicality and quantum coherence.Overall, this work marks a resolutive step in the ongoing debate by delivering a unifying conceptual and practical understanding of entanglement between identical particles.
These two accompanying papers treat 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 where the probabilities for joint measurements on the composite sub-systems are no longer determined from measurement probabilities on the separate sub-systems. We focus on the meaning of entanglement, the quantum paradoxes associated with entangled states, and important tests that allow an experimentalist to determine whether a quantum state - in particular, one for massive bosons is entangled. Our tests for entanglement fully utilise the symmetrisation principle and super-selection rules for bosonic massive particles. These papers provides detailed arguments necessary for the conclusions of a recent paper which presented results for rigorously demonstrating the entanglement of a two-mode Bose-Einstein condensate (BEC). In the first paper (I), we discuss the meaning of entanglement for systems of identical particles. For such systems, the relevant quantum density operators must satisfy the symmetrisation principle and global and local super-selection rules prohibiting states in which there are coherences between differing particle numbers. We fully justify these requirements. In the second quantisation approach, both the system and sub-systems are modes (or sets of modes) rather than particles, particles being associated with different mode occupancies. In the accompanying review paper (II), we consider spin squeezing and other tests for entanglement for two-mode bosonic systems. Starting from paper (I) we determine which tests are useful for detecting entanglement in massive bosonic (BEC), as opposed to photonic, systems. Several new inequalities are derived, and recent key experiments are analysed.