Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1311.3855 · Physical Review A

Cauchy-Schwarz inequality and particle entanglement

Tomasz Wasak, Piotr Szańkowski, Paweł Zin, Marek Trippenbach, Jan Chwedeńczuk

Published Sep 17, 2014• 28 citations
arXivHTMLPDF

Abstract

Recently, violation of the Cauchy-Schwarz inequality by the second-order correlation function was reported in a collection of ultracold bosons [Kheruntsyan et al., Phys. Rev. Lett. 108, 260401 (2012)]. We show that the observation of this effect proves the presence of particle entanglement in a many-body system of bosons. Our derivation is based on the analogy between the coherent states of electromagnetic fields and separable states of massive particles. The presented argument applies to any quantum system of identical bosons with either a fixed or a fluctuating number of particles, provided that there is no coherence between different number states.

DOI
10.1103/physreva.90.033616

Related papers

Criteria for particle entanglement in many-body systems of bosons

Tomasz Wasak, Piotr Szańkowski, Marek Trippenbach +1 more

Basing on the analogy between the coherent states of light and separable states of $N$ bosons, we demonstrate that the violation Cauchy-Schwarz inequality for any-order correlation function signals the entanglement among the constituent particles. Rather than restricting to the correlations between the positions of particles, we consider the broadest set of measurements allowed by quantum mechanics. Our result is general -- it applies to any quantum system of bosons, even when the number of particles is not fixed, provided that there is no coherence between different number states. We also demonstrate that the compact expression for the separable state of bosons can be used to relate some known metrological quantities to the particle entanglement in a very simple way.

arXiv:1504.06189•Apr 23, 2015•0 citations

Quantum Entanglement for Systems of Identical Bosons. I General Theory

Bryan J. Dalton, John Goold, B. M. Garraway +1 more

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.

arXiv:1506.06906•Jun 23, 2015•1 citation

Quantum entanglement for systems of identical bosons: I. General features

Bryan J. Dalton, John Goold, B. M. Garraway +1 more

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.

Physica Scripta•Jan 9, 2017•36 citations

Violation of the Cauchy-Schwarz Inequality with Matter Waves

Karen V. Kheruntsyan, Jean-Christophe Jaskula, Piotr Deuar +6 more

The Cauchy-Schwarz (CS) inequality-one of the most widely used and important inequalities in mathematics-can be formulated as an upper bound to the strength of correlations between classically fluctuating quantities. Quantum-mechanical correlations can, however, exceed classical bounds. Here we realize four-wave mixing of atomic matter waves using colliding Bose-Einstein condensates, and demonstrate the violation of a multimode CS inequality for atom number correlations in opposite zones of the collision halo. The correlated atoms have large spatial separations and therefore open new opportunities for extending fundamental quantum-nonlocality tests to ensembles of massive particles.

arXiv:1204.0058•Physical Review Letters•Jun 25, 2012•94 citations

Mixed state entanglement via the Cauchy-Schwarz inequality

Nishaant Jacobus, Paul W. Brumer, Chern Chuang

The entanglement properties of mixed states are of great importance in the study of open quantum systems and quantum information science, but common entanglement criteria such as the Peres-Horodecki condition can be difficult to relate to physical properties of the system. We introduce the Cauchy-Schwarz violation (CSV) condition for entanglement, which has a simple dependence on the populations and coherences of the density operator. A sufficient condition for entanglement, it provides a direct connection to the physical characteristics of the system such as its symmetries, yielding intuition that can be used to understand and engineer highly entangled states. We illustrate the often surprising insights gained from the CSV condition by applying it to the Jaynes-Cummings model, the quantum Rabi model, and an open-system quantum Rabi model.

arXiv:2508.00334•Physical Review Research•Jul 6, 2026•0 citations

Violations of Bell Inequality, Cauchy—Schwarz Inequality and Entanglement in a Two-Mode Three-Level Atomic System

Yunxia Ping, Lie Liu, Chaomin Zhang +1 more

Violations of Bell inequality, Cauchy—Schwarz inequality and entanglement in a two-mode three-level atomic system are investigated. It is shown that there are some states, which are entangled but do not violate Bell inequality in this system. Moreover, the relations of violations of Bell inequality, Cauchy—Schwarz inequality, and entanglement are discussed in detail.

Communications in Theoretical Physics•Jul 1, 2011•1 citation
Recently, violation of the Cauchy-Schwarz inequality by the second-order correlation function was reported in a collection of ultracold bosons [Kheruntsyan et al., Phys. Rev. Lett. 108, 260401 (2012)]. We show that the observation of this effect proves the presence of particle entanglement in a many-body system of bosons. Our derivation is based on the analogy between the coherent states of electromagnetic fields and separable states of massive particles. The presented argument applies to any quantum system of identical bosons with either a fixed or a fluctuating number of particles, provided that there is no coherence between different number states.
Basing on the analogy between the coherent states of light and separable states of NNN bosons, we demonstrate that the violation Cauchy-Schwarz inequality for any-order correlation function signals the entanglement among the constituent particles. Rather than restricting to the correlations between the positions of particles, we consider the broadest set of measurements allowed by quantum mechanics. Our result is general -- it applies to any quantum system of bosons, even when the number of particles is not fixed, provided that there is no coherence between different number states. We also demonstrate that the compact expression for the separable state of bosons can be used to relate some known metrological quantities to the particle entanglement in a very simple way.
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.
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 Cauchy-Schwarz (CS) inequality-one of the most widely used and important inequalities in mathematics-can be formulated as an upper bound to the strength of correlations between classically fluctuating quantities. Quantum-mechanical correlations can, however, exceed classical bounds. Here we realize four-wave mixing of atomic matter waves using colliding Bose-Einstein condensates, and demonstrate the violation of a multimode CS inequality for atom number correlations in opposite zones of the collision halo. The correlated atoms have large spatial separations and therefore open new opportunities for extending fundamental quantum-nonlocality tests to ensembles of massive particles.
The entanglement properties of mixed states are of great importance in the study of open quantum systems and quantum information science, but common entanglement criteria such as the Peres-Horodecki condition can be difficult to relate to physical properties of the system. We introduce the Cauchy-Schwarz violation (CSV) condition for entanglement, which has a simple dependence on the populations and coherences of the density operator. A sufficient condition for entanglement, it provides a direct connection to the physical characteristics of the system such as its symmetries, yielding intuition that can be used to understand and engineer highly entangled states. We illustrate the often surprising insights gained from the CSV condition by applying it to the Jaynes-Cummings model, the quantum Rabi model, and an open-system quantum Rabi model.