Generalizing the well-known spin-squeezing inequalities, we study the relation between squeezing of collective N -particle s u ( d ) operators and many-body entanglement geometry in multi-particle systems. For that aim, we define the set of pseudo-separable states, which are mixtures of products of single-particle states that lie in the ( d 2 − 1 ) -dimensional Bloch sphere but are not necessarily positive semidefinite. We obtain a set of necessary conditions for states of N qudits to be of the above form. Any state that violates these conditions is entangled. We also define a corresponding s u ( d ) -squeezing parameter that can be used to detect entanglement in large particle ensembles. Geometrically, this set of conditions defines a convex set of points in the space of first and second moments of the collective N -particle s u ( d ) operators. We prove that, in the limit N ≫ 1 , such set is filled by pseudo-separable states, while any state corresponding to a point outside of this set is necessarily entangled. We also study states that are detected by these inequalities: We show that states with a bosonic symmetry are detected if and only if the two-body reduced state violates the positive partial transpose (PPT) criterion. On the other hand, highly mixed states states close to the s u ( d ) singlet are detected which have a separable two-body reduced state and are also PPT with respect to all possible bipartitions. We also provide numerical examples of thermal equilibrium states that are detected by our set of inequalities, comparing the spin-squeezing inequalities with the s u ( 3 ) -squeezing inequalities.
Generalizing the well-known spin-squeezing inequalities, we study the relation between squeezing of collective N -particle s u ( d ) operators and many-body entanglement geometry in multi-particle systems. For that aim, we define the set of pseudo-separable states, which are mixtures of products of single-particle states that lie in the ( d 2 − 1 ) -dimensional Bloch sphere but are not necessarily positive semidefinite. We obtain a set of necessary conditions for states of N qudits to be of the above form. Any state that violates these conditions is entangled. We also define a corresponding s u ( d ) -squeezing parameter that can be used to detect entanglement in large particle ensembles. Geometrically, this set of conditions defines a convex set of points in the space of first and second moments of the collective N -particle s u ( d ) operators. We prove that, in the limit N ≫ 1 , such set is filled by pseudo-separable states, while any state corresponding to a point outside of this set is necessarily entangled. We also study states that are detected by these inequalities: We show that states with a bosonic symmetry are detected if and only if the two-body reduced state violates the positive partial transpose (PPT) criterion. On the other hand, highly mixed states states close to the s u ( d ) singlet are detected which have a separable two-body reduced state and are also PPT with respect to all possible bipartitions. We also provide numerical examples of thermal equilibrium states that are detected by our set of inequalities, comparing the spin-squeezing inequalities with the s u ( 3 ) -squeezing inequalities.
A complete set of generalized spin-squeezing inequalities is derived for an ensemble of particles with an arbitrary spin. Our conditions are formulated with the first and second moments of the collective angular momentum coordinates. A method for mapping the spin-squeezing inequalities for spin-21 particles to entanglement conditions for spin-j particles is also presented. We apply our mapping to obtain a generalization of the original spin-squeezing inequality to higher spins. We show that, for large particle numbers, a spin-squeezing parameter for entanglement detection based on one of our inequalities is strictly stronger than the original spin-squeezing parameter defined in S\o{}rensen et al. [Nature (London) 409, 63 (2001)]. We present a coordinate system independent form of our inequalities that contains, besides the correlation and covariance tensors of the collective angular momentum operators, the nematic tensor appearing in the theory of spin nematics. Finally, we discuss how to measure the quantities appearing in our inequalities in experiments.
Entanglement detection is crucial and a necessity in the context of quantum information and quantum computation. This important task has proved to be quite hard for quantum systems of dimensions higher than 2×3, in which case, there exists well established necessary and sufficient conditions like Peres-Horodecki criterion.To tackle this challenge for bipartite systems, we introduce a mathematical framework to reduce the problem to entanglement in a two qubit system. This is done by mapping each subsystem locally into a qubit without increasing entanglement. The mapping is expressed in terms of expectation values of three arbitrary operators in the original state. We give necessary and sufficient conditions for such mapping to be valid from physical point of view, providing thence a versatile tool for dimension reduction in various applications.Our main use of this formalism is as a gate way to derive entanglement criteria for bipartite or multi-partite systemas based on existing ones derived for qubit systems. By mapping each subsystem locally into a qubit, applying entanglement criteria known for qubits on the resulting state automatically gives us entanglement criteria in terms of the chosen operators used to implement the mapping.For the multi-partite case, we focus on spin squeezing inequalities for qubits to derive entanglement criteria for general systems. However, when applying our formalism to this case, an interesting situation arises where one is able to obtain coherent superposition of multi-partite qubit states with different particle number. Hence, to derive better entanglement criteria, we had to consider quantum and/or classical fluctuationsthat may be exhibited by the particle number operator. We derive generalized form of Sørensen-Mølmer’s criterion and of spin squeezing inequalities for fluctuating particle number in terms of arbitrary collective operators. We applied our results to study entanglement in a system of ultra-cold Chromium atoms with spin s = 3 trapped in a bi-dimensional optical lattice incollaboration with Quantum Dipolar Gazes team in Laboratoire de Physique de Laser at Paris Nord 13 university. We showed, in a numerical simulation, that our generalized inequalities are able to detect entanglement in their system using collective operators. Moreover, we show that such observables can be measured using available techniques.
What is the relation between spin squeezing and entanglement? To clarify this, we derive the full set of generalized-spin-squeezing inequalities for the detection of entanglement. These are inequalities for the mean values and variances of the collective angular-momentum components Jk. They can be used for the experimental detection of entanglement in a system of spin-21 particles in which the spins cannot be individually addressed. We present various sets of inequalities that can detect all entangled states that can be detected based on the knowledge of (i) the mean values and variances of Jk in three orthogonal directions, or (ii) the variances of Jk in three orthogonal directions, or (iii) the mean values of Jk2 in three orthogonal directions, or (iv) the mean values and variances of Jk in arbitrary directions. We compare our inequalities to known spin-squeezing entanglement criteria and discuss to which extent spin squeezing is related to entanglement in the reduced two-qubit states. Finally, we apply our criteria for the detection of entanglement in spin models, showing that they can be used to detect bound entanglement in these systems.
Abstract These two accompanying papers are concerned with entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. The main focus is on two mode entanglement, but multi-mode entanglement is also considered. The bosons may be atoms or molecules as in cold quantum gases. The previous paper I dealt with the general features of quantum entanglement and its specific definition in the case of systems of identical bosons. Entanglement is a property shared between two (or more) quantum sub-systems. In defining entanglement for systems of identical massive particles, it was concluded that the single particle states or modes are the most appropriate choice for sub-systems that are distinguishable, that the general quantum states must comply both with the symmetrization principle and the super-selection rules (SSR) that forbid quantum superpositions of states with differing total particle number (global SSR compliance). Further, it was concluded that (in the separable states) quantum superpositions of sub-system states with differing sub-system particle number (local SSR compliance) also do not occur. The present paper II determines possible tests for entanglement based on the treatment of entanglement set out in paper I. Several inequalities involving variances and mean values of operators have been previously proposed as tests for entanglement between two sub-systems. These inequalities generally involve mode annihilation and creation operators and include the inequalities that define spin squeezing. In this paper, spin squeezing criteria for two mode systems are examined, and spin squeezing is also considered for principle spin operator components where the covariance matrix is diagonal. The proof, which is based on our SSR compliant approach shows that the presence of spin squeezing in any one of the spin components requires entanglement of the relevant pair of modes. A simple Bloch vector test for entanglement is also derived. Thus we show that spin squeezing becomes a rigorous test for entanglement in a system of massive bosons, when viewed as a test for entanglement between two modes. In addition, other previously proposed tests for entanglement involving spin operators are considered, including those based on the sum of the variances for two spin components. All of the tests are still valid when the present concept of entanglement based on the symmetrization and SSR criteria is applied. These tests also apply in cases of multi-mode entanglement, though with restrictions in the case of sub-systems each consisting of pairs of modes. Tests involving quantum correlation functions are also considered and for global SSR compliant states these are shown to be equivalent to tests involving spin operators. A new weak correlation test is derived for entanglement based on local SSR compliance for separable states, complementing the stronger correlation test obtained previously when this is ignored. The Bloch vector test is equivalent to one case of this weak correlation test. Quadrature squeezing for single modes is also examined but not found to yield a useful entanglement test, whereas two mode quadrature squeezing proves to be a valid entanglement test, though not as useful as the Bloch vector test. The various entanglement tests are considered for well-known entangled states, such as binomial states, relative phase eigenstates and NOON states—sometimes the new tests are satisfied while than those obtained in other papers are not. The present paper II then outlines the theory for a simple two mode interferometer showing that such an interferometer can be used to measure the mean values and covariance matrix for the spin operators involved in entanglement tests for the two mode bosonic system. The treatment is also generalized to cover multi-mode interferometry. The interferometer involves a pulsed classical field characterized by a phase variable and an area variable defined by the time integral of the field amplitude, and leads to a coupling between the two modes. For simplicity the center frequency was chosen to be resonant with the inter-mode transition frequency. Measuring the mean and variance of the population difference between the two modes for the output state of the interferometer for various choices of interferometer variables is shown to enable the mean values and covariance matrix for the spin operators for the input quantum state of the two mode system to be determined. The paper concludes with a discussion of several key experimental papers on spin squeezing.
Particle number fluctuations, no matter how small, are present in experimental setups. One should rigorously take these fluctuations into account, especially for entanglement detection. In this context, we generalize the spin-squeezing inequalities introduced by G. T\'oth et al. [Phys. Rev. Lett. 99, 250405 (2007).]. These new inequalities are fulfilled by all separable states even when the number of particles is not constant and may present quantum fluctuations. These inequalities are useful for detecting entanglement in many-body systems when the superselection rule does not apply or when only a subspace of the total system Hilbert space is considered. We also define general dichotomic observables for which we obtain a coordinate-independent form of the generalized spin-squeezing inequalities. We give an example where our generalized coordinate-independent spin-squeezing inequalities present a clear advantage over the original ones.
We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors.
This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions.
We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.