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arXiv:2308.14913 · New Journal of Physics

Generalization of Gisin’s theorem to quantum fields

Konrad Schlichtholz, Marcin Markiewicz

Published Feb 1, 2024• 3 citations

Abstract

Abstract We generalize Gisin’s theorem on the relation between the entanglement of pure states and Bell non-classicality to the case of mode entanglement of separated groups of modes of quantum fields extending the theorem to cover also states with undefined particle number. We show that any pure state of the field which contains entanglement between two groups of separated modes violates some Clauser–Horne (CH) inequality. In order to construct the observables leading to a violation in the first step, we show an isomorphism between the Fock space built from a single-particle space involving two separated groups of modes and a tensor product of two abstract separable Hilbert spaces spanned by formal monomials of creation operators. In the second step, we perform a Schmidt decomposition of a given entangled state mapped to this tensor product space and then we map back the obtained Schmidt decomposition to the original Fock space of the system under consideration. Such obtained Schmidt decomposition in Fock space allows for construction of observables leading to a violation of the CH inequality. We also show that our generalization of Gisin’s theorem holds for the case of states on non-separable Hilbert spaces, which physically represent states with actually infinite number of particles. Such states emerge, for example, in the discussion of quantum phase transitions. Finally, we discuss the experimental feasibility of constructed Bell test and provide a necessary condition for realizability of this test within the realm of passive linear optics.

DOI: 10.1088/1367-2630/ad2821

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Abstract We generalize Gisin’s theorem on the relation between the entanglement of pure states and Bell non-classicality to the case of mode entanglement of separated groups of modes of quantum fields extending the theorem to cover also states with undefined particle number. We show that any pure state of the field which contains entanglement between two groups of separated modes violates some Clauser–Horne (CH) inequality. In order to construct the observables leading to a violation in the first step, we show an isomorphism between the Fock space built from a single-particle space involving two separated groups of modes and a tensor product of two abstract separable Hilbert spaces spanned by formal monomials of creation operators. In the second step, we perform a Schmidt decomposition of a given entangled state mapped to this tensor product space and then we map back the obtained Schmidt decomposition to the original Fock space of the system under consideration. Such obtained Schmidt decomposition in Fock space allows for construction of observables leading to a violation of the CH inequality. We also show that our generalization of Gisin’s theorem holds for the case of states on non-separable Hilbert spaces, which physically represent states with actually infinite number of particles. Such states emerge, for example, in the discussion of quantum phase transitions. Finally, we discuss the experimental feasibility of constructed Bell test and provide a necessary condition for realizability of this test within the realm of passive linear optics.

On classical Lie groups, which act by means of a unitary representation on finite dimensional Hilbert spaces H, we identify two classes of tensor field constructions. First, as pull-back tensor fields of order two from modified Hermitian tensor fields, constructed on Hilbert spaces by means of the property of having the vertical distributions of the C_0-principal bundle H_0 over the projective Hilbert space P(H) in the kernel. And second, directly constructed on the Lie group, as left-invariant representation-dependent operator-valued tensor fields (LIROVTs) of arbitrary order being evaluated on a quantum state. Within the NP-hard problem of deciding whether a given state in a n-level bi-partite quantum system is entangled or separable (Gurvits, 2003), we show that both tensor field constructions admit a geometric approach to this problem, which evades the traditional ambiguity on defining metrical structures on the convex set of mixed states. In particular by considering manifolds associated to orbits passing through a selected state when acted upon by the local unitary group U(n)xU(n) of Schmidt coefficient decomposition inducing transformations, we find the following results: In the case of pure states we show that Schmidt-equivalence classes which are Lagrangian submanifolds define maximal entangled states. This implies a stronger statement as the one proposed by Bengtsson (2007). Moreover, Riemannian pull-back tensor fields split on orbits of separable states and provide a quantitative characterization of entanglement which recover the entanglement measure proposed by Schlienz and Mahler (1995). In the case of mixed states we highlight a relation between LIROVTs of order two and a class of computable separability criteria based on the Bloch-representation (de Vicente, 2007).

Entanglement between two separate systems is a necessary resource to violate a Bell inequality in a test of local realism. We demonstrate that to overcome the Bell bound, this correlation must be accompanied by the entanglement between the constituent particles. This happens whenever a super-selection rule prohibits coherences between states with different total number of particles and thus imposes a constraint on feasible local operations in each sub-system. We show that the necessary entanglement between the particles might solely result from their indistinguishability. We also give an example of both mode and particle-entangled pure state, which does not violate any Bell inequality. Our result reveals a fundamental relation between the non-locality and the particle entanglement.

In this Rapid Communication, we show analytically that all pure entangled states of two

d

-dimensional systems (qudits) violate the Collins-Gisin-Linden-Masser-Popescu (CGLMP) inequality. This property was pointed out by Gisin in the qubit case and then generalized by Gisin-Peres and Popescu-Rohrlich to the qudit case based on the Clauser-Horne-Shimony-Holt (CHSH) inequality. We report the Gisin's theorem for two qudits by making use of the CGLMP inequality.

The characterization of entanglement is a fundamental issue for Quantum Information Theory. But the definition of entanglement depends on the notion of locality, and thus on the tensor product structure of the state space of the composite system. This notion is affected by the presence of superselection rules that restrict the accessible Hilbert space to a direct sum of subspaces. Indistinguishability of particles imposes one such restriction, namely to totally symmetric or totally antisymmetric states. The entanglement can in this case be defined with respect to partitions of modes in the second quantization formalism. For fermionic systems the Fock space of m modes is isomorphic to the space of m qubits, but the action of creation and annihilation operators is not local, due to their anticommutation. Conservation of the parity of fermion number imposes another relevant superselection rule. It requires that local physical observables commute with the local parity operator. Taking into account the considerations above, it is possible to define the set of separable states or equivalently the concept of entanglement for fermionic systems in a number of ways. Here we analyze systematically these possibilities and the relation among the various sets of separable states. We also discuss the behavior of the different classes when taking several copies of the state, as well as the characterization of the sets in terms of the usual criteria regarding the tensor product.