Tensor Product Structures, Entanglement, and Particle Scattering (original) (raw)

CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT II: MIXED STATES

International Journal of Geometric Methods in Modern Physics, 2011

Invariant operator-valued tensor fields on Lie groups are considered. These define classical tensor fields on Lie groups by evaluating them on a quantum state. This particular construction, applied on the local unitary group U (n) × U (n), may establish a method for the identification of entanglement monotone candidates by deriving invariant functions from tensors being by construction invariant under local unitary transformations. In particular, for n = 2, we recover the purity and a concurrence related function (Wootters 1998) as a sum of inner products of symmetric and anti-symmetric parts of the considered tensor fields. Moreover, we identify a distinguished entanglement monotone candidate by using a non-linear realization of the Lie algebra of SU (2) × SU (2). The functional dependence between the latter quantity and the concurrence is illustrated for a subclass of mixed states parametrized by two variables.

Entanglement and tensor product decomposition for two fermions

Journal of Physics A: Mathematical and General, 2005

The problem of the choice of tensor product decomposition in a system of two fermions with the help of Bogoliubov transformations of creation and annihilation operators is discussed. The set of physical states of the composite system is restricted by the superselection rule forbidding the superposition of fermions and bosons. It is shown that the Wootters concurrence is not the proper entanglement measure in this case. The explicit formula for the entanglement of formation is found. This formula shows that the entanglement of a given state depends on the tensor product decomposition of a Hilbert space. It is shown that the set of separable states is narrower than in the two-qubit case. Moreover, there exist states which are separable with respect to all tensor product decompositions of the Hilbert space.

Entanglement and Its Multipartite Extensions

International Journal of Modern Physics B, 2013

The aspects of many particle systems as far as their entanglement is concerned is highlighted. To this end we briefly review the bipartite measures of entanglement and the entanglement of pairs both for systems of distinguishable and indistinguishable particles. The analysis of these quantities in macroscopic systems shows that close to quantum phase transitions, the entanglement of many particles typically dominates that of pairs. This leads to an analysis of a method to construct many-body entanglement measures. SL-invariant measures are a generalization to quantities as the concurrence, and can be obtained with a formalism containing two (actually three) orthogonal antilinear operators. The main drawback of this antilinear framework, namely to measure these quantities in the experiment, is resolved by a formula linking the antilinear formalism to an equivalent linear framework.

Canonical entanglement for two indistinguishable particles

Journal of Physics A: Mathematical and General, 2005

We determine the degree of entanglement for two indistinguishable particles based on the twoqubit tensor product structure, which is a framework for emphasizing entanglement founded on observational quantities. Our theory connects canonical entanglement and entanglement based on occupation number for two fermions and for two bosons and shows that the degree of entanglement, based on linear entropy, is closely related to the correlation measure for both the bosonic and fermionic cases. PACS numbers: 03.65.Ta,03.67.-a,03.75.-b,05.30.-d

Entanglement and Particle Identity: A Unifying Approach

2013

It has been known for some years that entanglement entropy obtained from partial trace does not provide the correct entanglement measure when applied to systems of identical particles. Several criteria have been proposed that have the drawback of being different according to whether one is dealing with fermions, bosons, or distinguishable particles. In this Letter, we give a precise and mathematically natural answer to this problem. Our approach is based on the use of the more general idea of the restriction of states to subalgebras. It leads to a novel approach to entanglement, which is suitable to be used in general quantum systems and especially in systems of identical particles. This settles some recent controversy regarding entanglement for identical particles. The prospects for applications of our criteria are wide ranging, from spin chains in condensed matter to entropy of black holes.

Bipartite entanglement in systems of identical particles: The partial transposition criterion

We study bipartite entanglement in systems of N identical bosons distributed in M different modes. For such systems, a definition of separability not related to any a priori Hilbert space tensor product structure is needed and can be given in terms of commuting subalgebras of observables. Using this generalized notion of separability, we classify the states for which partial transposition turns out to be a necessary and sufficient condition for entanglement detection.

Entanglement for multipartite systems of indistinguishable particles

Journal of Physics A: Mathematical and Theoretical, 2011

We analyze the concept of entanglement for multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. We use the representation theory of symmetry groups to formulate a unified approach to this problem in terms of simple tensors with appropriate symmetry. For an arbitrary parastatistics, we define the S-rank generalizing the notion of the Schmidt rank. The S-rank, defined for all types of tensors, serves for distinguishing entanglement of pure states. In addition, for Bose and Fermi statistics, we construct an analog of the Jamio lkowski isomorphism.

Entanglement and scattering in quantum electrodynamics: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">mml:miS matrix information from an entangled spectator particle

Physical review, 2021

We consider a general quantum field relativistic scattering involving two half spin fermions, A and B, which are initially entangled with another fermion C that does not participate in the scattering dynamics. We construct general expressions for the reduced spin matrices for the out-state considering a general tripartite spin-entangled state. In particular we study an inelastic QED process at tree-level, namely e − e + → µ − µ + and a half spin fermion C as a spectator particle which can be entangled to the AB system in the following ways: W state, GHZ state, |A α ⊗ |Ψ ± BC and |A α ⊗ |Φ ± BC, where {|Ψ ± , |Φ ± } are the Bell basis states and |A α is a spin superposition state of system A. We calculate the von-Neumann entropy variation before and after the scattering for the particle C and show that spin measurements in C contain numerical information about the total cross section of the process. We compare the initial states W and GHZ as well as study the role played by the parameter α in the evaluation of the entropy variations and the cross section encoded in the spectator particle.

Observables and entanglement in the two-body system

AIP Conference Proceedings, 2012

Using the quantum two-body system as a familiar model, this talk will describe how entanglement can be used to select preferred observables for interrogating a physical system. The symmetries and dynamics of the quantum two-body system provide a backdrop for testing the relativity of entanglement with respect to observable-induced tensor product structures. We believe this exploration leads us to a general statement: the physically-meaningful observable subalgebras are the ones that minimize entanglement in typical states.