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AIP conference proceedings

Classical positivity, quantum positivity and entanglement of a multi-partite density matrix, with the example of polarized reactions

X. Artru

Published Jan 1, 2012• 0 citations
Publisher

Abstract

The spin correlations in a reaction involving initial and final polarized particles are encoded in a cross section matrix which is taken as an example of multipartite density matrix. Quantum positivity stipulates that the cross section is positive for any spin states - separable or entangled - of the particles. A weaker condition, classical positivity only requires that the cross section is positive for separable spin state. Classical and Quantum positivity are represented by diagrams. The domains of classical positivity and of separability are dual under a reciprocal polar transformation. This property can be usefull in quantum information theory.

DOI
10.1063/1.4715425

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The spin correlations in a reaction involving initial and final polarized particles are encoded in a cross section matrix which is taken as an example of multipartite density matrix. Quantum positivity stipulates that the cross section is positive for any spin states - separable or entangled - of the particles. A weaker condition, classical positivity only requires that the cross section is positive for separable spin state. Classical and Quantum positivity are represented by diagrams. The domains of classical positivity and of separability are dual under a reciprocal polar transformation. This property can be usefull in quantum information theory.
Constraints on spin observables coming from discrete symmetries such as P, C, T and identical particles may be divided in two types: 1) classical ones, which insure the invariance of the cross sections under the symmetry operation; 2) non-classical ones, which can only be obtained at the level of amplitudes. Similarly, positivity constraints can be divided into classical and non-classical constraints. The former insure the positivity of the cross section for arbitrary individual polarisations of the external particles, the latter extend this requirement to the case of entangled external spins. The domain of classical positivity is shown to be dual to the domain of separability
Relativistic effects affect nearly all notions of quantum information theory. The vacuum behaves as a noisy channel, even if the detectors are perfect. The standard definition of a reduced density matrix fails for photon polarization because the transversality condition behaves like a superselection rule. We can however define an effective reduced density matrix which corresponds to a restricted class of positive operator-valued measures. There are no pure photon qubits, and no exactly orthogonal qubit states. Reduced density matrices for the spin of massive particles are well-defined, but are not covariant under Lorentz transformations. The spin entropy is not a relativistic scalar and has no invariant meaning. The distinguishability of quantum signals and their entanglement depends on the relative motion of observers.
We derive a classification and a measure of classical- and quantum-correlation of multipartite qubit, qutrit, and in general, nnn-level systems, in terms of SU(n)(n)(n) representations of density matrices. We compare the measure for the case of bipartite correlation with concurrence and the entropy of entanglement. The characterization of correlation is in terms of the number of nonzero singular values of the correlation matrix, but that of mixed state entanglement requires additional invariant parameters in the density matrix. For the bipartite qubit case, the condition for mixed state entanglement is written explicitly in terms of the invariant paramters in the density matrix. For identical particle systems we analyze the effects of exchange symmetry on classical and quantum correlation.
A family of separability criteria based on correlation matrix (tensor) is provided. Interestingly, it unifies several criteria known before like, e.g., computable cross-norm or realignment criterion (CNNR), de Vicente criterion, and derived recently separability criterion based on symmetric informationally complete positive operator valued measures (SIC POVMs). It should be stressed that, unlike the well-known correlation matrix criterion or criterion based on local uncertainty relations, our criteria are linear in the density operator and hence one may find unexplored classes of entanglement witnesses and positive maps. Interestingly, there is a natural generalization to multipartite scenario using multipartite correlation matrix. We illustrate the detection power of the above criteria on several well-known examples of quantum states.
Lorentz transformations of spin density matrices for a particle with positive mass and spin 1∕21∕21∕2 are described by maps of the kind used in open quantum dynamics. They show how the Lorentz transformations of the spin depend on the momentum. Since the spin and momentum generally are not independent, the maps generally are not completely positive and act in limited domains. States with two momentum values are considered, so the maps are for the spin qubit correlated with the qubit made from the two momentum values, and results from the open quantum dynamics of two coupled qubits can be applied. Inverses are used to show that every Lorentz transformation completely removes the spin polarization, and so completely removes the information, from a number of spin density matrices. The size of the spin polarization that is removed is calculated for particular cases.
While entanglement is believed to underlie the power of quantum computation and communication, it is not generally well understood for multipartite systems. Recently, it has been appreciated that there exists proper no-signaling probability distributions derivable from operators that do not represent valid quantum states. Such systems exhibit supra-correlations that are stronger than allowed by quantum mechanics, but less than the algebraically allowed maximum in Bell-inequalities (in the bipartite case). Some of these probability distributions are derivable from an entanglement witness W, which is a non-positive Hermitian operator constructed such that its expectation value with a separable quantum state (positive density matrix) ρsep is non-negative (so that Tr[W ρ]< 0 indicates entanglement in quantum state ρ). In the bipartite case, it is known that by a modification of the local no-signaling measurements by spacelike separated parties A and B, the supra-correlations exhibited by any W can be modeled as derivable from a physically realizable quantum state ρ. However, this result does not generalize to the n-partite case for n>2. Supracorrelations can also be exhibited in 2- and 3-qubit systems by explicitly constructing "states" Ο (not necessarily positive quantum states) that exhibit PR correlations for a fixed, but arbitrary number, of measurements available to each party. In this paper we examine the structure of "states" that exhibit supra-correlations. In addition, we examine the affect upon the distribution of the correlations amongst the parties involved when constraints of positivity and purity are imposed. We investigate circumstances in which such "states" do and do not represent valid quantum states.