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arXiv:math-ph/0603048

Symmetries, group actions, and entanglement

Janusz Grabowski, Marek Kuś, Giuseppe Marmo

Published Mar 20, 2006• 0 citations
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Abstract

We address several problems concerning the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space, in particular the geometry of the space of density states and canonical group actions on it. For quantum composite systems we discuss and give examples of measures of entanglement.
DOI
10.48550/arxiv.math-ph/0603048

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Various problems concerning the geometry of the space u ∗ (H) of Hermitian operators on a Hilbert space H are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into Kähler submanifolds. It is also shown that the space D(H) of density states on an n-dimensional Hilbert space H is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space D k (H) of rank-k states, k = 1,...,n, is a smooth manifold of (real) dimension 2nk −k 2 −1 and this stratification is maximal in the sense that every smooth curve in D(H), viewed as a subset of the dual u ∗ (H) to the Lie algebra of the unitary group U(H), at every point must be tangent to the strata D k (H) it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition H = H 1 ⊗ H 2, an abstract criterion of entanglement is proved.
Various problems concerning the geometry of the space of Hermitian operators on a Hilbert space are addressed. In particular, we study the canonical Poisson and Riemann–Jordan tensors and the corresponding foliations into Kähler submanifolds. It is also shown that the space of density states on an n -dimensional Hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space of rank- k states, k = 1, ..., n , is a smooth manifold of (real) dimension 2 nk − k 2 − 1 and this stratification is maximal in the sense that every smooth curve in , viewed as a subset of the dual to the Lie algebra of the unitary group , at every point must be tangent to the strata it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition , an abstract criterion of entanglement is proved.
G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed
Finite-dimensional Quantum Mechanics can be geometrically formulated as a proper classical-like Hamiltonian theory in a projective Hilbert space. The description of composite quantum systems within the geometric Hamiltonian framework is discussed in this paper. As summarized in the first part of this work, in the Hamiltonian formulation the phase space of a quantum system is the Kähler manifold given by the complex projective space P(H) of the Hilbert space H of the considered quantum theory. However the phase space of a bipartite system must be P(H1 ⊗ H2) and not simply P(H1) × P(H2) as suggested by the analogy with Classical Mechanics. A part of this paper is devoted to manage this problem. In the second part of the work, a definition of quantum entanglement and a proposal of entanglement measure are given in terms of a geometrical point of view (a rather studied topic in recent literature). Finally two known separability criteria are implemented in the Hamiltonian formalism.
We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using the Kostant-Sternberg theorem, we show that separable states form a unique symplectic orbit, whereas orbits of entangled states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement. The above statements remain true for systems with an arbitrary number of components, moreover the presented method is general and can be applied also under different additional symmetry conditions stemming, e.g., from the indistinguishability of particles. We show how to calculate the degeneracy for various multiparticle systems providing also simple criteria of separability.