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Journal of Mathematical Physics

Duality versus dual flatness in quantum information geometry

Nihat Ay, Wilderich Tuschmann

Published Mar 20, 2003• 18 citations
Publisher

Abstract

We investigate questions in quantum information geometry which concern the existence and nonexistence of dual and dually flat structures on stratified sets of density operators on finite-dimensional Hilbert spaces. We show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal. We prove, moreover, that there is never a dually flat structure on the set of pure states. Thus any general theory of quantum information geometry that involves duality concepts must inevitably be based on dual structures which are nonflat.

DOI
10.1063/1.1556192

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We investigate questions in quantum information geometry which concern the existence and nonexistence of dual and dually flat structures on stratified sets of density operators on finite-dimensional Hilbert spaces. We show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal. We prove, moreover, that there is never a dually flat structure on the set of pure states. Thus any general theory of quantum information geometry that involves duality concepts must inevitably be based on dual structures which are nonflat.
Abstract The notion of dually flatness is of central importance in information geometry. Nevertheless, little is known about dually flat structures on quantum statistical manifolds except that the Bogoliubov metric admits a global dually flat structure on a quantum state space S(Cd){{\mathcal {S}}}({{\mathbb {C}}}^d)S(Cd) S ( C d ) for any d≥2d\ge 2d≥2 d ≥ 2 . In this paper, we show that every monotone metric on a two-level quantum state space S(C2){{\mathcal {S}}}({{\mathbb {C}}}^2)S(C2) S ( C 2 ) admits a local dually flat structure.
Let [Formula: see text] be a separable Hilbert space. We consider the manifold [Formula: see text] consisting of density operators ρ on [Formula: see text] such that ρp is of trace class for some p ɛ (0, 1). We say [Formula: see text] is nearby ρ if there exists C > 1 such that C−1 ρ < σ < Cρ. We show that the space of nearby points to ρ can be furnished with the two flat connections known as the (±)-affine structures, which are dual relative to the BKM metric. We furnish [Formula: see text] with a norm making it into a Banach manifold.
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.
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.