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arXiv:1907.00732 · Information Geometry

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

F. M. Ciaglia, A. Ibort, J. Jost, G. Marmo

Published Oct 24, 2019• 12 citations
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Abstract

Abstract The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of $$C^{*}$$ C∗ -algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given $$C^{*}$$ C∗ -algebra $$\mathscr {A}$$ A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $$\mathscr {S}$$ S of a possibly infinite-dimensional, unital $$C^{*}$$ C∗ -algebra $$\mathscr {A}$$ A is partitioned into the disjoint union of the orbits of an action of the group $$\mathscr {G}$$ G of invertible elements of $$\mathscr {A}$$ A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space $$\mathcal {H}$$ H are smooth, homogeneous Banach manifolds of $$\mathscr {G}=\mathcal {GL}(\mathcal {H})$$ G=GL(H) , and, when $$\mathscr {A}$$ A admits a faithful tracial state $$\tau $$ τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through $$\tau $$ τ is a smooth, homogeneous Banach manifold for $$\mathscr {G}$$ G .

DOI
10.1007/s41884-019-00022-1

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Abstract The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of C∗C^{*}C∗ C∗ -algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given C∗C^{*}C∗ C∗ -algebra A\mathscr {A}A A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states S\mathscr {S}S S of a possibly infinite-dimensional, unital C∗C^{*}C∗ C∗ -algebra A\mathscr {A}A A is partitioned into the disjoint union of the orbits of an action of the group G\mathscr {G}G G of invertible elements of A\mathscr {A}A A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space H\mathcal {H}H H are smooth, homogeneous Banach manifolds of G=GL(H)\mathscr {G}=\mathcal {GL}(\mathcal {H})G=GL(H) G=GL(H) , and, when A\mathscr {A}A A admits a faithful tracial state τ\tau τ τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through τ\tau τ τ is a smooth, homogeneous Banach manifold for G\mathscr {G}G G .
Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive density matrices to a manifold of faithful quantum states on the C*-algebra of bounded linear operators. In addition, ideas from the parameter-free approach to information geometry are adopted. The underlying Hilbert space is assumed to be finite-dimensional. In this way, technicalities are avoided so that strong results are obtained, which one can hope to prove later on in a more general context. Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection.
We review the geometry of the space of quantum states S(H)\mathscr{S}(\mathcal{H})S(H) of a finite-level quantum system with Hilbert space H\mathcal{H}H from a group-theoretical point of view. This space carries two stratifications generated by the action of two different Lie groups: the special unitary group SU(H)\mathcal{SU}(\mathcal{H})SU(H) and its complexification SL(H)\mathcal{SL}(\mathcal{H})SL(H), the complex special linear group. A stratum of the stratification generated by SU(H)\mathcal{SU}(\mathcal{H})SU(H) is composed of isospectral states, that is, density operators with the same spectrum, A stratum of the stratification generated by SL(H)\mathcal{SL}(\mathcal{H})SL(H) is composed of quantum states with the same rank. We prove that on every submanifold of isospectral quantum states there is also a canonical left action of SL(H)\mathcal{SL}(\mathcal{H})SL(H) which is related with the canonical Kähler structure on isospectral quantum states. The fundamental vector fields of this SL(H)\mathcal{SL}(\mathcal{H})SL(H)-action are divided into Hamiltonian and gradient vector fields. The former give rise to invertible maps on S(H)\mathscr{S}(\mathcal{H})S(H) that preserve the von Neumann entropy and the convex structure of S(H)\mathscr{S}(\mathcal{H})S(H), while the latter give rise to invertible maps on S(H)\mathscr{S}(\mathcal{H})S(H) that preserve the von Neumann entropy but not the convex structure of S(H)\mathscr{S}(\mathcal{H})S(H). A similar decomposition is given for the SL(H)\mathcal{SL}(\mathcal{H})SL(H)-action generating the stratification of S(H)\mathscr{S}(\mathcal{H})S(H) into manifolds of quantum states with the same rank, where gradient vector fields preserve the rank but do not preserve entropy. Some comments on multipartite quantum systems are made. It is proved that the sets of product states of a multipartite quantum system are homogeneous manifolds for the action of the complex special linear group associated with the partition.
Abstract In classical statistical mechanics, physical states (probability measures) are embedded in the Banach algebra of complex Borel measures on phase space, where the algebra product is realized by convolution. Convolution is state-preserving; namely, the convolution of two classical states is a state too. This is a special case of the convolution algebra of all complex measures on a locally compact group. A natural problem is whether an analogous structure may emerge in the quantum setting. By resorting to a group-theoretical construction, a quantum counterpart of the convolution of probability measures — the twirled product, or quantum convolution — can be introduced, yielding a group-covariant, associative binary operation on the states of a quantum system, that preserves the convex structure of this set. The analogy with the classical setting becomes striking in the case where the symmetry group is abelian. We focus, in particular, on the quantum convolution product stemming from the group of phase-space translations.
The subject of this thesis is the geometry of the space of quantum states. The aim of this thesis is to present a geometrical analysis of the structural properties of this space, being them of ``kinematical'' or ``dynamical'' character. We will see that the space of quantum states of finite-dimensional systems may be partitioned into the union of disjoint orbits of the complexification of the unitary group. These orbits are the manifolds of quantum states with fixed rank. On the one hand, we will compute the two-parameter family of quantum metric tensors associated with the two-parameter family of quantum q-z-Renyi relative entropies on the manifold of invertible quantum states (maximal rank). Using the powerful language of differential geometry we are able to perform all the computations in an arbitrary number of (finite) dimensions without the need to introduce explicit coordinate systems. On the other hand, we will develop a geometrization of the GKLS equation for the dynamical evolution of Markovian open quantum systems. Specifically, we will write the GKLS generator by means of an affine vector field on an affine space, and we will decompose this vector field into the sum of a Hamiltonian vector field, a gradient-like vector field, and a so-called Kraus vector field. This geometrization will be used in order to analyze and completely characterize the asymptotic behaviour of the dynamical evolutions known as quantum random unitary semigroups by means of the so-called purity function. Finally, we will comment on the possibility of extending the results presented to the infinite-dimensional case, and to the case of multipartite quantum systems.
The set Q\mathcal{Q}Q of quantum correlations is the collection of all possible probability distributions on measurement outcomes achievable by space-like separated parties sharing a quantum state. Since the original work of Tsirelson [Tsirelson, Lett. Math. Phys. 4, 93 (1980)], this set has mainly been studied through the means algebraic and convex geometry techniques. We introduce a perturbative method using Lie-theoretic tools for the unitary group to analyze the response of the evaluations of Bell functionals under infinitesimal unitary perturbations of quantum strategies. Our main result shows that, near classical deterministic points, an (n,2,d)(n, 2, d)(n,2,d) Bell operator decomposes into a direct sum of (k,2,d−1)(k, 2, d-1)(k,2,d−1) Bell operators which we call \emph{subset games}. We then derive three key insights: (1) in the (n,2,2)(n, 2, 2)(n,2,2) case, if p0p_0p0​ is classically optimal, it remains locally optimal even among 2-dimensional quantum strategies, implying in turn that the boundary of Q\mathcal{Q}Q is flat around classical deterministic points; (2) it suggests a proof strategy for Gisin's open problem on correlations in Q(D)\mathcal{Q}(D)Q(D) unattainable by projective strategies of the same dimension; and (3) it establishes that the Ansatz dimension is a critical resource for learning in distributed scenarios, even when the optimal solution admits a low-dimensional representation.
Abstract A notion of stochastic product of quantum states — a binary operation on the set of density operators preserving the convex structure — is discussed. We describe, in particular, a class of group-covariant, associative stochastic products: the twirled products. Each binary operation in this class can be constructed by means of a square integrable projective representation of a locally compact group, a probability measure on this group and a fiducial density operator in the Hilbert space of the representation. By suitably extending this operation from the convex set of density operators to the full Banach space of trace class operators, one obtains a Banach algebra, which is commutative in the case where the relevant group is abelian.