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Stratified manifold of quantum states, actions of the complex special linear group

Dariusz Chruściński, Ciaglia, F., Alberto Ibort, Giuseppe Marmo, F. Ventriglia

Published Jan 1, 2019• 10 citations
OpenAlex

Abstract

We review the geometry of the space of quantum states $\mathscr{S}(\mathcal{H})$ of a finite-level quantum system with Hilbert space $\mathcal{H}$ 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 $\mathcal{SU}(\mathcal{H})$ and its complexification $\mathcal{SL}(\mathcal{H})$, the complex special linear group. A stratum of the stratification generated by $\mathcal{SU}(\mathcal{H})$ is composed of isospectral states, that is, density operators with the same spectrum, A stratum of the stratification generated by $\mathcal{SL}(\mathcal{H})$ 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 $\mathcal{SL}(\mathcal{H})$ which is related with the canonical Kähler structure on isospectral quantum states. The fundamental vector fields of this $\mathcal{SL}(\mathcal{H})$-action are divided into Hamiltonian and gradient vector fields. The former give rise to invertible maps on $\mathscr{S}(\mathcal{H})$ that preserve the von Neumann entropy and the convex structure of $\mathscr{S}(\mathcal{H})$, while the latter give rise to invertible maps on $\mathscr{S}(\mathcal{H})$ that preserve the von Neumann entropy but not the convex structure of $\mathscr{S}(\mathcal{H})$. A similar decomposition is given for the $\mathcal{SL}(\mathcal{H})$-action generating the stratification of $\mathscr{S}(\mathcal{H})$ 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.

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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.
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.
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 .
On classical Lie groups, which act by means of a unitary representation on finite dimensional Hilbert spaces H, we identify two classes of tensor field constructions. First, as pull-back tensor fields of order two from modified Hermitian tensor fields, constructed on Hilbert spaces by means of the property of having the vertical distributions of the C_0-principal bundle H_0 over the projective Hilbert space P(H) in the kernel. And second, directly constructed on the Lie group, as left-invariant representation-dependent operator-valued tensor fields (LIROVTs) of arbitrary order being evaluated on a quantum state. Within the NP-hard problem of deciding whether a given state in a n-level bi-partite quantum system is entangled or separable (Gurvits, 2003), we show that both tensor field constructions admit a geometric approach to this problem, which evades the traditional ambiguity on defining metrical structures on the convex set of mixed states. In particular by considering manifolds associated to orbits passing through a selected state when acted upon by the local unitary group U(n)xU(n) of Schmidt coefficient decomposition inducing transformations, we find the following results: In the case of pure states we show that Schmidt-equivalence classes which are Lagrangian submanifolds define maximal entangled states. This implies a stronger statement as the one proposed by Bengtsson (2007). Moreover, Riemannian pull-back tensor fields split on orbits of separable states and provide a quantitative characterization of entanglement which recover the entanglement measure proposed by Schlienz and Mahler (1995). In the case of mixed states we highlight a relation between LIROVTs of order two and a class of computable separability criteria based on the Bloch-representation (de Vicente, 2007).
We investigate the relation between local unitary symmetries and entanglement invariants of multi-qubit systems. The Hilbert space of such systems can be stratified in terms of states with different types of symmetry. We review the connection between this stratification and the ring of entanglement invariants and the corresponding geometric description in terms of algebraic varieties. On a stratum of a non-trivial symmetry group the invariants of the symmetry preserving operations gives a sufficient description of entanglement. Finding these invariants is often a simpler problem than finding the invariants of the local unitary group. The conditions, as given by the Luna–Richardson theorem, for when the ring of such invariants is isomorphic to the ring of local unitary invariants on the stratum are discussed. As an example we consider symmetry groups that can be diagonalized by local unitary operations and for which the group action on each qubit is non-trivial. On the stratum of such a symmetry the entanglement can be described in terms of a canonical form and the invariants of the symmetry preserving operations. This canonical form and the invariants are directly determined by the symmetry group. Further, we briefly discuss how some recently proposed entanglement classification schemes capture symmetry properties.
In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.