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arXiv:1108.4134 · Journal of Physics A Mathematical and Theoretical

Geometry of the local equivalence of states

Adam Sawicki, Marek Kuś

Published Nov 15, 2011• 18 citations
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Abstract

We present a description of locally equivalent states in terms of symplectic geometry. Using the moment map between local orbits in the space of states and coadjoint orbits of the local unitary group, we reduce the problem of local unitary equivalence to an easy part consisting of identifying the proper coadjoint orbit and a harder problem of the geometry of fibers of the moment map. We give a detailed analysis of the properties of orbits of 'equally entangled states'. In particular, we show connections between certain symplectic properties of orbits such as their isotropy and coisotropy with effective criteria of local unitary equivalence.

DOI
10.1088/1751-8113/44/49/495301

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We present a description of locally equivalent states in terms of symplectic geometry. Using the moment map between local orbits in the space of states and coadjoint orbits of the local unitary group, we reduce the problem of local unitary equivalence to an easy part consisting of identifying the proper coadjoint orbit and a harder problem of the geometry of fibers of the moment map. We give a detailed analysis of the properties of orbits of 'equally entangled states'. In particular, we show connections between certain symplectic properties of orbits such as their isotropy and coisotropy with effective criteria of local unitary equivalence.
We use the geometry of the moment map to investigate properties of pure entangled states of composite quantum systems. The orbits of equally entangled states are mapped by the moment map onto coadjoint orbits of local transformations (unitary transformations which do not change entanglement). Thus, the geometry of coadjoint orbits provides a partial classification of different entanglement classes. To achieve the full classification, a further study of fibers of the moment map is needed. We show how this can be done effectively in the case of the bipartite entanglement by employing Brion's theorem. In particular, we presented the exact description of the partial symplectic structure of all local orbits for two bosons, fermions, and distinguishable particles putting a special emphasis on the generality of the approach allowing one to consider all three cases in completely parallel manners.
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.
In this work we explore the implications of applying the formalism of symplectic geometry to quantum mechanics, particularly focusing on many-particle systems. We extend the concept of a symplectic indicator of entanglement, originally introduced by Sawicki et al., Commun. Math. Phys. 305, 441 (2011), to these complex systems. Specifically, we demonstrate that the restriction of the symplectic structure to manifolds comprising all states characterized by isospectral reduced one-particle density matrices ${M}_{\ensuremath{\mu}(\ensuremath{\psi})}^{0}$ exhibits degeneracy for nonseparable states. We prove that the degree of degeneracy at any given state $|\ensuremath{\varphi}\ensuremath{\rangle}\ensuremath{\in}{M}_{\ensuremath{\mu}(\ensuremath{\psi})}^{0}$ corresponds to the degree of degeneracy of the symplectic form $\ensuremath{\omega}$ when restricted to the manifold of states that are locally unitary equivalent with $|\ensuremath{\varphi}\ensuremath{\rangle}$. Additionally, we provide a physical interpretation of this symplectic indicator of entanglement, articulating it as an inherent ambiguity within the associated classical dynamical framework. Our findings underscore the pivotal role of symplectic geometry in elucidating entanglement properties in quantum mechanics and suggest avenues for further exploration into the geometric structures underlying quantum state spaces.
Local unitary equivalence is central to entanglement quantification and classification. Identifying the local unitary equivalence remains a formidable challenge. We address this problem for a class of quantum states with one highly degenerate eigenvalue and the rest non-degenerate simple eigenvalues that are pivotal to structured resources in quantum resource theory. We introduce a ``reduction" procedure that maps each state to a ``reduced state" by nullifying the highest-multiplicity eigenvalue and prove that the local unitary equivalence of the original states is equivalent to that of their reduced counterparts. For the resulting pure or non-degenerate reduced states, we employ the existing invariants or fixed-point subgroup criteria to establish a complete discrimination framework, although the existing criteria can not directly identify the local unitary equivalence of the original states. We also verify the local unitary equivalence of two families of single-parameterized multipartite mixed states constructed by perturbing absolutely maximally entangled states from distinct combinatorial origins, demonstrating the efficacy and generality of our approach.
In this paper, we study the local unitary equivalence of quantum states on Cn⊗Cn\mathbb{C}^n\otimes\mathbb{C}^nCn⊗Cn, which is an important notion in quantum information theory. For the case n=2n=2n=2, the local unitary orbits of two-qubit states are completely determined by their local unitary Bargmann invariants. We show that the local unitary Bargmann invariants do not form a complete set of invariants of local unitary orbits for n⩾3n\geqslant 3n⩾3, which negatively answers a problem proposed by L. Zhang and B. Xie.
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.