Manifolds of interconvertible pure states (original) (raw)
Physical Review A, 2001
Geometric properties of the set of quantum entangled states are investigated. We propose an explicit method to compute the dimension of local orbits for any mixed state of the general KϫM problem and characterize the set of effectively different states ͑which cannot be related by local transformations͒. Thus, we generalize earlier results obtained for the simplest 2ϫ2 system, which lead to a stratification of the six-dimensional set of Nϭ4 pure states. We define the concept of absolutely separable states, for which all globally equivalent states are separable.
Orbits of quantum states and geometry of Bloch vectors for N -level systems
Journal of Physics A: Mathematical and General, 2004
Physical constraints such as positivity endow the set of quantum states with a rich geometry if the system dimension is greater than two. To shed some light on the complicated structure of the set of quantum states, we consider a stratification with strata given by unitary orbit manifolds, which can be identified with flag manifolds. The results are applied to study the geometry of the coherence vector for n-level quantum systems. It is shown that the unitary orbits can be naturally identified with spheres in R n 2 −1 only for n = 2. In higher dimensions the coherence vector only defines a non-surjective embedding into a closed ball. A detailed analysis of the three-level case is presented. Finally, a refined stratification in terms of symplectic orbits is considered.
Absolutely entangled sets of pure states for bipartitions and multipartitions
Physical Review A
A set of quantum states is said to be absolutely entangled when at least one state in the set remains entangled for any definition of subsystems, i.e., for any choice of the global reference frame. In this work we investigate the properties of absolutely entangled sets (AESs) of pure quantum states. For the case of a two-qubit system, we present a sufficient condition to detect an AES, and use it to construct families of N states such that N − 3 (the maximal possible number) remain entangled for any definition of subsystems. For a general bipartition d = d 1 d 2 , we prove that sets of N > (d 1 + 1)(d 2 + 1)/2 states are AESs with Haar measure 1. Then, we define AESs for multipartitions. We derive a general lower bound on the number of states in an AES for a given multipartition, and also construct explicit examples. In particular, we exhibit an AES with respect to any possible multipartitioning of the total system.
Bipartite entanglement, spherical actions, and geometry of local unitary orbits
Journal of Mathematical Physics, 2013
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. arXiv:1206.4200v2 [math-ph]
2007
We investigate the geometric characterization of pure state bipartite entanglement of (2 × D)-and (3 × D)-dimensional composite quantum systems. To this aim, we analyze the relationship between states and their images under the action of particular classes of local unitary operations. We find that invariance of states under the action of single-qubit and single-qutrit transformations is a necessary and sufficient condition for separability. We demonstrate that in the (2×D)-dimensional case the von Neumann entropy of entanglement is a monotonic function of the minimum squared Euclidean distance between states and their images over the set of single qubit unitary transformations. Moreover, both in the (2×D)-and in the (3×D)-dimensional cases the minimum squared Euclidean distance exactly coincides with the linear entropy (and thus as well with the tangle measure of entanglement in the (2 × D)-dimensional case). These results provide a geometric characterization of entanglement measures originally established in informational frameworks. Consequences and applications of the formalism to quantum critical phenomena in spin systems are discussed.
On bipartite pure-state entanglement structure in terms of disentanglement
Journal of Mathematical Physics, 2006
tation of bipartite state vectors, which is reviewed, and the relevant results of Cassinelli et al. [J. Math. Analys. and Appl., 210, 472 (1997)] in mathematical analysis, which are summed up. Linearly-independent bases (finite or infinite) are shown to be almost as useful in some quantum mechanical studies as orthonormal ones. Finally, it is shown that linearly-independent remote pure-state preparation carries the highest probability of occurrence. This singles out linearly-independent remote influence from all possible ones.
Tensor rank and entanglement of pure quantum states
arXiv: Quantum Physics, 2019
The rank of a tensor is analyzed in context of description of entanglement of pure states of multipartite quantum systems. We discuss the notions of the generic rank of a tensor with ddd indices and nnn levels in each mode and the maximal rank of a tensor of these dimensions. Other variant of this notion, called border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. As entanglement of a given quantum state depends on the way the total system is divided into subsystems, we introduce a notion of `partitioning rank' of a tensor, which depends on a way how the entries forming the tensor are treated. In particular, we analyze the tensor product of several copies of the nnn-qubit state ∣Wnrangle|W_n\rangle∣Wnrangle and analyze its partitioning rank for various splittings of the entire space. Some results concerning the generic rank of a tensor are also provided.
SEPARABILITY OF PURE N-QUBIT STATES: TWO CHARACTERIZATIONS
International Journal of Foundations of Computer Science, 2003
Given a pure state ψ N N ∈H of a quantum system composed of n qubits, where H N is the Hilbert space of dimension N n = 2 , this paper answers two questions: what conditions should the amplitudes in ψ N satisfy for this state to be separable (i) into a tensor product of n qubit states ψ ψ ψ 2 0 2 1 2 1 ⊗ ⊗ ⊗ − ... n , and (ii), into a tensor product of 2 subsystems states ψ ψ P Q ⊗ with P p = 2 and Q q = 2 such that p q n + = ? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. These conditions bear some relation with entanglement measures, and a number of more refined questions about separability in n qubit systems can be studied on the basis of these results.
Entanglement convertibility for infinite-dimensional pure bipartite states
Physical Review A, 2004
It is shown that the order property of pure bipartite states under SLOCC (stochastic local operations and classical communications) changes radically when dimensionality shifts from finite to infinite. In contrast to finite dimensional systems where there is no pure incomparable state, the existence of infinitely many mutually SLOCC incomparable states is shown for infinite dimensional systems even under the bounded energy and finite information exchange condition. These results show that the effect of the infinite dimensionality of Hilbert space, the "infinite workspace" property, remains even in physically relevant infinite dimensional systems. PACS numbers: 03.65.Ud, 03.67.-a, 03.67.Mn
On the volume of the set of mixed entangled states II
The problem of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of density matrices ̺ describing N -dimensional quantum systems affects the results obtained. We demonstrate that the link between the purity of the mixed states and the probability of entanglement is not sensitive to the measure chosen. Since the criterion of partial transposition is not sufficient to distinguish all separable states for N ≥ 8, we develop an efficient algorithm to calculate numerically the entanglement of formation of a given mixed quantum state, which allows us to compute the volume of separable states for N = 8 and to estimate the volume of the bound entangled states in this case. 03.65.Bz, 42.50.Dv, 89.70.+c