A Gramian approach to entanglement in bipartite finite dimensional systems: the case of pure states

It has been observed that the reduced density matrices of bipartite qudit pure states possess a Gram matrix structure. This observation has opened a possibility of analysing the entanglement in such systems from the purely geometrical point of view. In particular, a new quantitative measure of an entanglement of the geometrical nature, has been proposed. Using the invented Gram matrix approach, a version of a non-linear purification of mixed states describing the system analysed has been presented.

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Some Remarks on the Entanglement Number

Quanta ◽

10.12743/quanta.v9i1.140 ◽

2020 ◽

Vol 9 (1) ◽

pp. 22-36

Author(s):

George Androulakis ◽

Ryan McGaha

Keyword(s):

Point Of View ◽

Entanglement Measure ◽

Mixed States ◽

Interesting Point ◽

Finite Dimensional ◽

Entanglement Measures ◽

Pure States ◽

Tree Representations ◽

Natural Way ◽

Finite Dimensional Hilbert Space

Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement.Quanta 2020; 9: 22–36.

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Average of Uncertainty Product for Bounded Observables

Open Systems & Information Dynamics ◽

10.1142/s1230161218500087 ◽

2018 ◽

Vol 25 (02) ◽

pp. 1850008 ◽

Cited By ~ 1

Author(s):

Lin Zhang ◽

Jiamei Wang

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Quantum States ◽

Density Matrices ◽

Finite Dimensional ◽

Uncertainty Product ◽

Schmidt Norm ◽

Rank One ◽

Pure States ◽

Average Uncertainty

The goal of this paper is to calculate exactly the average of uncertainty product of two bounded observables and to establish its typicality over the whole set of finite dimensional quantum pure states. Here we use the uniform ensembles of pure and isospectral states as well as the states distributed uniformly according to the measure induced by the Hilbert-Schmidt norm. Firstly, we investigate the average uncertainty of an observable over isospectral density matrices. By letting the isospectral density matrices be of rank-one, we get the average uncertainty of an observable restricted to pure quantum states. These results can help us check how large is the gap between the uncertainty product and any lower bounds obtained for the uncertainty product. Although our method in the present paper cannot give a tighter lower bound of uncertainty product for bounded observables, it can help us drop any one that is not substantially tighter than the known one.

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Superselection Sectors, Pure States, Bose and Fermi Distributions by Completely Positive Quantum-Motions

Open Systems & Information Dynamics ◽

10.1007/s11080-005-0484-4 ◽

2005 ◽

Vol 12 (01) ◽

pp. 23-35 ◽

Cited By ~ 1

Author(s):

Klaus Dietz

Keyword(s):

Hilbert Spaces ◽

Mixed States ◽

Completely Positive ◽

Absolute Value ◽

Zero Modes ◽

Finite Dimensional ◽

Constants Of Motion ◽

Pure States ◽

Damped Oscillator

The connection of the operators V, building up the Kossakowski-Lindblad generator, with the asymptotic states of the corresponding completely positive quantum-maps is discussed. Maps leading to decoherence are constructed, the importance of zero-modes in the absolute value [Formula: see text] of V for the generation of pure states from arbitrary mixed states is illustrated. The universal rôle of equipartite states appears when unitary V are chosen. The 'damped oscillator model' is generalized to yield Bose and Fermi distributions as asymptotic states for systems described by a Hamiltonian and other constants of motion. Calculations are performed in finite dimensional Hilbert spaces.

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k-uniform maximally mixed states from multi-qudit phase states

Modern Physics Letters A ◽

10.1142/s0217732319501517 ◽

2019 ◽

Vol 34 (19) ◽

pp. 1950151 ◽

Cited By ~ 5

Author(s):

Mostafa Mansour ◽

Mohammed Daoud

Keyword(s):

Special Kind ◽

Mixed States ◽

Density Matrices ◽

Finite Dimensional ◽

Heisenberg Type ◽

Multipartite System ◽

Multipartite States ◽

Phase Operators

This paper concerns the construction of k-uniform maximally mixed multipartite states by using the formalism of phase states for finite dimensional systems (qudits). The k-uniform states are a special kind of entangled (n)-qudits states, such that after tracing out arbitrary (n[Formula: see text]k) subsystems, the remaining (k) subsystems are maximally mixed. We recall some basic elements about unitary phase operators of a multi-qudit system and we give the corresponding separable density matrices. Evolved density matrices arise when qudits of the multipartite system are allowed to interact via an Hamiltonian of Heisenberg type. The expressions of maximally mixed states are explicitly derived from multipartite evolved phase states and their properties are discussed.

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Classification of all alternatives to the Born rule in terms of informational properties

Quantum ◽

10.22331/q-2017-07-14-15 ◽

2017 ◽

Vol 1 ◽

pp. 15 ◽

Cited By ~ 8

Author(s):

Thomas D. Galley ◽

Lluis Masanes

Keyword(s):

Quantum Theory ◽

Two Dimensional ◽

Born Rule ◽

Mixed States ◽

Structure And Dynamics ◽

Finite Dimensional ◽

Information Causality ◽

Pure States ◽

Probabilistic Theories

The standard postulates of quantum theory can be divided into two groups: the first one characterizes the structure and dynamics of pure states, while the second one specifies the structure of measurements and the corresponding probabilities. In this work we keep the first group of postulates and characterize all alternatives to the second group that give rise to finite-dimensional sets of mixed states. We prove a correspondence between all these alternatives and a class of representations of the unitary group. Some features of these probabilistic theories are identical to quantum theory, but there are important differences in others. For example, some theories have three perfectly distinguishable states in a two-dimensional Hilbert space. Others have exotic properties such as lack of bit symmetry, the violation of no simultaneous encoding (a property similar to information causality) and the existence of maximal measurements without phase groups. We also analyze which of these properties single out the Born rule.

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AN EFFICIENT SEPARABILITY CRITERION FOR n-PARTITE ARBITRARILY DIMENSIONAL QUANTUM STATES

International Journal of Quantum Information ◽

10.1142/s0219749911007514 ◽

2011 ◽

Vol 09 (04) ◽

pp. 1101-1112

Author(s):

YINXIANG LONG ◽

DAOWEN QIU ◽

DONGYANG LONG

Keyword(s):

Hilbert Space ◽

Quantum State ◽

Pure State ◽

Coefficient Matrix ◽

Quantum States ◽

Mixed States ◽

Density Matrices ◽

Separability Criterion ◽

Dimensional Hilbert Space ◽

Pure States

In this paper, we obtain an efficient separability criterion for bipartite quantum pure state systems, which is based on the two-order minors of the coefficient matrix corresponding to quantum state. Then, we generalize this criterion to multipartite arbitrarily dimensional pure states. Our criterion is directly built upon coefficient matrices, but not density matrices or observables, so it has the advantage of being computed easily. Indeed, to judge separability for an arbitrary n-partite pure state in a d-dimensional Hilbert space, it only needs at most O(d) times operations of multiplication and comparison. Our criterion can be extended to mixed states. Compared with Yu's criteria, our methods are faster, and can be applied to any quantum state.

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