scholarly journals Completing bases in four dimensions

2022 ◽  
Author(s):  
Hans Havlicek ◽  
Karl Svozil
Keyword(s):  
Hilbert Space ◽  
Quantum States ◽  
Four Dimensions ◽  
Separable States ◽  
Orthogonal Representation ◽  
Relational Properties ◽  
Dimensional Hilbert Space ◽  
Physical Entanglement

Abstract Criteria for the completion of an incomplete basis of, or context in, a four-dimensional Hilbert space by (in)decomposable vectors are given. This, in particular, has consequences for the task of ``completing'' one or more bases or contexts of a (hyper)graph: find a complete faithful orthogonal representation (aka coordinatization) of a hypergraph when only a coordinatization of the intertwining observables is known. In general indecomposability and thus physical entanglement and the encoding of relational properties by quantum states ``prevails'' and occurs more often than separability associated with well defined individual, separable states.

Quantumness, generalized spherical 2-design, and symmetric informationally complete POVM

2007 ◽  
Vol 7 (8) ◽  
pp. 730-737
Author(s):  
I.H. Kim
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Equivalence Relation ◽  
Open Problem ◽  
Natural Generalization ◽  
Quantum States ◽  
Minimal Set ◽  
Dimensional Hilbert Space

Fuchs and Sasaki defined the quantumness of a set of quantum states in \cite{Quantumness}, which is related to the fidelity loss in transmission of the quantum states through a classical channel. In \cite{Fuchs}, Fuchs showed that in $d$-dimensional Hilbert space, minimum quantumness is $\frac{2}{d+1}$, and this can be achieved by all rays in the space. He left an open problem, asking whether fewer than $d^2$ states can achieve this bound. Recently, in a different context, Scott introduced a concept of generalized $t$-design in \cite{GenSphet}, which is a natural generalization of spherical $t$-design. In this paper, we show that the lower bound on the quantumness can be achieved if and only if the states form a generalized 2-design. As a corollary, we show that this bound can be only achieved if the number of states are larger or equal to $d^2$, answering the open problem. Furthermore, we also show that the minimal set of such ensemble is Symmetric Informationally Complete POVM(SIC-POVM). This leads to an equivalence relation between SIC-POVM and minimal set of ensemble achieving minimal quantumness.

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

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.

Simplifying Schmidt number witnesses via higher-dimensional embeddings

2004 ◽  
Vol 4 (3) ◽  
pp. 207-221
Author(s):  
F. Hulpke ◽  
D. Bruss ◽  
M. Levenstein ◽  
A. Sanpera
Keyword(s):  
Hilbert Space ◽  
Schmidt Number ◽  
Convex Sets ◽  
Entanglement Witness ◽  
Simple Method ◽  
Dimensional Hilbert Space ◽  
Higher Dimensional ◽  
Arbitrary Convex

We apply the generalised concept of witness operators to arbitrary convex sets, and review the criteria for the optimisation of these general witnesses. We then define an embedding of state vectors and operators into a higher-dimensional Hilbert space. This embedding leads to a connection between any Schmidt number witness in the original Hilbert space and a witness for Schmidt number two (i.e. the most general entanglement witness) in the appropriate enlarged Hilbert space. Using this relation we arrive at a conceptually simple method for the construction of Schmidt number witnesses in bipartite systems.

Optimal ambiguous discrimination between states given by two projections

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Krzysztof Kaniowski
Keyword(s):  
Hilbert Space ◽  
Quantum States ◽  
Prior Probabilities ◽  
Optimal Measurements

AbstractLet P 0 and P 1 be projections in a Hilbert space H. We shall construct a class of optimal measurements for the problem of discrimination between quantum states $$\rho _i = \tfrac{1} {{\dim P_i }}P_i$$, with prior probabilities π 0 and π 1. The probabilities of failure for such measurements will also be derived.

scholarly journals EXTENSION ALGEBRAS OF CUNTZ ALGEBRA, II

2009 ◽  
Vol 80 (1) ◽  
pp. 83-90 ◽  
Author(s):  
SHUDONG LIU ◽  
XIAOCHUN FANG
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Cuntz Algebra ◽  
Infinite Dimensional ◽  
Algebra Ii ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
K Theory ◽  
Extension Algebra

AbstractIn this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

scholarly journals Partial automorphisms of stable C*-algebras and Hilbert C*-bimodules

2005 ◽  
Vol 79 (3) ◽  
pp. 391-398
Author(s):  
Kazunori Kodaka
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Equivalence Classes ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Countably Infinite

AbstractLet A be a C*-algebra and K the C*-algebra of all compact operators on a countably infinite dimensional Hilbert space. In this note, we shall show that there is an isomorphism of a semigroup of equivalence classes of certain partial automorphisms of A ⊗ K onto a semigroup of equivalence classes of certain countably generated A-A-Hilbert bimodules.

The Set of Finite Operators is Nowhere Dense

1989 ◽  
Vol 32 (3) ◽  
pp. 320-326 ◽  
Author(s):  
Domingo A. Herrero
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Dense Subset ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractA bounded linear operator A on a complex, separable, infinite dimensional Hilbert space is called finite if for each . It is shown that the class of all finite operators is a closed nowhere dense subset of

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