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.

A NEW PROOF OF NONEXISTENCE OF PROBABILISTIC QUANTUM CLONING MACHINE

2007 ◽  
Vol 05 (04) ◽  
pp. 611-616
Author(s):  
SAMIR KUNKRI ◽  
SUJIT K. CHOUDHARY
Keyword(s):  
Hilbert Space ◽  
Simple Proof ◽  
Causality Principle ◽  
Quantum Cloning ◽  
Dimensional Hilbert Space ◽  
Higher Dimensional ◽  
Probabilistic Quantum Cloning

We give a simple proof of the impossibility of probabilistic exact 1 → 2 cloning for any three different states of a qubit. The simplicity of the proof is due to the use of a stronger version of no violation of the causality principle. The proof is also extended to higher-dimensional Hilbert space, but for a special ensemble of states.

EXACT REMOTE STATE PREPARATION FOR MULTIPARTIES USING DARK STATES

2003 ◽  
Vol 01 (03) ◽  
pp. 301-319 ◽  
Author(s):  
P. AGRAWAL ◽  
P. PARASHAR ◽  
A. K. PATI
Keyword(s):  
Hilbert Space ◽  
Higher Dimensions ◽  
Remote State Preparation ◽  
Space Systems ◽  
Quantum Channels ◽  
Classical Communication ◽  
State Preparation ◽  
Dimensional Hilbert Space ◽  
Higher Dimensional

We discuss the exact remote state preparation (RSP) protocol of special ensembles of qubits at multiple locations. Generalization of this protocol for higher dimensional Hilbert space systems for multiparties is also presented. Using the "dark states", for multiparties in higher dimensions as quantum channels, we show several instances of remote state preparation protocol using multiparticle measurement and classical communication. We find that not all dark states can be used for exact remote state preparation, nevertheless any superposition of dark states can be used for exact RSP in a probabilistic manner.

scholarly journals The distance between two convex sets in Hilbert space

2021 ◽  
Author(s):  
Yuri C. Sagala ◽  
Susilo Hariyanto ◽  
Y. D. Sumanto ◽  
Titi Udjiani
Keyword(s):  
Hilbert Space ◽  
Convex Sets

scholarly journals Remote operations and interactions for systems of arbitrary-dimensional Hilbert space: State-operator approach

2002 ◽  
Vol 65 (3) ◽  
Author(s):  
Benni Reznik ◽  
Yakir Aharonov ◽  
Berry Groisman
Keyword(s):  
Hilbert Space ◽  
Operator Approach ◽  
State Operator ◽  
Dimensional Hilbert Space ◽  
Space State ◽  
Remote Operations

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 On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

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.

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