scholarly journals REVERSIBILITY CONDITIONS FOR QUANTUM OPERATIONS

2012 ◽  
Vol 24 (07) ◽  
pp. 1250016 ◽  
Author(s):  
ANNA JENČOVÁ
Keyword(s):  
Large Class ◽  
Quantum States ◽  
Quantum Operations ◽  
The Given ◽  
Equivalent Conditions ◽  
Nikodym Derivative

We give a list of equivalent conditions for reversibility of the adjoint of a unital Schwarz map, with respect to a set of quantum states. A large class of such conditions is given by preservation of distinguishability measures: F-divergences, L1-distance, quantum Chernoff and Hoeffding distances. Here we summarize and extend the known results. Moreover, we prove a number of conditions in terms of the properties of a quantum Radon–Nikodym derivative and factorization of states in the given set. Finally, we show that reversibility is equivalent to preservation of a large class of quantum Fisher informations and χ2-divergences.

scholarly journals Neural network operations and Susuki–Trotter evolution of neural network states

2018 ◽  
Vol 16 (08) ◽  
pp. 1840008 ◽  
Author(s):  
Nahuel Freitas ◽  
Giovanna Morigi ◽  
Vedran Dunjko
Keyword(s):  
Neural Network ◽  
Optimization Techniques ◽  
Quantum States ◽  
Stochastic Methods ◽  
Many Body ◽  
Boltzmann Machines ◽  
Quantum Operations ◽  
Model Complex ◽  
Network States ◽  
Hidden Nodes

It was recently proposed to leverage the representational power of artificial neural networks, in particular Restricted Boltzmann Machines, in order to model complex quantum states of many-body systems [G. Carleo and M. Troyer, Science 355(6325) (2017) 602.]. States represented in this way, called Neural Network States (NNSs), were shown to display interesting properties like the ability to efficiently capture long-range quantum correlations. However, identifying an optimal neural network representation of a given state might be challenging, and so far this problem has been addressed with stöchastic optimization techniques. In this work, we explore a different direction. We study how the action of elementary quantum operations modifies NNSs. We parametrize a family of many body quantum operations that can be directly applied to states represented by Unrestricted Boltzmann Machines, by just adding hidden nodes and updating the network parameters. We show that this parametrization contains a set of universal quantum gates, from which it follows that the state prepared by any quantum circuit can be expressed as a Neural Network State with a number of hidden nodes that grows linearly with the number of elementary operations in the circuit. This is a powerful representation theorem (which was recently obtained with different methods) but that is not directly useful, since there is no general and efficient way to extract information from this unrestricted description of quantum states. To circumvent this problem, we propose a step-wise procedure based on the projection of Unrestricted quantum states to Restricted quantum states. In turn, two approximate methods to perform this projection are discussed. In this way, we show that it is in principle possible to approximately optimize or evolve Neural Network States without relying on stochastic methods such as Variational Monte Carlo, which are computationally expensive.

scholarly journals On quantum operations as quantum states

2004 ◽  
Vol 311 (1) ◽  
pp. 26-52 ◽  
Author(s):  
Pablo Arrighi ◽  
Christophe Patricot
Keyword(s):  
Quantum States ◽  
Quantum Operations

scholarly journals Representation functions of additive bases for abelian semigroups

2004 ◽  
Vol 2004 (30) ◽  
pp. 1589-1597 ◽  
Author(s):  
Melvyn B. Nathanson
Keyword(s):  
Large Class ◽  
Abelian Semigroup ◽  
Representation Function ◽  
Asymptotic Basis ◽  
Countably Infinite ◽  
The Given

A subset of an abelian semigroup is called an asymptotic basis for the semigroup if every element of the semigroup with at most finitely many exceptions can be represented as the sum of two distinct elements of the basis. The representation function of the basis counts the number of representations of an element of the semigroup as the sum of two distinct elements of the basis. Suppose there is given function from the semigroup into the set of nonnegative integers together with infinity such that this function has only finitely many zeros. It is proved that for a large class of countably infinite abelian semigroups, there exists a basis whose representation function is exactly equal to the given function for every element in the semigroup.

scholarly journals Entangling capacities and the geometry of quantum operations

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Jhih-Yuan Kao ◽  
Chung-Hsien Chou
Keyword(s):  
Quantum Information ◽  
Physical Significance ◽  
Quantum States ◽  
Important Class ◽  
Quantum Operation ◽  
Positive Partial Transpose ◽  
Quantum Operations ◽  
Partial Transpose ◽  
Ppt States

Abstract Quantum operations are the fundamental transformations on quantum states. In this work, we study the relation between entangling capacities of operations, geometry of operations, and positive partial transpose (PPT) states, which are an important class of states in quantum information. We show a method to calculate bounds for entangling capacity, the amount of entanglement that can be produced by a quantum operation, in terms of negativity, a measure of entanglement. The bounds of entangling capacity are found to be associated with how non-PPT (PPT preserving) an operation is. A length that quantifies both entangling capacity/entanglement and PPT-ness of an operation or state can be defined, establishing a geometry characterized by PPT-ness. The distance derived from the length bounds the relative entangling capability, endowing the geometry with more physical significance. We also demonstrate the equivalence of PPT-ness and separability for unitary operations.

scholarly journals Some Generalised Fixed Point Theorems Applied to Quantum Operations

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 759
Author(s):  
Umar Batsari Yusuf ◽  
Poom Kumam ◽  
Sikarin Yoo-Kong
Keyword(s):  
Fixed Point ◽  
Quantum State ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Quantum States ◽  
Contractive Condition ◽  
Quantum Operation ◽  
Quantum Operations ◽  
Finite Dimensional

In this paper, we consider an order-preserving mapping T on a complete partial b-metric space satisfying some contractive condition. We were able to show the existence and uniqueness of the fixed point of T. In the application aspect, the fidelity of quantum states was used to establish the existence of a fixed quantum state associated to an order-preserving quantum operation. The method we presented is an alternative in showing the existence of a fixed quantum state associated to quantum operations. Our method does not capitalise on the commutativity of the quantum effects with the fixed quantum state(s) (Luders’s compatibility criteria). The Luders’s compatibility criteria in higher finite dimensional spaces is rather difficult to check for any prospective fixed quantum state. Some part of our results cover the famous contractive fixed point results of Banach, Kannan and Chatterjea.

scholarly journals Multipartite Circulant States with Positive Partial Transposes

2008 ◽  
Vol 15 (03) ◽  
pp. 189-212 ◽  
Author(s):  
Dariusz Chruściński ◽  
Andrzej Kossakowski
Keyword(s):  
Hilbert Space ◽  
Large Class ◽  
Quantum States ◽  
Direct Sum ◽  
Circular Structure ◽  
The Family ◽  
Direct Sum Decomposition ◽  
New States

We construct a large class of multipartite qudit states which are positive under the family of partial transpositions. The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure and hence generalizes a class of bipartite circulant states proposed recently by the authors. This class contains many well-known examples of multipartite quantum states from the literature and gives rise to a huge family of completely new states.

ENTANGLEMENT OF FORMATION FOR A CLASS OF SPECIAL QUANTUM STATES

2007 ◽  
Vol 05 (03) ◽  
pp. 343-352 ◽  
Author(s):  
HUI ZHAO ◽  
ZHI-XI WANG
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Special Kind ◽  
Quantum States ◽  
The Other ◽  
High Dimensional ◽  
Entangled States ◽  
Mixed States ◽  
Density Matrices ◽  
Entanglement Of Formation

The entanglement of formation for a class of high-dimensional quantum mixed states is investigated. A special kind of D-computable states is defined and the lower bound of entanglement of formation for a large class of density matrices whose decompositions lie in these D-computable quantum states is obtained. Moreover we present a kind of construction for this special state which is defined by a class of special matrices with two non-zero different eigenvalues and the other eigenvalues are zero. Making use of the D-computable we construct a class of bound entangled states.

scholarly journals SPECTRAL SELF-AFFINE MEASURES IN $\mathbb{R}^N$

2007 ◽  
Vol 50 (1) ◽  
pp. 197-215 ◽  
Author(s):  
Jian-Lin Li
Keyword(s):  
Large Class ◽  
Sierpinski Gasket ◽  
Three Dimensional ◽  
Iterated Function Systems ◽  
Sierpiński Gasket ◽  
Spectral Pair ◽  
Iterated Function ◽  
Spectral Pairs ◽  
The Given ◽  
Check Condition

AbstractThe aim of this paper is to investigate and study the possible spectral pair $(\mu_{M,D},\varLambda(M,S))$ associated with the iterated function systems $\{\phi_{d}(x)= M^{-1}(x+d)\}_{d\in D}$ and $\{\psi_{s}(x)=M^{\ast}x+s\}_{s\in S}$ in $\mathbb{R}^n$. For a large class of self-affine measures $\mu_{M,D}$, we obtain an easy check condition for $\varLambda(M,S)$ not to be a spectrum, and answer a question of whether we have such a spectral pair $(\mu_{M,D},\varLambda(M,S))$ in the Eiffel Tower or three-dimensional Sierpinski gasket. Further generalization of the given condition as well as some elementary properties of compatible pairs and spectral pairs are discussed. Finally, we give several interesting examples to illustrate the spectral pair conditions considered here.

scholarly journals Operator and Graph Theoretic Techniques for Distinguishing Quantum States via One-Way LOCC

2021 ◽  
Vol 11 (20) ◽  
pp. 9542
Author(s):  
David W. Kribs ◽  
Comfort Mintah ◽  
Michael Nathanson ◽  
Rajesh Pereira
Keyword(s):  
Operator Theory ◽  
Operator Algebras ◽  
Quantum States ◽  
Classical Communication ◽  
Quantum Operations ◽  
Graph Theoretic ◽  
Local Quantum ◽  
First Time ◽  
Theoretic Description ◽  
Theory Operator

We bring together in one place some of the main results and applications from our recent work on quantum information theory, in which we have brought techniques from operator theory, operator algebras, and graph theory for the first time to investigate the topic of distinguishability of sets of quantum states in quantum communication, with particular reference to the framework of one-way local quantum operations and classical communication (LOCC). We also derive a new graph-theoretic description of distinguishability in the case of a single-qubit sender.

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