scholarly journals Convex decomposition of dimension-altering quantum channels

2016 ◽  
Vol 14 (08) ◽  
pp. 1650045 ◽  
Author(s):  
Dong-Sheng Wang
Keyword(s):  
Quantum Channel ◽  
Convex Set ◽  
Quantum Circuit ◽  
Quantum Channels ◽  
Completely Positive ◽  
Decomposition Problem ◽  
Convex Decomposition ◽  
Decomposition Scheme ◽  
Low Dimensional ◽  
Channel Decomposition

Quantum channels, which are completely positive and trace preserving mappings, can alter the dimension of a system, e.g. a quantum channel from a qubit to a qutrit. We study the convex set properties of dimension-altering quantum channels, and particularly the channel decomposition problem in terms of convex sum of extreme channels. We provide various quantum circuit representations of extreme and generalized extreme channels, which can be employed in an optimization to approximately decompose an arbitrary channel. Numerical simulations of low-dimensional channels are performed to demonstrate our channel decomposition scheme.

scholarly journals Explicit Construction of Optimal Witnesses for Input-Output Correlations Attainable by Quantum Channels

2020 ◽  
Vol 27 (04) ◽  
pp. 2050017
Author(s):  
Michele Dall’Arno ◽  
Sarah Brandsen ◽  
Francesco Buscemi
Keyword(s):  
Quantum Channel ◽  
Explicit Construction ◽  
Quantum Channels ◽  
Input Output ◽  
Noisy Channels ◽  
Completely Positive ◽  
Communication Games ◽  
Single Use ◽  
Classical Quantum ◽  
Communication Resource

Given a quantum channel — that is, a completely positive trace-preserving linear map — as the only communication resource available between two parties, we consider the problem of characterizing the set of classical noisy channels that can be obtained from it by means of suitable classical-quantum encodings and quantum-classical decodings, respectively, on the sender’s and the receiver’s side. We consider various classes of linear witnesses and compute their optimum values in closed form for several classes of quantum channels. The witnesses that we consider here are formulated as communication games, in which Alice’s aim is to exploit a single use of a given quantum channel to help Bob guess some information she has received from an external referee.

scholarly journals Necessary and sufficient conditions on measurements of quantum channels

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190832 ◽  
Author(s):  
John Burniston ◽  
Michael Grabowecky ◽  
Carlo Maria Scandolo ◽  
Giulio Chiribella ◽  
Gilad Gour
Keyword(s):  
Quantum Measurement ◽  
Quantum Channel ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Necessary And Sufficient Conditions ◽  
Quantum Channels ◽  
Completely Positive ◽  
Principle Of Causality ◽  
Necessary And Sufficient

Quantum supermaps are a higher-order genera- lization of quantum maps, taking quantum maps to quantum maps. It is known that any completely positive and trace non-increasing (CPTNI) map can be performed as part of a quantum measurement. By providing an explicit counterexample we show that, instead, not every quantum supermap sending a quantum channel to a CPTNI map can be realized in a measurement on quantum channels. We find that the supermaps that can be implemented in this way are exactly those transforming quantum channels into CPTNI maps even when tensored with the identity supermap. We link this result to the fact that the principle of causality fails in the theory of quantum supermaps.

scholarly journals Probabilistic Resumable Quantum Teleportation of a Two-Qubit Entangled State

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 352 ◽  
Author(s):  
Zhan-Yun Wang ◽  
Yi-Tao Gou ◽  
Jin-Xing Hou ◽  
Li-Ke Cao ◽  
Xiao-Hui Wang
Keyword(s):  
Quantum Channel ◽  
Quantum Teleportation ◽  
The State ◽  
Entangled States ◽  
Entangled State ◽  
Quantum Channels ◽  
Unknown State ◽  
Optimal Probability

We explicitly present a generalized quantum teleportation of a two-qubit entangled state protocol, which uses two pairs of partially entangled particles as quantum channel. We verify that the optimal probability of successful teleportation is determined by the smallest superposition coefficient of these partially entangled particles. However, the two-qubit entangled state to be teleported will be destroyed if teleportation fails. To solve this problem, we show a more sophisticated probabilistic resumable quantum teleportation scheme of a two-qubit entangled state, where the state to be teleported can be recovered by the sender when teleportation fails. Thus the information of the unknown state is retained during the process. Accordingly, we can repeat the teleportion process as many times as one has available quantum channels. Therefore, the quantum channels with weak entanglement can also be used to teleport unknown two-qubit entangled states successfully with a high number of repetitions, and for channels with strong entanglement only a small number of repetitions are required to guarantee successful teleportation.

Performance characterization of Pauli channels assisted by indefinite causal order and post-measurement

2020 ◽  
Vol 20 (15&16) ◽  
pp. 1261-1280
Author(s):  
Francisco Delgado ◽  
Carlos Cardoso-Isidoro
Keyword(s):  
Quantum Channel ◽  
Communication Channel ◽  
Quantum Communication ◽  
Quantum Channels ◽  
Causal Order ◽  
Combined Process ◽  
Performance Characterization ◽  
Output State ◽  
Transmission Of Information

Indefinite causal order has introduced disruptive procedures to improve the fidelity of quantum communication by introducing the superposition of { orders} on a set of quantum channels. It has been applied to several well characterized quantum channels as depolarizing, dephasing and teleportation. This work analyses the behavior of a parametric quantum channel for single qubits expressed in the form of Pauli channels. Combinatorics lets to obtain affordable formulas for the analysis of the output state of the channel when it goes through a certain imperfect quantum communication channel when it is deployed as a redundant application of it under indefinite causal order. In addition, the process exploits post-measurement on the associated control to select certain components of transmission. Then, the fidelity of such outputs is analysed to characterize the generic channel in terms of its parameters. As a result, we get notable enhancement in the transmission of information for well characterized channels due to the combined process: indefinite causal order plus post-measurement.

Solution to time-energy costs of quantum channels

2015 ◽  
Vol 15 (7&8) ◽  
pp. 685-693
Author(s):  
Chi-Hang F. Fung ◽  
H. F. Chau ◽  
Chi-Kwong Li ◽  
Nung-Sing Sze
Keyword(s):  
Lower Bound ◽  
Semidefinite Programming ◽  
Energy Cost ◽  
Quantum Channel ◽  
Positive Semidefinite ◽  
Energy Costs ◽  
Quantum Channels ◽  
General Quantum ◽  
Special Cases

We derive a formula for the time-energy costs of general quantum channels proposed in [Phys. Rev. A {\bf 88}, 012307 (2013)]. This formula allows us to numerically find the time-energy cost of any quantum channel using positive semidefinite programming. We also derive a lower bound to the time-energy cost for any channels and the exact the time-energy cost for a class of channels which includes the qudit depolarizing channels and projector channels as special cases.

On Completely Positive Maps Defined by an Irreducible Correspondence

1990 ◽  
Vol 33 (4) ◽  
pp. 434-441 ◽  
Author(s):  
C. Anantharaman-Delaroche
Keyword(s):  
Von Neumann Algebras ◽  
Convex Set ◽  
Extreme Points ◽  
Completely Positive Maps ◽  
Complex Matrices ◽  
Completely Positive ◽  
Von Neumann ◽  
Positive Maps

AbstractCompletely positive maps defined by an irreducible correspondence between two von Neumann algebras M and N are introduced. We give results about their structure and characterize, among them, those which are extreme points in the convex set of all unital completely positive maps from M to N. As particular cases we obtain known results of M. D. Choi [4] on completely positive maps between complex matrices and of J. A. Mingo [8] on inner completely positive maps.

scholarly journals Quantum Generative Adversarial Networks for learning and loading random distributions

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Christa Zoufal ◽  
Aurélien Lucchi ◽  
Stefan Woerner
Keyword(s):  
Quantum State ◽  
Quantum Channel ◽  
Quantum Algorithms ◽  
Probability Distributions ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Simulation ◽  
Quantum States ◽  
Generative Adversarial Networks ◽  
Adversarial Networks

AbstractQuantum algorithms have the potential to outperform their classical counterparts in a variety of tasks. The realization of the advantage often requires the ability to load classical data efficiently into quantum states. However, the best known methods require $${\mathcal{O}}\left({2}^{n}\right)$$O2n gates to load an exact representation of a generic data structure into an $$n$$n-qubit state. This scaling can easily predominate the complexity of a quantum algorithm and, thereby, impair potential quantum advantage. Our work presents a hybrid quantum-classical algorithm for efficient, approximate quantum state loading. More precisely, we use quantum Generative Adversarial Networks (qGANs) to facilitate efficient learning and loading of generic probability distributions - implicitly given by data samples - into quantum states. Through the interplay of a quantum channel, such as a variational quantum circuit, and a classical neural network, the qGAN can learn a representation of the probability distribution underlying the data samples and load it into a quantum state. The loading requires $${\mathcal{O}}\left(poly\left(n\right)\right)$$Opolyn gates and can thus enable the use of potentially advantageous quantum algorithms, such as Quantum Amplitude Estimation. We implement the qGAN distribution learning and loading method with Qiskit and test it using a quantum simulation as well as actual quantum processors provided by the IBM Q Experience. Furthermore, we employ quantum simulation to demonstrate the use of the trained quantum channel in a quantum finance application.

The extreme points of SU(2)-irreducibly covariant channels

2014 ◽  
Vol 25 (06) ◽  
pp. 1450048 ◽  
Author(s):  
M. Al Nuwairan
Keyword(s):  
Extreme Points ◽  
Quantum Channels ◽  
Completely Positive ◽  
Positive Map ◽  
Covariant Quantum ◽  
Kraus Representation ◽  
Dual Map ◽  
Complementary Channel

In this paper, we introduce EPOSIC channels, a class of SU(2)-covariant quantum channels. For each of them, we give a Kraus representation, its Choi matrix, a complementary channel, and its dual map. We show that they are the extreme points of all SU(2)-irreducibly covariant channels. As an application of these channels, we get an example of a positive map that is not completely positive.

scholarly journals Infinite dimensional generalizations of Choi’s Theorem

2019 ◽  
Vol 7 (1) ◽  
pp. 67-77
Author(s):  
Shmuel Friedland
Keyword(s):  
Hilbert Spaces ◽  
Quantum Channel ◽  
Trace Class ◽  
Natural Generalization ◽  
Linear Operators ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Bounded Linear ◽  
Positive Map ◽  
Finite Dimensional

Abstract In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

QUANTUM CHANNELS OF THE QUTRIT STATE TELEPORTATION

2011 ◽  
Vol 09 (03) ◽  
pp. 893-901 ◽  
Author(s):  
XIU-LAO TIAN ◽  
GUO-FANG SHI ◽  
Yong ZHAO
Keyword(s):  
Quantum Information ◽  
Quantum System ◽  
Quantum Channel ◽  
Sufficient Condition ◽  
Quantum Channels ◽  
Necessary And Sufficient Condition ◽  
Measurement Matrix ◽  
Parameter Matrix ◽  
Channel Parameter ◽  
Necessary And Sufficient

Qudit quantum system can carry more information than that of qubit, the teleportation of qudit state has significance in quantum information task. We propose a method to teleport a general qutrit state (three-level state) and discuss the necessary and sufficient condition for realizing a successful and perfect teleportation, which is determined by the measurement matrix Tα and the quantum channel parameter matrix (CPM) X. By using this method, we study the channels of two-qutrit state and three-qutrit state teleportation.

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