Mixing doubly stochastic quantum channels with the completely depolarizing channel

2009 ◽  
Vol 9 (5&6) ◽  
pp. 406-413
Author(s):  
J. Watrous
Keyword(s):  
Quantum Channel ◽  
Convex Combination ◽  
Quantum Channels ◽  
Doubly Stochastic ◽  
Depolarizing Channel

It is proved that every doubly stochastic quantum channel that is properly averaged with the completely depolarizing channel can be written as a convex combination of unitary channels. It follows that within the space of doubly stochastic quantum channels, there is a ball with positive radius around the completely depolarizing channel within which all channels are convex combinations of unitary channels.

scholarly journals Relating Entropies of Quantum Channels

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1028
Author(s):  
Dariusz Kurzyk ◽  
Łukasz Pawela ◽  
Zbigniew Puchała
Keyword(s):  
Relative Entropy ◽  
Upper Bound ◽  
Quantum Channel ◽  
Von Neumann Entropy ◽  
Quantum Channels ◽  
Von Neumann ◽  
Channel One ◽  
Depolarizing Channel

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi–Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.

scholarly journals Depolarizing behavior of quantum channels in higher dimensions

2011 ◽  
Vol 11 (5&6) ◽  
pp. 466-484
Author(s):  
Easwar Magesan
Keyword(s):  
Quantum Channel ◽  
Higher Dimensions ◽  
Uniqueness Result ◽  
Quantum Gate ◽  
Distance Measures ◽  
Independent Interest ◽  
Quantum Channels ◽  
Quantum Operations ◽  
Pure States ◽  
Depolarizing Channel

The paper analyzes the behavior of quantum channels, particularly in large dimensions, by proving various properties of the quantum gate fidelity. Many of these properties are of independent interest in the theory of distance measures on quantum operations. A non-uniqueness result for the gate fidelity is proven, a consequence of which is the existence of non-depolarizing channels that produce a constant gate fidelity on pure states. Asymptotically, the gate fidelity associated with any quantum channel is shown to converge to that of a depolarizing channel. Methods for estimating the minimum of the gate fidelity are also presented.

scholarly journals Detecting mixed-unitary quantum channels is NP-hard

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 253
Author(s):  
Colin Do-Yan Lee ◽  
John Watrous
Keyword(s):  
Polynomial Time ◽  
Quantum Channel ◽  
Convex Combination ◽  
Np Hard ◽  
Quantum Channels

A quantum channel is said to be a mixed-unitary channel if it can be expressed as a convex combination of unitary channels. We prove that, given the Choi representation of a quantum channel Φ, it is NP-hard with respect to polynomial-time Turing reductions to determine whether or not Φ is a mixed-unitary channel. This hardness result holds even under the assumption that Φ is not within an inverse-polynomial distance (in the dimension of the space upon which Φ acts) of the boundary of the mixed-unitary channels.

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.

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.

scholarly journals Quantum key distribution with mismatched measurements over arbitrary channels

2017 ◽  
Vol 17 (3&4) ◽  
pp. 209-241
Author(s):  
Walter O. Krawec
Keyword(s):  
Quantum Channel ◽  
Quantum Key Distribution ◽  
Key Distribution ◽  
Rate Equations ◽  
Quantum Channels ◽  
Multiple Channel ◽  
Symmetric Quantum ◽  
Channel Statistics ◽  
Quantum Protocol ◽  
Measurement Bases

In this paper, we derive key-rate expressions for different quantum key distribution protocols. Our key-rate equations utilize multiple channel statistics, including those gathered from mismatched measurement bases - i.e., when Alice and Bob choose incompatible bases. In particular, we will consider an Extended B92 and a two-way semi-quantum protocol. For both these protocols, we demonstrate that their tolerance to noise is higher than previously thought - in fact, we will show the semi-quantum protocol can actually tolerate the same noise level as the fully quantum BB84 protocol. Along the way, we will also consider an optimal QKD protocol for various quantum channels. Finally, all the key-rate expressions which we derive in this paper are applicable to any arbitrary, not necessarily symmetric, quantum channel.

On the Extreme Rays of the Metric Cone

1980 ◽  
Vol 32 (1) ◽  
pp. 126-144 ◽  
Author(s):  
David Avis
Keyword(s):  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Convex Polyhedra ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Image Position ◽  
Polyhedral Convex Set ◽  
Extreme Rays

A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equationsBirkhoff [2] and Von Neuman have shown that the extreme points of this bounded polyhedron are just the n × n permutation matrices. The importance of this result for mathematical programming is that it tells us that the maximum of any linear form over P will occur for a permutation matrix X.

Quantum Polarization Codes for Capacity-Achieving in Discrete Memoryless Quantum Channel

2010 ◽  
Vol 44-47 ◽  
pp. 2978-2982
Author(s):  
Da Zu Huang ◽  
Zhi Gang Chen ◽  
Xin Li ◽  
Ying Guo
Keyword(s):  
Quantum Channel ◽  
Quantum Channels ◽  
Channel Codes ◽  
Quantum Polarization ◽  
Channel Polarization

Quantum channel combining and splitting, called quantum channel polarization, is suggested to design qubit sequences that achieve the symmetric capacity for any given discrete memoryless quantum channels. The polarized quantum channels can be well-conditioned for quantum channel codes, through which one need to send data at rate 1 by employing quantum channels with capacity near 1 and at rate 0 by employing the remaining quantum channels.

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