scholarly journals Defining quantum divergences via convex optimization

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 387
Author(s):  
Hamza Fawzi ◽  
Omar Fawzi
Keyword(s):  
Convex Optimization ◽  
Semidefinite Programming ◽  
Quantum Channel ◽  
Upper Bounds ◽  
Chain Rule ◽  
Quantum Channels ◽  
Optimization Program ◽  
Rényi Divergence ◽  
Strong Converse ◽  
A Chain

We introduce a new quantum Rényi divergence Dα# for α∈(1,∞) defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum Rényi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched α-Rényi divergence between quantum channels for α>1. Second it allows us to prove a chain rule property for the sandwiched α-Rényi divergence for α>1 which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.

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.

scholarly journals Geometric Rényi Divergence and its Applications in Quantum Channel Capacities

2021 ◽  
Author(s):  
Kun Fang ◽  
Hamza Fawzi
Keyword(s):  
Quantum Information ◽  
Quantum Channel ◽  
Distance Measure ◽  
Point Of View ◽  
Information Measures ◽  
Rényi Divergence ◽  
A Chain ◽  
The Right ◽  
Almost All ◽  
Open Question

AbstractHaving a distance measure between quantum states satisfying the right properties is of fundamental importance in all areas of quantum information. In this work, we present a systematic study of the geometric Rényi divergence (GRD), also known as the maximal Rényi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties, which are not satisfied by other quantum Rényi divergences. For example we prove a chain rule inequality that immediately implies the “amortization collapse” for the geometric Rényi divergence, addressing an open question by Berta et al. [Letters in Mathematical Physics 110:2277–2336, 2020, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric Rényi divergence, sharpening the previously best-known bounds based on the max-relative entropy while still keeping the new bounds single-letter and efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.

scholarly journals Fundamental limitations on distillation of quantum channel resources

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Bartosz Regula ◽  
Ryuji Takagi
Keyword(s):  
Lower Bounds ◽  
Quantum Channel ◽  
Quantum Communication ◽  
Fault Tolerant ◽  
State Of The Art ◽  
Quantum Systems ◽  
Noisy Channels ◽  
General Transformation ◽  
Fundamental Limitations ◽  
Strong Converse

AbstractQuantum channels underlie the dynamics of quantum systems, but in many practical settings it is the channels themselves that require processing. We establish universal limitations on the processing of both quantum states and channels, expressed in the form of no-go theorems and quantitative bounds for the manipulation of general quantum channel resources under the most general transformation protocols. Focusing on the class of distillation tasks — which can be understood either as the purification of noisy channels into unitary ones, or the extraction of state-based resources from channels — we develop fundamental restrictions on the error incurred in such transformations, and comprehensive lower bounds for the overhead of any distillation protocol. In the asymptotic setting, our results yield broadly applicable bounds for rates of distillation. We demonstrate our results through applications to fault-tolerant quantum computation, where we obtain state-of-the-art lower bounds for the overhead cost of magic state distillation, as well as to quantum communication, where we recover a number of strong converse bounds for quantum channel capacity.

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.

A chain rule for multivariate divided differences

2010 ◽  
Vol 50 (3) ◽  
pp. 577-586 ◽  
Author(s):  
Michael S. Floater
Keyword(s):  
Chain Rule ◽  
Divided Differences ◽  
A Chain

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.

A Chain Rule for Use with Determinants and Permutations

1947 ◽  
Vol 31 (297) ◽  
pp. 279 ◽  
Author(s):  
N. Chater ◽  
W. J. Chater
Keyword(s):  
Chain Rule ◽  
A Chain
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