scholarly journals Trade-off relations for operation entropy of complementary quantum channels

2019 ◽  
Vol 17 (05) ◽  
pp. 1950046
Author(s):  
Jakub Czartowski ◽  
Daniel Braun ◽  
Karol Życzkowski
Keyword(s):  
Quantum Channel ◽  
Lower Boundary ◽  
Von Neumann Entropy ◽  
Input State ◽  
Quantum Channels ◽  
Quantum Operation ◽  
Trade Off ◽  
Von Neumann ◽  
State Of The Environment ◽  
Complementary Channel

The entropy of a quantum operation, defined as the von Neumann entropy of the corresponding Choi–Jamiołkowski state, characterizes the coupling of the principal system with the environment. For any quantum channel acting on a state of a given size, one defines the complementary channel, which sends the input state into the state of the environment after the operation. Making use of subadditivity of entropy, we show that for any dimension the sum of both entropies is bounded from below. This result characterizes the trade-off between the information on the initial quantum state accessible to the principal system and the information leaking to the environment. For one qubit maps we describe the interpolating family of depolarizing maps, for which the sum of both entropies gives the lower boundary of the region allowed in the space spanned by both entropies.

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 On Entangled Information and Quantum Capacity

2001 ◽  
Vol 08 (01) ◽  
pp. 1-18 ◽  
Author(s):  
Viacheslav P. Belavkin
Keyword(s):  
Quantum Channel ◽  
Simple Algebra ◽  
Von Neumann Entropy ◽  
Entangled States ◽  
Input State ◽  
Von Neumann ◽  
Quantum Capacity ◽  
Different Types ◽  
Mixed Compound ◽  
True Quantum

The pure quantum entanglement is generalized to the case of mixed compound states to include the classical and quantum encodings as particular cases. The true quantum entanglements are characterized as transpose-CP but not CP maps. The entangled information is introduced as the relative entropy of the mutual and the input state and total information of the entangled states leads to two different types of entropy for a given quantum state: the von Neumann entropy, which is achieved as the supremum of the information over all c-entanglements, and the true quantum entropy, which is achieved at the standard entanglement. The q-capacity, defined as the supremum over all entanglements, doubles the c-capacity in the case of the simple algebra. The conditional q-entropy is positive, and q-information of a quantum channel is additive.

LIMITATION ON THE ACCESSIBLE INFORMATION FOR QUANTUM CHANNELS WITH INEFFICIENT MEASUREMENTS

2005 ◽  
Vol 03 (supp01) ◽  
pp. 87-95
Author(s):  
KURT JACOBS
Keyword(s):  
Quantum System ◽  
Von Neumann Entropy ◽  
Quantum Channels ◽  
Initial State ◽  
Von Neumann ◽  
Classical Information ◽  
Quantum Operations ◽  
Amount Of Information ◽  
Accessible Information

To transmit classical information using a quantum system, the sender prepares the system in one of a set of possible states and sends it to the receiver. The receiver then makes a measurement on the system to obtain information about the senders choice of state. The amount of information which is accessible to the receiver depends upon the encoding and the measurement. Here we derive a bound on this information which generalizes the bound derived by Schumacher, Westmoreland and Wootters [Schumacher, Westmoreland and Wootters, Phys. Rev. Lett. 76, 3452 (1996)] to include inefficient measurements, and thus all quantum operations. This also allows us to obtain a generalization of a bound derived by Hall [Hall, Phys. Rev. A 55, 100 (1997)], and to show that the average reduction in the von Neumann entropy which accompanies a measurement is concave in the initial state, for all quantum operations.

scholarly journals ENTROPIC CHARACTERIZATION OF QUANTUM OPERATIONS

2011 ◽  
Vol 09 (04) ◽  
pp. 1031-1045 ◽  
Author(s):  
WOJCIECH ROGA ◽  
KAROL ŻYCZKOWSKI ◽  
MARK FANNES
Keyword(s):  
Von Neumann Entropy ◽  
Input State ◽  
Dimensional System ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Output Entropy ◽  
The Individual ◽  
Depolarizing Channel ◽  
Minimal Output Entropy

We investigate decoherence induced by a quantum channel in terms of minimal output entropy and map entropy. The latter is the von Neumann entropy of the Jamiołkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1 acting on a finite dimensional system, there exists a class of channels Φ2 sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1 ⊗ Ψ2 holds.

scholarly journals On the optimal certification of von Neumann measurements

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Paulina Lewandowska ◽  
Aleksandra Krawiec ◽  
Ryszard Kukulski ◽  
Łukasz Pawela ◽  
Zbigniew Puchała
Keyword(s):  
Numerical Range ◽  
Statistical Significance ◽  
Measurement Procedure ◽  
Input State ◽  
Single Element ◽  
Single Shot ◽  
Quantum Channels ◽  
Von Neumann ◽  
Alternative Hypotheses ◽  
Von Neumann Measurements

AbstractIn this report we study certification of quantum measurements, which can be viewed as the extension of quantum hypotheses testing. This extension involves also the study of the input state and the measurement procedure. Here, we will be interested in two-point (binary) certification scheme in which the null and alternative hypotheses are single element sets. Our goal is to minimize the probability of the type II error given some fixed statistical significance. In this report, we begin with studying the two-point certification of pure quantum states and unitary channels to later use them to prove our main result, which is the certification of von Neumann measurements in single-shot and parallel scenarios. From our main result follow the conditions when two pure states, unitary operations and von Neumann measurements cannot be distinguished perfectly but still can be certified with a given statistical significance. Moreover, we show the connection between the certification of quantum channels or von Neumann measurements and the notion of q-numerical range.

scholarly journals Local extrema of entropy functions under tensor products

2011 ◽  
Vol 11 (11&12) ◽  
pp. 1028-1044
Author(s):  
Shmuel Friedland ◽  
Gilad Gour ◽  
Aidan Roy
Keyword(s):  
Quantum Channel ◽  
Tensor Products ◽  
Von Neumann Entropy ◽  
Local Minima ◽  
Von Neumann ◽  
Local Commutativity ◽  
Local Extrema ◽  
Dimension 2 ◽  
Entropy Functions

We show that under a certain condition of local commutativity the minimum von-Neumann entropy output of a quantum channel is locally additive. We also show that local minima of the 2-norm entropy functions are closed under tensor products if one of the subspaces has dimension 2.

Classical capacity of a noiseless quantum channel assisted by noisy entanglement

2001 ◽  
Vol 1 (3) ◽  
pp. 70-78
Author(s):  
M Horodecki ◽  
P Horodecki ◽  
R l Horodecki ◽  
D Leung ◽  
B Terha
Keyword(s):  
Mutual Information ◽  
General Formula ◽  
Quantum Channel ◽  
Classical Case ◽  
Von Neumann Entropy ◽  
Entangled States ◽  
Von Neumann ◽  
Classical Capacity ◽  
Quantum Mutual Information

We derive the general formula for the capacity of a noiseless quantum channel assisted by an arbitrary amount of noisy entanglement. In this capacity formula, the ratio of the quantum mutual information and the von Neumann entropy of the sender's share of the noisy entanglement plays the role of mutual information in the completely classical case. A consequence of our results is that bound entangled states cannot increase the capacity of a noiseless quantum channel.

scholarly journals Islands in linear dilaton black holes

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Georgios K. Karananas ◽  
Alex Kehagias ◽  
John Taskas
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Entanglement Entropy ◽  
Von Neumann Entropy ◽  
Two Dimensional ◽  
Extremal Case ◽  
Von Neumann ◽  
Dimensional Black Hole ◽  
Linear Dilaton ◽  
Linear Dilaton Black Hole

Abstract We derive a novel four-dimensional black hole with planar horizon that asymptotes to the linear dilaton background. The usual growth of its entanglement entropy before Page’s time is established. After that, emergent islands modify to a large extent the entropy, which becomes finite and is saturated by its Bekenstein-Hawking value in accordance with the finiteness of the von Neumann entropy of eternal black holes. We demonstrate that viewed from the string frame, our solution is the two-dimensional Witten black hole with two additional free bosons. We generalize our findings by considering a general class of linear dilaton black hole solutions at a generic point along the σ-model renormalization group (RG) equations. For those, we observe that the entanglement entropy is “running” i.e. it is changing along the RG flow with respect to the two-dimensional worldsheet length scale. At any fixed moment before Page’s time the aforementioned entropy increases towards the infrared (IR) domain, whereas the presence of islands leads the running entropy to decrease towards the IR at later times. Finally, we present a four-dimensional charged black hole that asymptotes to the linear dilaton background as well. We compute the associated entanglement entropy for the extremal case and we find that an island is needed in order for it to follow the Page curve.

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