scholarly journals Physical Implementability of Linear Maps and Its Application in Error Mitigation

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 600
Author(s):  
Jiaqing Jiang ◽  
Kun Wang ◽  
Xin Wang
Keyword(s):  
General Linear ◽  
Local Error ◽  
Decomposition Technique ◽  
Completely Positive ◽  
Linear Map ◽  
Quantum Operations ◽  
Error Mitigation ◽  
Linear Maps ◽  
Analytic Expressions ◽  
Sampling Cost

Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.

scholarly journals Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 522
Author(s):  
Bartosz Regula ◽  
Ryuji Takagi ◽  
Mile Gu
Keyword(s):  
Quantum Dynamics ◽  
Distance Measure ◽  
Quantum States ◽  
Quantum Channels ◽  
Linear Combinations ◽  
Operational Approach ◽  
Completely Positive ◽  
Error Mitigation ◽  
Linear Maps ◽  
The Cost

Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.

Computing stabilized norms for quantum operations

2009 ◽  
Vol 9 (1&2) ◽  
pp. 16-36
Author(s):  
N. Johnston ◽  
D.W. Kribs ◽  
V.I. Paulsen
Keyword(s):  
Quantum Information ◽  
Information Science ◽  
Distance Measures ◽  
Quantum Information Science ◽  
Linear Map ◽  
Quantum Operations ◽  
Linear Maps

The diamond and completely bounded norms for linear maps play an increasingly important role in quantum information science, providing fundamental stabilized distance measures for differences of quantum operations. We give a brief introduction to the theory of completely bounded maps. Based on this theory, we formulate an algorithm to compute the norm of an arbitrary linear map. We present an implementation of the algorithm via MATLAB, discuss its efficiency, and consider the case of differences of unitary maps.

scholarly journals A Simple Proof of the Jamiołkowski Criterion for Complete Positivity of Linear Maps

2005 ◽  
Vol 12 (01) ◽  
pp. 55-64 ◽  
Author(s):  
D. Salgado ◽  
J. L. Sánchez-Gómez ◽  
M. Ferrero
Keyword(s):  
Simple Proof ◽  
Direct Proof ◽  
Complete Positivity ◽  
Systematic Method ◽  
Completely Positive ◽  
Linear Map ◽  
Matrix Algebras ◽  
Linear Maps ◽  
Kraus Decomposition

We give a simple direct proof of the Jamiołkowski criterion to check whether a linear map between matrix algebras is completely positive or not. This proof is more accessible for physicists than other ones found in the literature and provides a systematic method to give any set of Kraus matrices of the Kraus decomposition.

scholarly journals Extreme n-positive linear maps

1993 ◽  
Vol 36 (1) ◽  
pp. 123-131 ◽  
Author(s):  
Sze-Kai Tsui
Keyword(s):  
General Procedure ◽  
Irreducible Representations ◽  
Completely Positive ◽  
Linear Map ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Positive Linear Map ◽  
Positive Linear Maps

In this article we prove that if a completely positive linear map Φ of a unital C*-algebra A into another B with only finite dimensional irreducible representations is pure, then we have NΦ = Φker + kerΦ, where NΦ={x∈A|Φ(x) = 0}, Φker = {x∈A|Φ(x*x) = 0}, and kerΦ={x∈A|Φ(xx*) = 0}. We also prove that for every unital strongly positive and n-positive linear map Φ of a C*-algebra A onto another B with n≧2, if NΦ = Φker + kerΦ, then Φ is extreme in Pn(A, B, IB). By this null-kernel condition, many new extreme n-positive linear maps are identified. A general procedure for constructing extreme n-positive linear maps is suggested and discussed.

scholarly journals How Many Inflections are There in the Lyapunov Spectrum?

2021 ◽  
Author(s):  
O. Jenkinson ◽  
M. Pollicott ◽  
P. Vytnova
Keyword(s):  
Piecewise Linear ◽  
Lyapunov Spectrum ◽  
Stat Phys ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Expanding Map ◽  
Lyapunov Spectra ◽  
Linear Maps ◽  
Inflection Points ◽  
Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

Positive Linear Maps on C*-Algebras

1972 ◽  
Vol 24 (3) ◽  
pp. 520-529 ◽  
Author(s):  
Man-Duen Choi
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Complex Matrices ◽  
Completely Positive ◽  
C Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

scholarly journals Linear maps on *-algebras acting on orthogonal elements like derivations or anti-derivations

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4543-4554 ◽  
Author(s):  
H. Ghahramani ◽  
Z. Pan
Keyword(s):  
Linear Map ◽  
Unital Algebra ◽  
C Algebras ◽  
Orthogonality Conditions ◽  
Linear Maps ◽  
Unital Simple

Let U be a unital *-algebra and ? : U ? U be a linear map behaving like a derivation or an anti-derivation at the following orthogonality conditions on elements of U: xy = 0, xy* = 0, xy = yx = 0 and xy* = y*x = 0. We characterize the map ? when U is a zero product determined algebra. Special characterizations are obtained when our results are applied to properly infinite W*-algebras and unital simple C*-algebras with a non-trivial idempotent.

scholarly journals On the surjectivity of linear maps on locally convex spaces

1982 ◽  
Vol 32 (2) ◽  
pp. 187-211 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Basic Notion ◽  
Linear Map ◽  
Linear Maps ◽  
Locally Convex ◽  
Do So ◽  
Convex Spaces

AbstractThe aim of this note is to investigate the structure of general surjectivity problem for a continuous linear map between locally convex spaces. We shall do so by using the method introduced in Yamamuro (1980). Its basic notion is that of calibrations which has been introduced in Yamamuro (1975), studied in detail in Yamamuro (1979) and appliced to several problems in Yamamuro (1978) and Yamamuro (1979a).

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