Restricted numerical range: A versatile tool in the theory of quantum information

Gawron, Piotr; Puchała, Zbigniew; Miszczak, Jarosław Adam; Skowronek, Łukasz; Życzkowski, Karol

doi:10.1063/1.3496901

Cited by 43 publications

(52 citation statements)

References 59 publications

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“…where the infimum is taken over all states τ A such that I A ⊗ τ A is invertible on the support of Ω, and Λ ⊗ is the product numerical range [54,55], defined as…”

Section: Appendix E: Proof Of Theoremmentioning

confidence: 99%

Test One to Test Many: A Unified Approach to Quantum Benchmarks

Bai

Chiribella

2018

Phys. Rev. Lett.

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Quantum benchmarks are routinely used to validate the experimental demonstration of quantum information protocols. Many relevant protocols, however, involve an infinite set of input states, of which only a finite subset can be used to test the quality of the implementation. This is a problem, because the benchmark for the finitely many states used in the test can be higher than the original benchmark calculated for infinitely many states. This situation arises in the teleportation and storage of coherent states, for which the benchmark of 50% fidelity is commonly used in experiments, although finite sets of coherent states normally lead to higher benchmarks. Here, we show that the average fidelity over all coherent states can be indirectly probed with a single setup, requiring only two-mode squeezing, a 50-50 beam splitter, and homodyne detection. Our setup enables a rigorous experimental validation of quantum teleportation, storage, amplification, attenuation, and purification of noisy coherent states. More generally, we prove that every quantum benchmark can be tested by preparing a single entangled state and measuring a single observable.

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“…where the infimum is taken over all states τ A such that I A ⊗ τ A is invertible on the support of Ω, and Λ ⊗ is the product numerical range [54,55], defined as…”

Section: Appendix E: Proof Of Theoremmentioning

confidence: 99%

Test One to Test Many: A Unified Approach to Quantum Benchmarks

Bai

Chiribella

2018

Phys. Rev. Lett.

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“…Due to these and many other such properties, numerical ranges and related objects have found numerous applications in diverse areas including differential equations, numerical analysis, and quantum computing, see for example [5,28,34,36,50,51,60]. As the topic of numerical ranges is both natural and useful, it has been extensively studied and the current body of research is quite vast.…”

Section: Introductionmentioning

confidence: 99%

Numerical range and compressions of the shift

Bickel¹,

Gorkin²

2020

Complex Analysis and Spectral Theory

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The numerical range of a bounded, linear operator on a Hilbert space is a set in C that encodes important information about the operator. In this survey paper, we first consider numerical ranges of matrices and discuss several connections with envelopes of families of curves. We then turn to the shift operator, perhaps the most important operator on the Hardy space H 2 (D), and compressions of the shift operator to model spaces, i.e. spaces of the form H 2 θH 2 where θ is inner. For these compressions of the shift operator, we provide a survey of results on the connection between their numerical ranges and the numerical ranges of their unitary dilations. We also discuss related results for compressed shift operators on the bidisk associated to rational inner functions and conclude the paper with a brief discussion of the Crouzeix conjecture. Primary 47A12; Secondary 47A13, 30C15 Date: October 30, 2018.

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“…And each H i can be written in the Pauli basis as (27) With this parameterization, our joint product numerical range of H 1 , H 2 , H 3 becomes the set of points in R 3 given by (f 1 (r, s, t), f 2 (r, s, t), f 3 (r, s, t)) ,…”

Section: The Symmetric Case and Bosonic Systemsmentioning

confidence: 99%

“…This simplification to only separable states then allows us to study the geometry of 2-RDMs with a mathematical concept, called joint product numerical range [22][23][24][25][26][27], denoted by Π, of the Hamiltonian interaction terms. Π includes all the extreme points of the three-dimensional projections of 2-RDMs, and the projection itself, denoted by Θ, is a convex hull of Π.…”

Section: Introductionmentioning

confidence: 99%

Joint product numerical range and geometry of reduced density matrices

Chen

Guo

et al. 2016

Sci. China Phys. Mech. Astron.

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The reduced density matrices of a many-body quantum system form a convex set, whose threedimensional projection Θ is convex in R 3 . The boundary ∂Θ of Θ may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces: symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti's theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range Π of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that ruled surface emerge naturally when taking a convex hull of Π. We show that, a ruled surface on ∂Θ sitting in Π has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of Θ, with two boundary pieces of symmetry breaking origin separated by two gapless lines.

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Restricted numerical range: A versatile tool in the theory of quantum information

Cited by 43 publications

References 59 publications

Test One to Test Many: A Unified Approach to Quantum Benchmarks

Test One to Test Many: A Unified Approach to Quantum Benchmarks

Numerical range and compressions of the shift

Joint product numerical range and geometry of reduced density matrices

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