scholarly journals Minimal sufficient positive-operator valued measure on a separable Hilbert space

2015 ◽  
Vol 56 (10) ◽  
pp. 102205 ◽  
Author(s):  
Yui Kuramochi
Keyword(s):  
Hilbert Space ◽  
Positive Operator ◽  
Separable Hilbert Space ◽  
Positive Operator Valued Measure

scholarly journals Self-testing nonprojective quantum measurements in prepare-and-measure experiments

2020 ◽  
Vol 6 (16) ◽  
pp. eaaw6664 ◽  
Author(s):  
Armin Tavakoli ◽  
Massimiliano Smania ◽  
Tamás Vértesi ◽  
Nicolas Brunner ◽  
Mohamed Bourennane
Keyword(s):  
Experimental Data ◽  
Hilbert Space ◽  
Quantum System ◽  
Positive Operator ◽  
Space Dimension ◽  
Quantum Measurements ◽  
Quantum Devices ◽  
Self Testing ◽  
Robust To Noise ◽  
Positive Operator Valued Measure

Self-testing represents the strongest form of certification of a quantum system. Here, we theoretically and experimentally investigate self-testing of nonprojective quantum measurements. That is, how can one certify, from observed data only, that an uncharacterized measurement device implements a desired nonprojective positive-operator valued measure (POVM). We consider a prepare-and-measure scenario with a bound on the Hilbert space dimension and develop methods for (i) robustly self-testing extremal qubit POVMs and (ii) certifying that an uncharacterized qubit measurement is nonprojective. Our methods are robust to noise and thus applicable in practice, as we demonstrate in a photonic experiment. Specifically, we show that our experimental data imply that the implemented measurements are very close to certain ideal three- and four-outcome qubit POVMs and hence non-projective. In the latter case, the data certify a genuine four-outcome qubit POVM. Our results open interesting perspective for semi–device-independent certification of quantum devices.

PROBABILITY STRUCTURES FOR QUANTUM STATE SPACES

1995 ◽  
Vol 07 (07) ◽  
pp. 1105-1121 ◽  
Author(s):  
PAUL BUSCH ◽  
GIANNI CASSINELLI ◽  
PEKKA J. LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Quantum State ◽  
Positive Operator ◽  
Trace Class ◽  
Measurable Space ◽  
Probability Measures ◽  
State Spaces ◽  
Trace Class Operators ◽  
Positive Operator Valued Measure

The theme of this paper is to represent the states of a quantum system by means of probability measures. We fix a positive operator valued measure E on a measurable space (Ω, ℬ(Ω)) acting in a Hilbert space ℋ, and we study the properties of the mapping that it induces from the set of trace class operators on ℋ to the set of measures on (Ω, ℬ(Ω)). In particular, the injectivity and the surjectivity of this map are characterised in terms of the properties of E.

scholarly journals The Frame Potential, on Average

2009 ◽  
Vol 16 (02n03) ◽  
pp. 145-156 ◽  
Author(s):  
Ingemar Bengtsson ◽  
Helena Granström
Keyword(s):  
Hilbert Space ◽  
Heisenberg Group ◽  
Positive Operator ◽  
Global Minimum ◽  
Dimensional Hilbert Space ◽  
Clifford Group ◽  
Frame Potential ◽  
Positive Operator Valued Measure ◽  
Unit Vectors

A Symmetric Informationally Complete Positive Operator Valued Measure (SIC) consists of N2 equiangular unit vectors in an N dimensional Hilbert space. The frame potential is a function of N2 unit vectors. It has a unique global minimum if the vectors form a SIC, and this property has been made use of in numerical searches for SICs. When the vectors form an orbit of the Weyl-Heisenberg group the frame potential becomes a function of a single fiducial vector. We analytically compute the average of this function over Hilbert space. We also compute averages when the fiducial vector is placed in certain special subspaces defined by the Clifford group.

scholarly journals Absorption in Invariant Domains for Semigroups of Quantum Channels

2021 ◽  
Author(s):  
Raffaella Carbone ◽  
Federico Girotti
Keyword(s):  
Hilbert Space ◽  
Ergodic Theory ◽  
Markov Processes ◽  
Separable Hilbert Space ◽  
Quantum Channels ◽  
Quantum Markov Processes ◽  
Positive Recurrent ◽  
Markov Evolution ◽  
Absorption Probabilities ◽  
Invariant Domains

AbstractWe introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

scholarly journals Hilbert space valued Gabor frames in weighted amalgam spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 377-394
Author(s):  
Anirudha Poria ◽  
Jitendriya Swain
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor Frames ◽  
Frame Operator ◽  
Wiener Amalgam Space ◽  
Amalgam Spaces ◽  
Special Case

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

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