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.

Quantum Subspace Measurements

2007 ◽  
Vol 14 (01) ◽  
pp. 117-126 ◽  
Author(s):  
Neal G. Anderson
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Upper Bound ◽  
Quantum Measurements ◽  
Positive Operators ◽  
Accessible Information ◽  
Measured System ◽  
Characteristic Features

Fundamental studies of quantum measurements and their capacity to acquire information are typically based on scenarios in which the full Hilbert space of the measured quantum system is open to measurement interactions. In this work, we consider a class of incomplete quantum measurements — quantum subspace measurements (QSM's) — for which all measurement interactions are restricted to an arbitrary but specified subspace of the measured system Hilbert space. We define QSM's formally through a condition on the measurement Hamiltonian, obtain forms for the post-measurement states and positive operators (POVM elements) associated with QSM's acting in a specified subspace, and upper bound the accessible information for such measurements. Characteristic features of QSM's are identified and discussed.

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 A universal scheme for robust self-testing in the prepare-and-measure scenario

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 424
Author(s):  
Nikolai Miklin ◽  
Michał Oszmaniec
Keyword(s):  
Hilbert Space ◽  
Quantum States ◽  
Practical Application ◽  
Quantum Devices ◽  
Self Testing ◽  
Pure States ◽  
Target States ◽  
Self Test ◽  
Universal Scheme ◽  
Projective Measurements

We consider the problem of certification of arbitrary ensembles of pure states and projective measurements solely from the experimental statistics in the prepare-and-measure scenario assuming the upper bound on the dimension of the Hilbert space. To this aim, we propose a universal and intuitive scheme based on establishing perfect correlations between target states and suitably-chosen projective measurements. The method works in all finite dimensions and allows for robust certification of the overlaps between arbitrary preparation states and between the corresponding measurement operators. Finally, we prove that for qubits, our technique can be used to robustly self-test arbitrary configurations of pure quantum states and projective measurements. These results pave the way towards the practical application of the prepare-and-measure paradigm to certification of quantum devices.

scholarly journals RESTRICTIONS ON INFORMATION TRANSFER IN QUANTUM MEASUREMENTS

2007 ◽  
Vol 05 (01n02) ◽  
pp. 279-285 ◽  
Author(s):  
S. MAYBUROV
Keyword(s):  
Hilbert Space ◽  
Information System ◽  
Quantum System ◽  
Information Transfer ◽  
Quantum Measurements ◽  
The Individual

Information-theoretical restrictions on the information transfer in quantum measurements are studied for practical systems. For the measurement of quantum system S by information system O such restrictions are described by a formalism of inference maps in Hilbert space, the resulting O restricted states ξO calculated from the agreement with Schrödinger S, O dynamics. It is shown that the principal S information losses stipulate the stochasticity of measurement outcomes; consequently ξO describes the random "pointer" outcomes qj observed by O in the individual events.

Quantum measurements and entropic bounds

2006 ◽  
Vol 6 (1) ◽  
pp. 16-45
Author(s):  
A. Barchielli ◽  
G. Lupieri
Keyword(s):  
Quantum Measurement ◽  
Positive Operator ◽  
Upper Bound ◽  
Quantum Channel ◽  
Quantum Measurements ◽  
A Posteriori ◽  
Classical Information ◽  
Positive Operator Valued Measure

While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the monotonicity theorem for relative entropies many bounds on the classical information extracted in a quantum measurement are obtained in a unified manner. In particular, it is shown that such bounds can all be stated as inequalities between mutual entropies. This approach based on channels gives rise to a unified picture of known and new bounds on the classical information (the bounds by Holevo, by Shumacher, Westmoreland and Wootters, by Hall, by Scutaru, a new upper bound and a new lower one). Some examples clarify the mutual relationships among the various bounds.

scholarly journals Ergodicity probes: using time-fluctuations to measure the Hilbert space dimension

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 207
Author(s):  
Charlie Nation ◽  
Diego Porras
Keyword(s):  
Hilbert Space ◽  
Space Dimension ◽  
Quantum Systems ◽  
Quantum States ◽  
Quantum Computers ◽  
Quantum Devices ◽  
Fluctuation Dissipation ◽  
Fluctuation Dissipation Theorem ◽  
Quantum Device ◽  
Meaningful Measure

Quantum devices, such as quantum simulators, quantum annealers, and quantum computers, may be exploited to solve problems beyond what is tractable with classical computers. This may be achieved as the Hilbert space available to perform such `calculations' is far larger than that which may be classically simulated. In practice, however, quantum devices have imperfections, which may limit the accessibility to the whole Hilbert space. We thus determine that the dimension of the space of quantum states that are available to a quantum device is a meaningful measure of its functionality, though unfortunately this quantity cannot be directly experimentally determined. Here we outline an experimentally realisable approach to obtaining the required Hilbert space dimension of such a device to compute its time evolution, by exploiting the thermalization dynamics of a probe qubit. This is achieved by obtaining a fluctuation-dissipation theorem for high-temperature chaotic quantum systems, which facilitates the extraction of information on the Hilbert space dimension via measurements of the decay rate, and time-fluctuations.

scholarly journals Quantum system compression: A Hamiltonian guided walk through Hilbert space

2021 ◽  
Vol 103 (1) ◽  
Author(s):  
Robert L. Kosut ◽  
Tak-San Ho ◽  
Herschel Rabitz
Keyword(s):  
Hilbert Space ◽  
Quantum System
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