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 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.

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.

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

scholarly journals Coexistency on Hilbert Space Effect Algebras and a Characterisation of Its Symmetry Transformations

2020 ◽  
Vol 379 (3) ◽  
pp. 1077-1112 ◽  
Author(s):  
György Pál Gehér ◽  
Peter Šemrl
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Mathematical Structure ◽  
Quantum Measurements ◽  
The Other ◽  
Effect Algebras ◽  
Natural Question ◽  
Fuzzy Event ◽  
Space Effect ◽  
Symmetry Transformations

Abstract The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper’s first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig’s theorem. As a byproduct of our methods we also strengthen a theorem of Molnár.

A note on nth roots of positive operators

1969 ◽  
Vol 66 (1) ◽  
pp. 27-30
Author(s):  
John H. Jowett
Keyword(s):  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Spectral Representation ◽  
Bounded Operator ◽  
Positive Operators

The existence and uniqueness of a positive self-adjoint nth. root of a positive, self-adjoint, not necessarily bounded operator on a Hilbert Space H can be readily demonstrated using the spectral representation of the transformation.

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