Optimal ambiguous discrimination between states given by two projections

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Krzysztof Kaniowski
Keyword(s):  
Hilbert Space ◽  
Quantum States ◽  
Prior Probabilities ◽  
Optimal Measurements

AbstractLet P 0 and P 1 be projections in a Hilbert space H. We shall construct a class of optimal measurements for the problem of discrimination between quantum states $$\rho _i = \tfrac{1} {{\dim P_i }}P_i$$, with prior probabilities π 0 and π 1. The probabilities of failure for such measurements will also be derived.

Quantumness, generalized spherical 2-design, and symmetric informationally complete POVM

2007 ◽  
Vol 7 (8) ◽  
pp. 730-737
Author(s):  
I.H. Kim
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Equivalence Relation ◽  
Open Problem ◽  
Natural Generalization ◽  
Quantum States ◽  
Minimal Set ◽  
Dimensional Hilbert Space

Fuchs and Sasaki defined the quantumness of a set of quantum states in \cite{Quantumness}, which is related to the fidelity loss in transmission of the quantum states through a classical channel. In \cite{Fuchs}, Fuchs showed that in $d$-dimensional Hilbert space, minimum quantumness is $\frac{2}{d+1}$, and this can be achieved by all rays in the space. He left an open problem, asking whether fewer than $d^2$ states can achieve this bound. Recently, in a different context, Scott introduced a concept of generalized $t$-design in \cite{GenSphet}, which is a natural generalization of spherical $t$-design. In this paper, we show that the lower bound on the quantumness can be achieved if and only if the states form a generalized 2-design. As a corollary, we show that this bound can be only achieved if the number of states are larger or equal to $d^2$, answering the open problem. Furthermore, we also show that the minimal set of such ensemble is Symmetric Informationally Complete POVM(SIC-POVM). This leads to an equivalence relation between SIC-POVM and minimal set of ensemble achieving minimal quantumness.

Coherent manipulations of atoms using laser light

2008 ◽  
Vol 58 (3) ◽  
Author(s):  
Bruce Shore
Keyword(s):  
Hilbert Space ◽  
Pulse Propagation ◽  
Laser Light ◽  
Equations Of Motion ◽  
Pulse Sequences ◽  
Three Dimensional ◽  
Analytic Solutions ◽  
Quantum States ◽  
Excitation Energies ◽  
Mathematical Concepts

Coherent manipulations of atoms using laser lightThe internal structure of a particle - an atom or other quantum system in which the excitation energies are discrete - undergoes change when exposed to pulses of near-resonant laser light. This tutorial review presents basic concepts of quantum states, of laser radiation and of the Hilbert-space statevector that provides the theoretical portrait of probability amplitudes - the tools for quantifying quantum properties not only of individual atoms and molecules but also of artificial atoms and other quantum systems. It discusses the equations of motion that describe the laser-induced changes (coherent excitation), and gives examples of laser-pulse effects, with particular emphasis on two-state and three-state adiabatic time evolution within the rotating-wave approximation. It provides pictorial descriptions of excitation based on the Bloch equations that allow visualization of two-state excitation as motion of a three-dimensional vector (the Bloch vector). Other visualization techniques allow portrayal of more elaborate systems, particularly the Hilbert-space motion of adiabatic states subject to various pulse sequences. Various more general multilevel systems receive treatment that includes degeneracies, chains and loop linkages. The concluding sections discuss techniques for creating arbitrary pre-assigned quantum states, for manipulating them into alternative coherent superpositions and for analyzing an unknown superposition. Appendices review some basic mathematical concepts and provide further details of the theoretical formalism, including photons, pulse propagation, statistical averages, analytic solutions to the equations of motion, exact solutions of periodic Hamiltonians, and population-trapping "dark" states.

scholarly journals Collapse dynamics and Hilbert-space stochastic processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Daniele Bajoni ◽  
Oreste Nicrosini ◽  
Alberto Rimini ◽  
Simone Rodini
Keyword(s):  
Hilbert Space ◽  
Beam Splitter ◽  
Measurement Problem ◽  
Single Photon ◽  
Numerical Approach ◽  
Quantum States ◽  
Photon Detector ◽  
State Vector Reduction ◽  
Collapse Models ◽  
Physical Features

AbstractSpontaneous collapse models of state vector reduction represent a possible solution to the quantum measurement problem. In the present paper we focus our attention on the Ghirardi–Rimini–Weber (GRW) theory and the corresponding continuous localisation models in the form of a Brownian-driven motion in Hilbert space. We consider experimental setups in which a single photon hits a beam splitter and is subsequently detected by photon detector(s), generating a superposition of photon-detector quantum states. Through a numerical approach we study the dependence of collapse times on the physical features of the superposition generated, including also the effect of a finite reaction time of the measuring apparatus. We find that collapse dynamics is sensitive to the number of detectors and the physical properties of the photon-detector quantum states superposition.

scholarly journals Minimum Time for the Evolution to a Nonorthogonal Quantum State and Upper Bound of the Geometric Efficiency of Quantum Evolutions

2021 ◽  
Vol 3 (3) ◽  
pp. 444-457
Author(s):  
Carlo Cafaro ◽  
Paul M. Alsing
Keyword(s):  
Hilbert Space ◽  
Maximum Energy ◽  
Quantum States ◽  
Minimum Time ◽  
Quantum Evolution ◽  
Efficiency Measure ◽  
Geometric Framework ◽  
Constant Energy ◽  
Projective Hilbert Space ◽  
Arbitrary Quantum

We present a simple proof of the fact that the minimum time TAB for quantum evolution between two arbitrary states A and B equals TAB=ℏcos−1A|B/ΔE with ΔE being the constant energy uncertainty of the system. This proof is performed in the absence of any geometrical arguments. Then, being in the geometric framework of quantum evolutions based upon the geometry of the projective Hilbert space, we discuss the roles played by either minimum-time or maximum-energy uncertainty concepts in defining a geometric efficiency measure ε of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the inequality ε≤1 even when the system only passes through nonorthogonal quantum states.

scholarly journals Completing bases in four dimensions

2022 ◽  
Author(s):  
Hans Havlicek ◽  
Karl Svozil
Keyword(s):  
Hilbert Space ◽  
Quantum States ◽  
Four Dimensions ◽  
Separable States ◽  
Orthogonal Representation ◽  
Relational Properties ◽  
Dimensional Hilbert Space ◽  
Physical Entanglement

Abstract Criteria for the completion of an incomplete basis of, or context in, a four-dimensional Hilbert space by (in)decomposable vectors are given. This, in particular, has consequences for the task of ``completing'' one or more bases or contexts of a (hyper)graph: find a complete faithful orthogonal representation (aka coordinatization) of a hypergraph when only a coordinatization of the intertwining observables is known. In general indecomposability and thus physical entanglement and the encoding of relational properties by quantum states ``prevails'' and occurs more often than separability associated with well defined individual, separable states.

scholarly journals Quantum States of Hierarchic Systems

2003 ◽  
Vol 01 (02) ◽  
pp. 269-278 ◽  
Author(s):  
Mikhail V. Altaisky
Keyword(s):  
Hilbert Space ◽  
Density Matrix ◽  
Degrees Of Freedom ◽  
Quantum Systems ◽  
Quantum States ◽  
Matrix Formalism ◽  
Density Matrix Formalism ◽  
Theory Of Measurements ◽  
Novel Method ◽  
Chemical And Biological Systems

The density matrix formalism which is widely used in the theory of measurements, quantum computing, quantum description of chemical and biological systems always implies the averaging over all states of the environment. In practice this is impossible because the environment of the system is the complement of this system to the whole Universe and contains infinitely many degrees of freedom. A novel method of construction density matrix which implies the averaging only over the direct environment is proposed. The Hilbert space of state vectors for the hierarchic quantum systems is constructed.

scholarly journals Obscure Qubits and Membership Amplitudes

2021 ◽  
Author(s):  
Steven Duplij ◽  
Raimund Vogl
Keyword(s):  
Hilbert Space ◽  
Quantum Computing ◽  
Density Matrix ◽  
Tensor Product ◽  
Quantum Computation ◽  
Membership Function ◽  
Quantum Probability ◽  
Quantum States ◽  
Born Rule ◽  
Quantum Amplitude

We propose a concept of quantum computing which incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), in a natural way by introducing new entities, obscure qudits (e.g. obscure qubits), which are characterized simultaneously by a quantum probability and by a membership function. To achieve this, a membership amplitude for quantum states is introduced alongside the quantum amplitude. The Born rule is used for the quantum probability only, while the membership function can be computed from the membership amplitudes according to a chosen model. Two different versions of this approach are given here: the “product” obscure qubit, where the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the “Kronecker” obscure qubit, where quantum and vagueness computations are to be performed independently (i.e. quantum computation alongside truth evaluation). The latter is called a double obscure-quantum computation. In this case, the measurement becomes mixed in the quantum and obscure amplitudes, while the density matrix is not idempotent. The obscure-quantum gates act not in the tensor product of spaces, but in the direct product of quantum Hilbert space and so called membership space which are of different natures and properties. The concept of double (obscure-quantum) entanglement is introduced, and vector and scalar concurrences are proposed, with some examples being given.

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