scholarly journals Optimal quantum tomography with constrained measurements arising from unitary bases

2021 ◽  
pp. 2130005
Author(s):  
S. Chaturvedi ◽  
S. Ghosh ◽  
K. R. Parthasarathy ◽  
Ajit Iqbal Singh
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Measurement Techniques ◽  
Quantum Tomography ◽  
Galois Field ◽  
Latin Squares ◽  
Quantum Mechanical ◽  
Dimension Formula ◽  
Rank One ◽  
Projective Representations

The purpose of this paper is to introduce techniques of obtaining optimal ways to determine a [Formula: see text]-level quantum state or distinguish such states. It entails designing constrained elementary measurements extracted from maximal abelian subsets of a unitary basis [Formula: see text] for the operator algebra [Formula: see text] of a Hilbert space [Formula: see text] of finite dimension [Formula: see text] or, after choosing an orthonormal basis for [Formula: see text], for the ⋆-algebra [Formula: see text] of complex matrices of order [Formula: see text]. Illustrations are given for the techniques. It is shown that the Schwinger basis [Formula: see text] of unitary operators can give for [Formula: see text], a product of primes [Formula: see text] and [Formula: see text], the ideal number [Formula: see text] of rank one projectors that have a few quantum mechanical overlaps (or, for that matter, a few angles between the corresponding unit vectors). Finally, we give a combination of the tensor product and constrained elementary measurement techniques to deal with all [Formula: see text], though with more overlaps or angles depending on the factorization of [Formula: see text] as a product of primes or their powers like [Formula: see text] with [Formula: see text], all primes, [Formula: see text] for [Formula: see text], or other types. A comparison is drawn for different forms of unitary bases for the Hilbert space factors of the tensor product like [Formula: see text] or [Formula: see text], where [Formula: see text] is the Galois field of size [Formula: see text] and [Formula: see text] is the ring of integers modulo [Formula: see text]. Even though as Hilbert spaces they are isomorphic, but quantum mechanical system-wise, these tensor products are different. In the process, we also study the equivalence relation on unitary bases defined by R. F. Werner [J. Phys. A: Math. Gen. 34 (2001) 7081–7094], connect it to local operations on maximally entangled vectors bases, find an invariant for equivalence classes in terms of certain commuting systems, called fan representations, and, relate it to mutually unbiased bases and Hadamard matrices. Illustrations are given in the context of Latin squares and projective representations as well.

scholarly journals Nonlinear Generalisation of Quantum Mechanics

2021 ◽  
Author(s):  
Alireza Jamali
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Lorentz Invariance ◽  
Current Theory ◽  
Quantum Mechanical ◽  
Dynamical Condition ◽  
Current Definition ◽  
Klein Gordon ◽  
Definition Of ◽  
Mechanical Momentum

It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.

Quantum Interference

2019 ◽  
pp. 66-88
Author(s):  
Jeffrey A. Barrett
Keyword(s):  
Hilbert Space ◽  
Wave Function ◽  
Quantum Mechanics ◽  
Quantum Interference ◽  
Standard Theory ◽  
Quantum Decoherence ◽  
Entangled States ◽  
Quantum Mechanical ◽  
Standard Formulation ◽  
Complex Valued

Moving to more subtle experiments, we consider how the standard formulation of quantum mechanics predicts and explains interference phenomena. Tracking the conditions under which one observes interference phenomena leads to the notion of quantum decoherence. We see why one must sharply distinguish between collapse phenomena and decoherence phenomena on the standard formulation of quantum mechanics. While collapses explain determinate measurement records, environmental decoherence just produces more complex, entangled states where the physical systems involved lack ordinary physical properties. We characterize the quantum-mechanical wave function as both an element of a Hilbert space and a complex-valued function over a configuration space. We also discuss how the wave function is interpreted in the standard theory.

scholarly journals Mutually unbiased unitary bases of operators on d-dimensional hilbert space

2020 ◽  
Vol 18 (01) ◽  
pp. 1941026 ◽  
Author(s):  
Rinie N. M. Nasir ◽  
Jesni Shamsul Shaari ◽  
Stefano Mancini
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Unitary Operator ◽  
Dimensional Subspace ◽  
Mutually Unbiased Bases ◽  
Unitary Operators ◽  
Dimension Formula ◽  
Dimensional Hilbert Space ◽  
Prime Dimension

Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].

scholarly journals When do Projections Commute?

1980 ◽  
Vol 35 (4) ◽  
pp. 437-441 ◽  
Author(s):  
W. Rehder
Keyword(s):  
Hilbert Space ◽  
Conditional Probability ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantum Mechanical ◽  
Probability Operator ◽  
Quantum Mechanical Systems ◽  
Necessary And Sufficient ◽  
New Criteria

Abstract Necessary and sufficient conditions for commutativity of two projections in Hilbert space are given through properties of so-called conditional connectives which are derived from the conditional probability operator PQP. This approach unifies most of the known proofs, provides a few new criteria, and permits certain suggestive interpretations for compound properties of quantum-mechanical systems.

scholarly journals THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED

2019 ◽  
Vol 7 ◽  
Author(s):  
WILLIAM SLOFSTRA
Keyword(s):  
Hilbert Space ◽  
Linear System ◽  
Tensor Product ◽  
Quantum Correlations ◽  
Embedding Theorem ◽  
Product Strategy ◽  
Finite Dimensional ◽  
Universal Embedding ◽  
Quantum Strategies ◽  
Finite Dimensional Hilbert Space

We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.

scholarly journals Groupoids and Coherent States

2019 ◽  
Vol 26 (04) ◽  
pp. 1950017 ◽  
Author(s):  
F. di Cosmo ◽  
A. Ibort ◽  
G. Marmo
Keyword(s):  
Hilbert Space ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Quantum Mechanical ◽  
Natural Setting ◽  
Invariant Subset ◽  
Quantum Mechanical System ◽  
Generalized Coherent States ◽  
Invertible Elements ◽  
Groupoid Algebra

Schwinger’s algebra of selective measurements has a natural interpretation in terms of groupoids. This approach is pushed forward in this paper to show that the theory of coherent states has a natural setting in the framework of groupoids. Thus given a quantum mechanical system with associated Hilbert space determined by a representation of a groupoid, it is shown that any invariant subset of the group of invertible elements in the groupoid algebra determines a family of generalized coherent states provided that a completeness condition is satisfied. The standard coherent states for the harmonic oscillator as well as generalized coherent states for f-oscillators are exemplified in this picture.

scholarly journals A COMPLETELY ENTANGLED SUBSPACE OF MAXIMAL DIMENSION

2006 ◽  
Vol 04 (02) ◽  
pp. 325-330 ◽  
Author(s):  
B. V. RAJARAMA BHAT
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Product Formula ◽  
Maximal Dimension ◽  
Maximum Dimension ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Minimum Dimension ◽  
Orthogonal Product ◽  
Product Vector

Consider a tensor product [Formula: see text] of finite-dimensional Hilbert spaces with dimension [Formula: see text], 1 ≤ i ≤ k. Then the maximum dimension possible for a subspace of [Formula: see text] with no non-zero product vector is known to be d1 d2…dk - (d1 + d2 + … + dk + k - 1. We obtain an explicit example of a subspace of this kind. We determine the set of product vectors in its orthogonal complement and show that it has the minimum dimension possible for an unextendible product basis of not necessarily orthogonal product vectors.

scholarly journals Tomography in Abstract Hilbert Spaces

2006 ◽  
Vol 13 (03) ◽  
pp. 239-253 ◽  
Author(s):  
V. I. Man'ko ◽  
G. Marmo ◽  
A. Simoni ◽  
F. Ventriglia
Keyword(s):  
Hilbert Space ◽  
Quantum State ◽  
Trace Formula ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Rank One

The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.

scholarly journals THE SOq(N, R)-SYMMETRIC HARMONIC OSCILLATOR ON THE QUANTUM EUCLIDEAN SPACE ${\bf R}_q^N$ AND ITS HILBERT SPACE STRUCTURE

1993 ◽  
Vol 08 (26) ◽  
pp. 4679-4729 ◽  
Author(s):  
GAETANO FIORE
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Harmonic Oscillator ◽  
Mechanical Model ◽  
Scalar Product ◽  
Space Structure ◽  
Quantum Mechanical ◽  
Quantum Space ◽  
Isotropic Harmonic Oscillator ◽  
Definition Of

We show that the isotropic harmonic oscillator in the ordinary Euclidean space RN (N≥3) admits a natural q-deformation into a new quantum-mechanical model having a q-deformed symmetry (in the sense of quantum groups), SO q(N, R). The q-deformation is the consequence of replacing RN by [Formula: see text] (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over [Formula: see text], which we use for the definition of the scalar product of states.

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