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Journal of Physics Conference Series

Frames and fusion frames in quantum optics

Andrzej Jamiołkowski

Published Mar 1, 2010• 3 citations
Publisher

Abstract

The main purpose of this paper is to show that the notions of frames and fusion frames introduced in non-harmonic Fourier analysis are also very natural in discussion of some basic problems in theory of open quantum systems and, in particular, in quantum optics. Frames are collections of vectors in a Hilbert space which assure a natural representation of each vector in the space, but may have infinitely many different representations for any given vector. For a given quantum system represented in a Hilbert space the question of minimal number η of observables Q 1 ,..., Q η , whose expectation values at some instants t 1 ,...,t p determine the statistical state of the system is discussed. We assume that the time evolution of the system in question is governed by a semigroup of linear transformations with generator .

DOI
10.1088/1742-6596/213/1/012002

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The main purpose of this paper is to show that the notions of frames and fusion frames introduced in non-harmonic Fourier analysis are also very natural in discussion of some basic problems in theory of open quantum systems and, in particular, in quantum optics. Frames are collections of vectors in a Hilbert space which assure a natural representation of each vector in the space, but may have infinitely many different representations for any given vector. For a given quantum system represented in a Hilbert space the question of minimal number η of observables Q 1 ,..., Q η , whose expectation values at some instants t 1 ,...,t p determine the statistical state of the system is discussed. We assume that the time evolution of the system in question is governed by a semigroup of linear transformations with generator .
A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.
The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder–Berezin–Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.
We often hear at seminars and presentations ‘Show me the Hamiltonian of your problem and I will tell you what problem you are talking about’. Therefore, to understand what quantum optics is about, we start from the description of a standard Hamiltonian of the problems considered in quantum optics. We illustrate the method of derivation of the explicit form of the Hamiltonian for a simple system that is composed of two subsystems that interact (communicate) with each other. As we shall see, the explicit form of the Hamiltonian is essential to explicitly calculate energy levels of a given combined system and the temporal evolution of an arbitrary operator representing the system.
In this paper, we discuss the minimal number η of observables Q1,…, Qη, where expectation values at some time instants t1,…,tr determine the trajectory of a d-level quantum system (“qudit”) governed by the Gaussian semigroup [Formula: see text] We assume that the macroscopic information about the system in question is given by the mean values [Formula: see text] of n selfadjoint operators Q1,…, Qn at some time instants t1< t2 < … <tr, where n < d2 − 1 and r ≤ deg μ(λ, 𝕃). Here μ(λ, 𝕃) stands for the minimal polynomial of the generator [Formula: see text] of the Gaussian flow Φ(t).
The standard formalism of quantum mechanics is extended to describe a total system including the reference system (RS), with respect to which the total system is described. The RS is assumed to be able to act as a measuring apparatus, with measurement records given by the values of some reference properties of the RS. In order to describe the total system, we define a frame of reference (FR) as a set of states that can be used to express all other states of the total system. The theory is based on four basic postulates, which have, loosely speaking, the following contents. (i) A reference property of a RS has a definite value and is sufficiently stable in the FR directly related to the reference property. (ii) States of the total system are associated with vectors in the Hilbert space. (iii) Schrodinger equation is the dynamical law in each valid FR. (iv) Under certain condition a property of a system can be regarded as a reference property; vector descriptions of the total system given in different FRs of the same RS may have a probabilistic relationship like in Born's rule.
A dynamical problem in quantum optics is considered to be that of solving Schrodinger's equation with the initial state of the radiation field as a coherent state By cairying out a particular unitary transformation it is shown how one can arrive at a representation of the radiation field operators which is particulaily convenient for Proc.esses in which the interaction with the material system only involves one photon or a few photons. It is shown in which sense this representation is a counterpart of the Fock representation. In the derivation of the representation, which is exact throughout, some features are brought out, which may be interpreted as being semi-classical.