Universal Quantum Measurements

Dorje C Brody,Lane P Hughston

doi:10.1088/1742-6596/624/1/012002

Abstract

We introduce a family of operations in quantum mechanics that one can regard as “universal quantum measurements” (UQMs). These measurements are applicable to all finite dimensional quantum systems and entail the specification of only a minimal amount of structure. The first class of UQM that we consider involves the specification of the initial state of the system—no further structure is brought into play. We call operations of this type “tomographic measurements”, since given the statistics of the outcomes one can deduce the original state of the system. Next, we construct a disentangling operation, the outcome of which, when the procedure is applied to a general mixed state of an entangled composite system, is a disentangled product of pure constituent states. This operation exists whenever the dimension of the Hilbert space is not a prime, and can be used to model the decay of a composite system. As another example, we show how one can make a measurement of the direction along which the spin of a particle of spin s is oriented (s = 1/2, 1,...). The required additional structure in this case involves the embedding of CP1 as a rational curve of degree 2s in CP2s.

Highlights

As we enter into what may be the dawning of an age of quantum engineering, the widespread interest in quantum information, quantum communication, quantum cryptography, and quantum computation entailed has had the effect of reawakening research in finite dimensional quantum systems
As a way of honouring the scientific career of Professor Bogdan Mielnik we propose in the present paper to construct a variety of generalized measurement operations arising in the finitedimensional case that only involve a minimal amount of structure on the Hilbert space
No attempt will be made to model the universe as a whole, or to address the “measurement problem”, and the models we look at will be mainly nonrelativistic—or, more precisely, pre-relativistic, since we do not bring the geometry of space and time into play

Summary

Introduction

As we enter into what may be the dawning of an age of quantum engineering, the widespread interest in quantum information, quantum communication, quantum cryptography, and quantum computation entailed has had the effect of reawakening research in finite dimensional quantum systems. No attempt will be made to model the universe as a whole, or to address the “measurement problem”, and the models we look at will be mainly nonrelativistic—or, more precisely, pre-relativistic, since we do not bring the geometry of space and time into play We find it convenient to use an index notation for Hilbert space operations in our development of the theory of quantum state transformations. If measurements are performed on a large number of independent identical copies of a quantum system, by gathering the data of the resulting pure output states one can determine the input state. The additional structure involved in this class of generalized measurements takes the form of an embedding of CPn−1 as a rational variety of degree d in CPN−1, where (n + d − 1)!.

States and operations

Tomographic measurements

Disentangling operations

Coherent measurements