Measures on Hilbert space and quantum states

Stanley Gudder, Gottfried T. Rüttimann, Richard J. Greechie

We consider single and multiple measurements on a quantum logic (P,S) as well as states and propositions conditioned by a measurement. We show that corresponding to any measurement A, there is a canonically associated Hilbert space HA. Algebraic and statistical properties of (P,S) that are preserved in HA are found. We then study the problem of embedding a quantum logic in Hilbert space.

Gudrun Kalmbach

Terence William Barrett

A method of defining structural information in Hilbert space is described. Instead of commencing an information theory analysis with a simple signal and progressing to an n-dimensional information space, the reverse procedure can be followed: the signal is defined completely in both real and imaginary spaces, and is then represented as a dimensionless number in Hilbert space. Thus, a signal can be analyzed into a Hilbert space measure of information. This measure has similarities to the relativity of states of Everett's (1957) reformulation of quantum mechanics involving observer-observed interaction. Whereas Everett's theory deals with the totality of all possible ways in which a state function can be decomposed into the sum of products of state functions for subsystems of the overall system, the theory presented here deals with the totality of possible ways in which a Hilbert space measure of signal state can be decomposed by signal definition into reciprocally related measures of the state for subsystems of the overall system. In the overall system presented here, a signal is given a concise representation either as a quadratic form on a Hilbert space, or as a Fourier coefficients in isomorphic function space. The total information content of a signal qualified as a dimensionless number in Hilbert space may be given a representation in familiar units by the use of such methods. A primary postulate of quantum mechanics is obeyed in that all physically relevant information about a system is derivable from the knowledge of the state function of the system. Thus, a signal is defined firstly as an information measure, and secondly in the familiar units of frequency, bandwidth, midperiod, and duration.

Charlie Nation, Diego Porras

Quantum devices, such as quantum simulators, quantum annealers, and quantum computers, may be exploited to solve problems beyond what is tractable with classical computers. This may be achieved as the Hilbert space available to perform such `calculations' is far larger than that which may be classically simulated. In practice, however, quantum devices have imperfections, which may limit the accessibility to the whole Hilbert space. We thus determine that the dimension of the space of quantum states that are available to a quantum device is a meaningful measure of its functionality, though unfortunately this quantity cannot be directly experimentally determined. Here we outline an experimentally realisable approach to obtaining the required Hilbert space dimension of such a device to compute its time evolution, by exploiting the thermalization dynamics of a probe qubit. This is achieved by obtaining a fluctuation-dissipation theorem for high-temperature chaotic quantum systems, which facilitates the extraction of information on the Hilbert space dimension via measurements of the decay rate, and time-fluctuations.

S. Gudder