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International Journal of Production Economics

An Example of Non-Attainability of Expected Quantum Information

Ole Eiler Barndorff-Nielsen, Richard David Gill

Published Aug 5, 1998• 0 citations
OpenAlex

Abstract

Braunstein and Caves showed how quantum information can be used to define a metric for quantum states, relying on attainability of a quantum information bound. We show that the bound is not generally attainable. But a two-stage procedure of repeated measurements achieves the bound in the limit. We connect to the question of whether a generalised measurement on $n$ independent copies of a quantum system can yield more information than $n$ separate measurements. Separate measurements are in the limit as informative as joint, and measurement of classical observables suffices.

Related papers

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The attainability of the quantum Cram\'er-Rao bound (QCR), the ultimate limit in the precision of the estimation of a physical parameter, requires the saturation of the quantum information bound (QIB). This occurs when the Fisher information associated to a given measurement on the quantum state of a system which encodes the information about the parameter coincides with the quantum Fisher information associated to that quantum state. Braunstein and Caves [Phys. Rev. Lett. 72, 3439 (1994)] have shown that the QIB can always be achieved via a projective measurement in the eigenvectors basis of an observable called the symmetric logarithmic derivative. However, such projective measurement depends, in general, on the value of the parameter to be estimated, therefore requiring previous knowledge of the quantity one is trying to estimate. For this reason, it is important to investigate under which situation it is possible to saturate the QCR without previous information about the parameter to be estimated. Here, we show the complete solution to the problem of which are the initial pure states and which projective measurements allow the global saturation of the QIB, without the knowledge of the true value of the parameter, when the information about the parameter is encoded in the system by a unitary process.

arXiv:1701.09144•Physical Review A•Apr 20, 2017•7 citations
Braunstein and Caves showed how quantum information can be used to define a metric for quantum states, relying on attainability of a quantum information bound. We show that the bound is not generally attainable. But a two-stage procedure of repeated measurements achieves the bound in the limit. We connect to the question of whether a generalised measurement on nnn independent copies of a quantum system can yield more information than nnn separate measurements. Separate measurements are in the limit as informative as joint, and measurement of classical observables suffices.
Braunstein and Caves (Braunstein S L and Caves C M 1994 Phys. Rev. Lett. 72 3439-43) proposed to use Helstrom's quantum information number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing other than the maximal Fisher information in a measurement of the quantum system, maximized over all possible measurements. Combining this fact with classical statistical results, they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished, based on arbitrary measurements on n identical copies of the given quantum system. We show that the measurement which maximizes the Fisher information typically depends on the true, unknown, state of the quantum system. We close the resulting loophole in the argument by showing that one can still achieve the same, optimal, rate of distinguishability, by a two-stage adaptive measurement procedure. When we consider states lying not on a smooth curve, but on a manifold of higher dimension, the situation becomes much more complex. We show that the notion of `distinguishability of close-by states' depends strongly on the measurement resources one allows oneself, and on a further specification of the task at hand. The quantum information matrix no longer seems to play a central role.
We address the "major open problem" of evaluating how much increased efficiency in estimation is possible using non-separable measurements, rather than separable ones, of N copies of two-level quantum systems. To proceed, we compute the the 3 x 3 Fisher information matrices for the optimal measurements recently devised by Vidal et al (quant-ph/9812068) for N = 2,...,7. We obtain simple polynomial expressions for the traces of the products of the inverse of the quantum Helstrom information matrix and the computed Fisher information matrices. These traces all have minima of 2 N - 1 in the pure state limit, while for separable measurements, as established by Gill and Massar (quant-ph/9902063), they are bound above by N. This indicates both that there is a large discontinuity at the pure states (where the computed Fisher information matrices are indeterminate) and that the gain in using joint measurements over separate measurements is substantial for all impure states. In seeking to extend our results to N > 7, we investigate possible relationships between the Fisher information matrices generated here and the class of monotone Riemannian metrics, studied by Petz and Sudar as well as others.
Naive attempts to put together relativity and quantum measurements lead to signaling between space-like separated regions. In QFT, these are known as impossible measurements . We show that the same problem arises in non-relativistic quantum physics, where joint nonlocal measurements (i.e., between systems kept spatially separated) in general lead to signaling, while one would expect no-signaling (based for instance on the principle of no-nonphysical communication ). This raises the question: Which nonlocal quantum measurements are physically possible? We review and develop further a non-relativistic quantum information approach developed independently of the impossible measurements in QFT, and show that these two have been addressing virtually the same problem. The non-relativistic solution shows that all nonlocal measurements are localizable (i.e., they can be carried out at a distance without violating no-signaling) but they (i) may require arbitrarily large entangled resources and (ii) cannot in general be ideal , i.e., are not immediately reproducible. These considerations could help guide the development of a complete theory of measurement in QFT.
The attainability of the quantum Cram\'er-Rao bound (QCR), the ultimate limit in the precision of the estimation of a physical parameter, requires the saturation of the quantum information bound (QIB). This occurs when the Fisher information associated to a given measurement on the quantum state of a system which encodes the information about the parameter coincides with the quantum Fisher information associated to that quantum state. Braunstein and Caves [Phys. Rev. Lett. 72, 3439 (1994)] have shown that the QIB can always be achieved via a projective measurement in the eigenvectors basis of an observable called the symmetric logarithmic derivative. However, such projective measurement depends, in general, on the value of the parameter to be estimated, therefore requiring previous knowledge of the quantity one is trying to estimate. For this reason, it is important to investigate under which situation it is possible to saturate the QCR without previous information about the parameter to be estimated. Here, we show the complete solution to the problem of which are the initial pure states and which projective measurements allow the global saturation of the QIB, without the knowledge of the true value of the parameter, when the information about the parameter is encoded in the system by a unitary process.