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Entropy

Optimal Information Transfer and the Uniform Measure over Probability Space

William K. Wootters

Published May 30, 2023• 2 citations
Publisher

Abstract

-dimensional Hilbert space, suppose a pure state |ψ⟩ is subjected to a complete orthogonal measurement. The measurement effectively maps |ψ⟩ to a point (p1,…,pd) in the appropriate probability simplex. It is a known fact-which depends crucially on the complex nature of the system's Hilbert space-that if |ψ⟩ is distributed uniformly over the unit sphere, then the resulting ordered set (p1,…,pd) is distributed uniformly over the probability simplex; that is, the resulting measure on the simplex is proportional to dp1⋯dpd-1. In this paper we ask whether there is some foundational significance to this uniform measure. In particular, we ask whether it is the optimal measure for the transmission of information from a preparation to a measurement in some suitably defined scenario. We identify a scenario in which this is indeed the case, but our results suggest that an underlying real-Hilbert-space structure would be needed to realize the optimization in a natural way.

DOI
10.3390/e25060875

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-dimensional Hilbert space, suppose a pure state |ψ⟩ is subjected to a complete orthogonal measurement. The measurement effectively maps |ψ⟩ to a point (p1,…,pd) in the appropriate probability simplex. It is a known fact-which depends crucially on the complex nature of the system's Hilbert space-that if |ψ⟩ is distributed uniformly over the unit sphere, then the resulting ordered set (p1,…,pd) is distributed uniformly over the probability simplex; that is, the resulting measure on the simplex is proportional to dp1⋯dpd-1. In this paper we ask whether there is some foundational significance to this uniform measure. In particular, we ask whether it is the optimal measure for the transmission of information from a preparation to a measurement in some suitably defined scenario. We identify a scenario in which this is indeed the case, but our results suggest that an underlying real-Hilbert-space structure would be needed to realize the optimization in a natural way.
The main result of the paper can be summarized as follows. The number of distinguishable quantum states in a 2-dimensional Hilbert space is proportional to the square root of the number of identical copies of each state measured and to the total length of the angle intervals occupied by the state vectors. Surprisingly, it does not depend on the position of the arcs comprising the range on the unit circle. These results can be generalized for the N-dimensional Hilbert space of states of a quantum system.As in the 2-dimensional case, the unit sphere can be reduced to the non-negative orthant of the unit sphere in the real N-dimensional Euclidean space.It turns out [6] that the number of distinguishable states depends only on the area Ω of the domain on the unit sphere from which the states can be chosen, but does not depend on the shape and position of this domain. The optimal distribution is uniform over the domain in angular (polar) coordinates, and the number of distinguishable states is W(n,Ω)=Ω(cn)N−12 W(n,\Omega ) = \Omega (cn)^{\tfrac{{N - 1}} {2}} W(n,Ω)=Ω(cn)2N−1​ where c is a constant.
Recent theoretical developments are reviewed concentrating on the description of optimal generalised quantum measurement processes in physical systems with finite dimensional Hilbert spaces. Based on a self contained presentation of the required underlying quantum theoretical concepts this review particularly addresses recently explored basic theoretical properties and possible applications of informationally complete symmetric generalised measurements, a concept unifying the description of numerous so far largely separately explored measurement scenarios, such as measurements based on complete systems of mutually unbiased bases or symmetric informationally complete measurements. The recent results concerning the construction and existence of these informationally complete symmetric generalised measurements may also shed new light on still open questions concerning some of their special cases, such as the existence of complete sets of mutually unbiased bases in Hilbert spaces of non-prime power dimensions.
Using the Shannon information theory and the Bayesian methodology for inverting quantum data [K. R. W. Jones, Ann. Phys. (N.Y.) 207, 140 (1991)] we prove a fundamental bound upon the measurability of finite-dimensional quantum states. To do so we imagine a thought experiment for the quantum communication of a pure state \ensuremath{\psi}, known to one experimenter, to his colleague via the transmission of N identical copies of it in the limit of zero temperature. Initial information available to the second experimenter is merely that of the allowed manifold of superpositions upon which the chosen \ensuremath{\psi} may lie. Her efforts to determine it, in an optimal way, subject to the fundamental constraints imposed by quantum noise, define a statistical uncertainty principle. This limits the accuracy with which \ensuremath{\psi} can be measured according to the number N of transmitted copies. The general result is illustrated in the physically realizable case of polarized photons.
Consider a quantum system prepared in state ψ, a unit vector in a d -dimensional Hilbert space. Let b 1 , ..., b d be an orthonormal basis and suppose that, with some probability 0 < p < 1, ψ 'collapses', i.e., gets replaced by b k (possibly times a phase factor) with Born's probability |〈 b k |ψ〉| 2 . The question we investigate is: how well can any quantum experiment on the system determine afterwards whether a collapse has occurred? The answer depends on how much is known about the initial vector ψ. We provide a number of different results addressing several variants of the question. In each case, no experiment can provide more than rather limited probabilistic information. In case ψ is drawn randomly with uniform distribution over the unit sphere in Hilbert space, no experiment performs better than a blind guess without measurement; that is, no experiment provides any useful information.
A quantum measurement can be regarded as a communication channel, in which the parameters of the state are expressed only in the probabilities of the outcomes of the measurement. We begin this paper by considering, in a non-quantum-mechanical setting, the problem of communicating through probabilities. For example, a sender, Alice, wants to convey to a receiver, Bob, the value of a continuous variable, θ, but her only means of conveying this value is by sending Bob a coin in which the value of θ is encoded in the probability of heads. We ask what the optimal encoding is when Bob will be allowed to flip the coin only a finite number of times. As the number of tosses goes to infinity, we find that the optimal encoding is the same as what nature would do if we lived in a world governed by real-vector-space quantum theory. We then ask whether the problem might be modified, so that the optimal communication strategy would be consistent with standard, complex-vector-space quantum theory.
Let n>1 be a positive integer, {H1,…,Hn} be a finite collection of complex Hilbert spaces with dim⁡(Hk)≥2, and P1(Hk) be the set of all rank-1 self-adjoint projections on Hk, k=1,…,n. Set DP1⨂k=1nHk={A1⊗⋯⊗An:Ak∈P1(Hk),k=1,…,n}.We characterize the maps ϕ from D(P1(⨂k=1nHk)) to P1(⨂k=1nHk) preserving transition probability, i.e. tr(AB)=tr(ϕ(A)ϕ(B)),for all A,B∈DP1⨂k=1nHk.A particular case corresponding to n = 1 is well known as (non-surjective version) Wigner's theorem. Our result may be considered as a generalization of Wigner's theorem.