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arXiv:quant-ph/9911009 · Physical Review A

Distinguishability of states and von Neumann entropy

Richard Jozsa, J. Schlienz

Published Jun 5, 2000• 101 citations
arXivHTMLPDF

Abstract

Let ${|{\ensuremath{\psi}}_{1}〉,\dots{},|{\ensuremath{\psi}}_{n}〉{;p}_{1},\dots{}{,p}_{n}}$ be an ensemble of pure quantum states. We show that it is possible to increase all of the pairwise overlaps $|〈{\ensuremath{\psi}}_{i}|{\ensuremath{\psi}}_{j}〉|,$ i.e., make each constituent pair of the states more parallel (while keeping the prior probabilities the same), in such a way that the von Neumann entropy S is increased, and dually, make all pairs more orthogonal while decreasing S. We show that this phenomenon cannot occur for ensembles in two dimensions but that it is a feature of almost all ensembles of three states in three dimensions. It is known that the von Neumann entropy characterizes the classical and quantum information capacities of the ensemble and we argue that information capacity, in turn, is a manifestation of the distinguishability of the signal states. Hence, our result shows that the notion of distinguishability within an ensemble is a global property that cannot be reduced to considering distinguishability of each constituent pair of states.

DOI
10.1103/physreva.62.012301

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Let ∣\ensuremathψ1〉,…,∣\ensuremathψn〉;p1,…,pn{|{\ensuremath{\psi}}_{1}〉,\dots{},|{\ensuremath{\psi}}_{n}〉{;p}_{1},\dots{}{,p}_{n}}∣\ensuremathψ1​〉,…,∣\ensuremathψn​〉;p1​,…,pn​ be an ensemble of pure quantum states. We show that it is possible to increase all of the pairwise overlaps ∣〈\ensuremathψi∣\ensuremathψj〉∣,|〈{\ensuremath{\psi}}_{i}|{\ensuremath{\psi}}_{j}〉|,∣〈\ensuremathψi​∣\ensuremathψj​〉∣, i.e., make each constituent pair of the states more parallel (while keeping the prior probabilities the same), in such a way that the von Neumann entropy S is increased, and dually, make all pairs more orthogonal while decreasing S. We show that this phenomenon cannot occur for ensembles in two dimensions but that it is a feature of almost all ensembles of three states in three dimensions. It is known that the von Neumann entropy characterizes the classical and quantum information capacities of the ensemble and we argue that information capacity, in turn, is a manifestation of the distinguishability of the signal states. Hence, our result shows that the notion of distinguishability within an ensemble is a global property that cannot be reduced to considering distinguishability of each constituent pair of states.
A basic property of distinguishability is that it is non-increasing under further quantum operations. Following this, we generalize two measures of distinguishability of pure states--fidelity and von Neumann entropy, to mixed states as self-consistent measures. Then we extend these two measures to quantum operations. The information-theoretic point of the generalized Holevo quantity of an ensemble of quantum operations is constructed. Preferably it is an additive measure. The exact formula for SU(2) ensemble is presented. With the aid of the formula, we show Jozsa-Schlienz paradox that states as a whole are less distinguishable while all pairwise are more distinguishable in an ensemble of quantum states, also occurs in an ensemble of quantum operations, even in the minimal dimensional case SU(2) ensemble.
Consider a source of pure quantum states with density matrix ρ and von Neumann entropy S . It is known that the quantum information of the source may be faithfully compressed to 5 qubits/signal (asymptotically for long sequences) and that this limit is optimal if we demand high fidelity for long strings of signals. The usual method for achieving this compression, based on the typical subspace of the source, requires knowledge of the density matrix. We describe a new “universal” compression scheme which does not even require knowledge of ρ. More precisely, suppose that the source is known to have von Neumann entropy less than some given bound S 0 but is otherwise completely unspecified. Our method will faithfully compress the quantum information of any such source to S 0 qubits/signal. It is known that S is always less than or equal to the Shannon entropy H of the prior distribution of the source. According to Shannon’s theorem, H provides the limit for compression of classical signals, so one may ask the question: what is the physical origin of the extra compression available in the quantum case compared to the classical case (where the prior probabilities are the same)? One may suggest that this effect is a consequence of possible non-orthogonality of signal states. However we demonstrate that it is possible to make each pair of signal states of a source more parallel (keeping the prior probabilities the same) while increasing the von Neumann entropy. This casts doubt on the conventional wisdom that overlap of quantum states is a measure of their (non-)distinguishability. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The von Neumann entropy plays a vital role in quantum information theory. The von Neumann entropy determines, e.g., the capacities of quantum channels. Also, entropies of composite quantum systems are important for future quantum networks, and their characterization is related to the quantum marginal problem. Furthermore, they play a role in quantum thermodynamics. In this thesis the set of quantum entropies of multipartite quantum systems is studied. The problem of its characterization is not new -- however, progress has been sparse, indicating that the problem might be hard and that new methods might be needed. Here, a variety of different and complementary approaches are taken. First, I look at global properties. It is known that the von Neumann entropy region -- just like its classical counterpart -- forms a convex cone. I describe the symmetries of this cone and highlight geometric similarities and differences to the classical entropy cone. In a different approach, I utilize the local geometric properties of extremal rays of a cone. I show that quantum states whose entropy lies on such an extremal ray of the quantum entropy cone have a very simple structure. As the set of all quantum states is very complicated, I look at a simple subset called stabilizer states. I improve on previously known results by showing that under a technical condition on the local dimension, entropies of stabilizer states respect an additional class of information inequalities that is valid for random variables from linear codes. In a last approach I find a representation-theoretic formulation of the classical marginal problem simplifying the comparison with its quantum mechanical counterpart. This novel correspondence yields a simplified formulation of the group characterization of classical entropies (IEEE Trans. Inf. Theory, 48(7):1992-1995, 2002) in purely combinatorial terms.
Entanglement is well understood concept for distinguishable particles. However fundamental particles are inherently indistinguishable and when they occupy orthogonal modes, i.e. have orthogonal wave functions, they are effectively distinguishable. Thus there should exist a formulation for entanglement which works in the generic case, i.e. even when wave functions overlap. To achieve this, we consider R\`{e}nyi entropies which are measurable quantities. Outcomes of such measurements for a generic case are conjectured in terms of R\`{e}nyi entropies of spatial regions in a field theory. To leading order naive quantum mechanical answers are reproduced. Corrections depend on overlap of wave functions and exhibit novel features. In a generic case the Von Neumann entropy computed from these R\`{e}nyi entropies is argued to be a measure of entanglement of pure states and a formula for this is conjectured in terms of Von Neumann entropies of spatial regions in field theory. It seems, due to the corrections it is possible for the entanglement to exceed maximal quantum mechanical value, e.g. log2log 2log2 for a single qubit.
Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.
A theorem is proven for quantum information theory that is analogous to the noiseless coding theorem of classical information theory. In the quantum result, the von Neumann entropy S of the density operator describing an ensemble of pure quantum signal states is equal to the number of spin-1/2 systems (``quantum bits'' or ``qubits'') necessary to represent the signal faithfully. The theorem holds whether or not the signal states are orthogonal. Related results are also presented about the fidelity of quantum coding and about representing entangled quantum states.