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arXiv:quant-ph/0405135 · Journal of Physics A Mathematical and General

On compatibility and improvement of different quantum state assignments

F Herbut

Published Apr 28, 2004• 7 citations
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Abstract

When Alice and Bob have different quantum knowledges or state assignments (density operators) ρ A and ρ B , respectively, for one and the same specific individual system, then the problems of compatibility and pooling arise. The so-called first Brun–Finkelstein–Mermin (BFM) condition for compatibility is reobtained in terms of possessed or sharp (i.e., probability one) properties. The second BFM condition is shown to be generally invalid in an infinite-dimensional state space. An argument leading to a procedure of improvement of ρ A on account of ρ B and vice versa is presented.

DOI
10.1088/0305-4470/37/19/011

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When Alice and Bob have different quantum knowledges or state assignments (density operators) ρ A and ρ B , respectively, for one and the same specific individual system, then the problems of compatibility and pooling arise. The so-called first Brun–Finkelstein–Mermin (BFM) condition for compatibility is reobtained in terms of possessed or sharp (i.e., probability one) properties. The second BFM condition is shown to be generally invalid in an infinite-dimensional state space. An argument leading to a procedure of improvement of ρ A on account of ρ B and vice versa is presented.
When Alice and Bob have different quantum knowledges or state assignments (density operators) for one and the same specific individual system, then the problems of compatibility and pooling arise. The so-called first Brun-Finkelstein-Mermin (BFM) condition for compatibility is reobtained in terms of possessed or sharp (i. e., probability one) properties. The second BFM condition is shown to be generally invalid in an infinite-dimensional state space. An argument leading to a procedure of improvement of one state assifnment on account of the other and vice versa is presented.
Suppose N parties describe the state of a quantum system by N possibly different density operators. These N state assignments represent the beliefs of the parties about the system. We examine conditions for determining whether the N state assignments are compatible. We distinguish two kinds of procedures for assessing compatibility, the first based on the compatibility of the prior beliefs on which the N state assignments are based and the second based on the compatibility of predictive measurement probabilities they define. The first procedure leads to a compatibility criterion proposed by Brun, Finkelstein, and Mermin [BFM, Phys. Rev. A 65, 032315 (2002)]. The second procedure leads to a hierarchy of measurement-based compatibility criteria which is fundamentally different from the corresponding classical situation. Quantum mechanically none of the measurement-based compatibility criteria is equivalent to the BFM criterion.
In approaches to quantum theory in which the quantum state is regarded as a representation of knowledge, information, or belief, two agents can assign different states to the same quantum system. This raises two questions: when are such state assignments compatible? And how should the state assignments of different agents be reconciled? In this paper, we address these questions from the perspective of the recently developed conditional states formalism for quantum theory (Leifer M S and Spekkens R W 2013 Phys. Rev. A 88 052310). Specifically, we derive a compatibility criterion proposed by Brun, Finkelstein and Mermin from the requirement that, upon acquiring data, agents should update their states using a quantum generalization of Bayesian conditioning. We provide two alternative arguments for this criterion, based on the objective and subjective Bayesian interpretations of probability theory. We then apply the same methodology to the problem of quantum state improvement, i.e. how to update your state when you learn someone else's state assignment, and to quantum state pooling, i.e. how to combine the state assignments of several agents into a single assignment that accurately represents the views of the group. In particular, we derive a pooling rule previously proposed by Spekkens and Wiseman under much weaker assumptions than those made in the original derivation. All of our results apply to a much broader class of experimental scenarios than have been considered previously in this context.
The state that an observer attributes to a quantum system depends on the information available to that observer. If two or more observers have different information about a single system, they will in general assign different states. Is there any restriction on what states can be assigned, given reasonable assumptions about how the observers use their information? We derive necessary and sufficient conditions for a group of general density matrices to characterize what different people may know about one and the same physical system. These conditions are summarized by a single criterion, which we term compatibility.
We consider the pooling of quantum states when Alice and Bob both have one part of a tripartite system and, on the basis of measurements on their respective parts, each infers a quantum state for the third part SSS. We denote the conditioned states which Alice and Bob assign to SSS by $\ensuremath{\alpha}$ and $\ensuremath{\beta}$, respectively, while the unconditioned state of SSS is $\ensuremath{\rho}$. The state assigned by an overseer, who has all the data available to Alice and Bob, is $\ensuremath{\omega}$. The pooler is told only $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\rho}$. We show that for certain classes of tripartite states, this information is enough for her to reconstruct $\ensuremath{\omega}$ by the formula $\ensuremath{\omega}\ensuremath{\propto}\ensuremath{\alpha}{\ensuremath{\rho}}^{\ensuremath{-}1}\ensuremath{\beta}$. Specifically, we identify two classes of states for which this pooling formula works: (i) all pure states for which the rank of $\ensuremath{\rho}$ is equal to the product of the ranks of the states of Alice's and Bob's subsystems; (ii) all mixtures of tripartite product states that are mutually orthogonal on SSS.