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Physical Review Letters

Reversible Quantum Measurements on a Spin ½ and Measuring the State of a Single System

Antoine Royer

Published Feb 6, 1995• 25 citations
Publisher

Abstract

In [1],a reversible measurement on a spin 1/2 was introduced, and it was concluded that this would allow us, with a very small but finite probability of success, to measure the state of a single system.This conclusion is not correct.Let me recapitulate the argument: The measurement yields one of two outcomes, Tl or Tq, with respective probabilities P]

DOI
10.1103/physrevlett.74.1040

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Cambridge University Press eBooks•Mar 16, 2017•0 citations
In [1],a reversible measurement on a spin 1/2 was introduced, and it was concluded that this would allow us, with a very small but finite probability of success, to measure the state of a single system.This conclusion is not correct.Let me recapitulate the argument: The measurement yields one of two outcomes, Tl or Tq, with respective probabilities P]
A measurement procedure on a spin 1/2 system is described, whose effect on the (possibly unknown) state of the system can be reversed, by means of other similar measurements, with a sizable probability of success Prev{P}^{\mathrm{rev}}Prev (e.g., 12<Prev<1\frac{1}{2}<{P}^{\mathrm{rev}}<121​<Prev<1, depending on the state). Repeated such measurements followed by reversal, on a single system, allow us to determine its (a priori unknown) state operator with a probability of success which is nonzero, though extremely small (because Prev<1{P}^{\mathrm{rev}}<1Prev<1).
An argument, due originally to J. S. Bell, is somewhat simplified and made more specific. It deals primarily with a quantum mechanical system consisting of the spins of two spin-12 particles. It shows that a description of the quantum mechanical measurement of the spin components of these two particles by means of hidden parameters is impossible if we assume that the parameters determining the outcome of the measurement of the spin of each particle are independent of the direction in which we decide to measure the spin of the other particle. The mathematical reason for the impossibility is analyzed.
We address the following state comparison problem: is it possible to design an experiment enabling us to unambiguously decide (based on the observed outcome statistics) on the sameness or difference of two unknown state preparations without revealing complete information about the states? We find that the claim ``the same'' can never be concluded without any doubts unless the information is complete. Moreover, we prove that a universal comparison (that perfectly distinguishes all states) also requires complete information about the states. Nevertheless, for some measurements, the probability distribution of outcomes still allows one to make an unambiguous conclusion regarding the difference between the states even in the case of incomplete information. We analyze an efficiency of such a comparison of qudit states when it is based on the swap measurement. For qubit states, we consider in detail the performance of special families of two-valued measurements, enabling us to successfully compare at most half of the pairs of states. Finally, we introduce almost-universal comparison measurements which can distinguish almost all nonidentical states (up to a set of measure zero). The explicit form of such measurements with two and more outcomes is found in any dimension.
Using established probabilistic measurement-reversal theory, we study recovery placement in encoded-qutrit teleportation under monitored decay. We characterize the success advantage of reversing a no-jump map only on the encoded support and compare two receiver capabilities. The restricted-map construction applies to general finite-dimensional supports; explicit timing results are obtained for a coherent V-type sender, a ΛΛΛ-type receiver, and S={0,2}S=\{0,2\}S={0,2}, within a prescribed family of maximal exact sender reversals. With one outcome-independent full-space receiver filter, we maximize success probability at fixed total time subject to a minimum success-weighted conditional fidelity. A single preassigned recovery time yields up to three duration-dependent regimes: no sender reversal, the earliest feasible intermediate reversal, and final-time reversal. If recovery times may be randomized independently of the input and only the pooled successful ensemble must meet the fidelity requirement, the optimum over all such timing distributions mixes the two endpoints for ΓT≥(ln⁡3)/2ΓT \ge (\ln 3)/2ΓT≥(ln3)/2. With Bell-outcome-dependent nonunitary receiver recovery, each accepted full-column-rank transfer branch can instead be reversed exactly. For the same model and sender-operation family, omitting sender reversal uniquely maximizes the exact-teleportation success probability for every finite T>0T>0T>0 at positive decay rate. Partial pathway coherence further distinguishes relative support gain from absolute recovery success: the long-time dark-state plateau survives only for fully coherent decay. These results show how receiver control capabilities and the meaning of the fidelity constraint determine whether intermediate sender recovery is useful.
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. – Max Planck, 1936 The only way quantum theory allows us to interact with quantum systems is via quantum processes. Thus, the only way we can extract information about the state of a quantum system is to apply some non-deterministic process and observe which of the branches happened. For that reason, many people refer to the act of applying certain quantum processes as measurement and sometimes even as observation . A consequence of this unfortunate terminology is that one of the most touted ‘strange’ features of quantum theory is often misleadingly described as follows: ‘The mere act of observing a quantum state changes it.’ ‘Misleading’, since the concept of measurement described in the previous paragraph is far from the passive concept of observation familiar from our macroscopic world, but is rather a non-trivial process that will almost always drastically affect the quantum state. So, the mysterious aspect of quantum theory is not that ‘observation’ alters the quantum state but, rather, that it is impossible, even in principle, to ‘observe’ a quantum system in the classical sense. Given this fundamental restriction on how we can extract information from a quantum system, what can we learn about that system by means of a quantum process? The answer is, in fact, very little! First, performing a particular measurement could, for the vast majority of quantum states, yield any outcome i with a non-zero probability. In that case, obtaining an outcome i doesn't tell us much about what the state of the system was. Second, we say ‘was’, because the measurement will moreover irreversibly change the state according to the outcome. So rather than revealing the state of a system, a quantum measurement typically erases that state from history! Nonetheless, these quantum measurements are crucial to quantum theory since they constitute the only interface between us, in our classical world, and the quantum world. The great insight of the quantum computing community is that the non-deterministic changes induced by these quantum measurements are not a nuisance, but rather an extremely useful resource.