Measuring measurement--disturbance relationships with weak values

Using formal definitions for measurement precision {\epsilon} and disturbance (measurement backaction) {\eta}, Ozawa [Phys. Rev. A 67, 042105 (2003)] has shown that Heisenberg's claimed relation between these quantities is false in general. Here we show that the quantities introduced by Ozawa can be determined experimentally, using no prior knowledge of the measurement under investigation --- both quantities correspond to the root-mean-squared difference given by a weak-valued probability distribution. We propose a simple three-qubit experiment which would illustrate the failure of Heisenberg's measurement--disturbance relation, and the validity of an alternative relation proposed by Ozawa

Quantum measurement and the first law of thermodynamics: the energy cost of measurement is the work value of the acquired information

The energy cost of measurement is an interesting fundamental question, and may have profound implications for quantum technologies. In the context of Maxwell's demon, it is often stated that measurement has no minimum energy cost, while information has a work value, even though these statements can appear contradictory. However, as we elucidate, these statements do no refer to the cost paid by the measuring device. Here we show that it is only when a measuring device has access to a zero temperature reservoir - that is, never - that the measurement requires no energy. All real measuring devices pay the cost that a heat engine pays to obtain the work value of the information they acquire.Comment: 4 pages, revtex4-1. v2: added a referenc

On quantum error-correction by classical feedback in discrete time

We consider the problem of correcting the errors incurred from sending quantum information through a noisy quantum environment by using classical information obtained from a measurement on the environment. For discrete time Markovian evolutions, in the case of fixed measurement on the environment, we give criteria for quantum information to be perfectly corrigible and characterize the related feedback. Then we analyze the case when perfect correction is not possible and, in the qubit case, we find optimal feedback maximizing the channel fidelity.Comment: 11 pages, 1 figure, revtex

Reconsidering Rapid Qubit Purification by Feedback

This paper reconsiders the claimed rapidity of a scheme for the purification of the quantum state of a qubit, proposed recently in Jacobs 2003 Phys. Rev. A67 030301(R). The qubit starts in a completely mixed state, and information is obtained by a continuous measurement. Jacobs' rapid purification protocol uses Hamiltonian feedback control to maximise the average purity of the qubit for a given time, with a factor of two increase in the purification rate over the no-feedback protocol. However, by re-examining the latter approach, we show that it mininises the average time taken for a qubit to reach a given purity. In fact, the average time taken for the no-feedback protocol beats that for Jacobs' protocol by a factor of two. We discuss how this is compatible with Jacobs' result, and the usefulness of the different approaches.Comment: 11 pages, 3 figures. Final version, accepted for publication in New J. Phy

Adaptive Quantum Measurements of a Continuously Varying Phase

We analyze the problem of quantum-limited estimation of a stochastically varying phase of a continuous beam (rather than a pulse) of the electromagnetic field. We consider both non-adaptive and adaptive measurements, and both dyne detection (using a local oscillator) and interferometric detection. We take the phase variation to be \dot\phi = \sqrt{\kappa}\xi(t), where \xi(t) is \delta-correlated Gaussian noise. For a beam of power P, the important dimensionless parameter is N=P/\hbar\omega\kappa, the number of photons per coherence time. For the case of dyne detection, both continuous-wave (cw) coherent beams and cw (broadband) squeezed beams are considered. For a coherent beam a simple feedback scheme gives good results, with a phase variance \simeq N^{-1/2}/2. This is \sqrt{2} times smaller than that achievable by nonadaptive (heterodyne) detection. For a squeezed beam a more accurate feedback scheme gives a variance scaling as N^{-2/3}, compared to N^{-1/2} for heterodyne detection. For the case of interferometry only a coherent input into one port is considered. The locally optimal feedback scheme is identified, and it is shown to give a variance scaling as N^{-1/2}. It offers a significant improvement over nonadaptive interferometry only for N of order unity.Comment: 11 pages, 6 figures, journal versio

Phase measurements at the theoretical limit

It is well known that the result of any phase measurement on an optical mode made using linear optics has an introduced uncertainty in addition to the intrinsic quantum phase uncertainty of the state of the mode. The best previously published technique [H. M. Wiseman and R.B. Killip, Phys. Rev. A 57, 2169 (1998)] is an adaptive technique that introduces a phase variance that scales as n^{-1.5}, where n is the mean photon number of the state. This is far above the minimum intrinsic quantum phase variance of the state, which scales as n^{-2}. It has been shown that a lower limit to the phase variance that is introduced scales as ln(n)/n^2. Here we introduce an adaptive technique that attains this theoretical lower limit.Comment: 9 pages, 5 figures, updated with better feedback schem

Adiabatic Elimination in Compound Quantum Systems with Feedback

Feedback in compound quantum systems is effected by using the output from one sub-system (``the system'') to control the evolution of a second sub-system (``the ancilla'') which is reversibly coupled to the system. In the limit where the ancilla responds to fluctuations on a much shorter time scale than does the system, we show that it can be adiabatically eliminated, yielding a master equation for the system alone. This is very significant as it decreases the necessary basis size for numerical simulation and allows the effect of the ancilla to be understood more easily. We consider two types of ancilla: a two-level ancilla (e.g. a two-level atom) and an infinite-level ancilla (e.g. an optical mode). For each, we consider two forms of feedback: coherent (for which a quantum mechanical description of the feedback loop is required) and incoherent (for which a classical description is sufficient). We test the master equations we obtain using numerical simulation of the full dynamics of the compound system. For the system (a parametric oscillator) and feedback (intensity-dependent detuning) we choose, good agreement is found in the limit of heavy damping of the ancilla. We discuss the relation of our work to previous work on feedback in compound quantum systems, and also to previous work on adiabatic elimination in general.Comment: 18 pages, 12 figures including two subplots as jpeg attachment

Entanglement under restricted operations: Analogy to mixed-state entanglement

We show that the classification of bi-partite pure entangled states when local quantum operations are restricted yields a structure that is analogous in many respects to that of mixed-state entanglement. Specifically, we develop this analogy by restricting operations through local superselection rules, and show that such exotic phenomena as bound entanglement and activation arise using pure states in this setting. This analogy aids in resolving several conceptual puzzles in the study of entanglement under restricted operations. In particular, we demonstrate that several types of quantum optical states that possess confusing entanglement properties are analogous to bound entangled states. Also, the classification of pure-state entanglement under restricted operations can be much simpler than for mixed-state entanglement. For instance, in the case of local Abelian superselection rules all questions concerning distillability can be resolved.Comment: 10 pages, 2 figures; published versio

Inequivalence of pure state ensembles for open quantum systems: the preferred ensembles are those that are physically realizable

An open quantum system in steady state $\hat\rho_{ss}$ can be represented by a weighted ensemble of pure states $\hat\rho_{ss}=\sum_{k}\wp_{k}\ket{\psi_k} \bra{\psi_k}$ in infinitely many ways. A physically realizable (PR) ensemble is one for which some continuous measurement of the environment will collapse the system into a pure state $\ket{\psi(t)}$, stochastically evolving such that the proportion of time for which $\ket{\psi(t)} = \ket{\psi_{k}}$ equals $\wp_{k}$. Some, but not all, ensembles are PR. This constitutes the preferred ensemble fact, with the PR ensembles being the preferred ensembles. We present the necessary and sufficient conditions for a given ensemble to be PR, and illustrate the method by showing that the coherent state ensemble is not PR for an atom laser.Comment: 5 pages, no figure