Precision bounds for noisy nonlinear quantum metrology

We derive the ultimate bounds on the performance of nonlinear measurement schemes in the presence of noise. In particular, we investigate the precision of the second-order estimation scheme in the presence of the two most detrimental types of noise, photon loss and phase diffusion. We find that the second-order estimation scheme is affected by both types of noise in an analogous way as the linear one. Moreover, we observe that for both types of noise the gain in the phase sensitivity with respect to the linear estimation scheme is given by a multiplicative term $\mathcal{O}(1/N)$. Interestingly, we also find that under certain circumstances, a careful engineering of the environment can, in principle, improve the performance of measurement schemes affected by phase diffusion.Comment: 9 pages, 2 figures, 1 table, 1 appendix; v3 contains an improved analysis and a stronger precision bound for the case of photon loss; published versio

Symmetries and physically realizable ensembles for open quantum systems

A $D$-dimensional Markovian open quantum system will undergo stochastic evolution which preserves pure states, if one monitors without loss of information the bath to which it is coupled. If a finite ensemble of pure states satisfies a particular set of constraint equations then it is possible to perform the monitoring in such a way that the (discontinuous) trajectory of the conditioned system state is, at all long times, restricted to those pure states. Finding these physically realizable ensembles (PREs) is typically very difficult, even numerically, when the system dimension is larger than 2. In this paper, we develop symmetry-based techniques that potentially greatly reduce the difficulty of finding a subset of all possible PREs. The two dynamical symmetries considered are an invariant subspace and a Wigner symmetry. An analysis of previously known PREs using the developed techniques provides us with new insights and lays the foundation for future studies of higher dimensional systems.Comment: 30 pages, 4 figures, comments welcome. Published versio

Open quantum systems are harder to track than open classical systems

For a Markovian open quantum system it is possible, by continuously monitoring the environment, to know the stochastically evolving pure state of the system without altering the master equation. In general, even for a system with a finite Hilbert space dimension $D$, the pure state trajectory will explore an infinite number of points in Hilbert space, meaning that the dimension $K$ of the classical memory required for the tracking is infinite. However, Karasik and Wiseman [Phys. Rev. Lett., 106(2):020406, 2011] showed that tracking of a qubit ($D=2$) is always possible with a bit ($K=2$), and gave a heuristic argument implying that a finite $K$ should be sufficient for any $D$, although beyond $D=2$ it would be necessary to have $K>D$. Our paper is concerned with rigorously investigating the relationship between $D$ and $K_{\rm min}$, the smallest feasible $K$. We confirm the long-standing conjecture of Karasik and Wiseman that, for generic systems with $D>2$, $K_{\rm min}>D$, by a computational proof (via Hilbert Nullstellensatz certificates of infeasibility). That is, beyond $D=2$, $D$-dimensional open quantum systems are provably harder to track than $D$-dimensional open classical systems. Moreover, we develop, and better justify, a new heuristic to guide our expectation of $K_{\rm min}$ as a function of $D$, taking into account the number $L$ of Lindblad operators as well as symmetries in the problem. The use of invariant subspace and Wigner symmetries makes it tractable to conduct a numerical search, using the method of polynomial homotopy continuation, to find finite physically realizable ensembles (as they are known) in $D=3$. The results of this search support our heuristic. We thus have confidence in the most interesting feature of our heuristic: in the absence of symmetries, $K_{\rm min} \sim D^2$, implying a quadratic gap between the classical and quantum tracking problems.Comment: 35 pages, 3 figures, Accepted in Quantum Journal, minor change

Maximum information gain in weak or continuous measurements of qudits: complementarity is not enough

To maximize average information gain for a classical measurement, all outcomes of an observation must be equally likely. The condition of equally likely outcomes may be enforced in quantum theory by ensuring that one's state $\rho$ is maximally different, or complementary, to the measured observable. This requires the ability to perform unitary operations on the state, conditioned on the results of prior measurements. We consider the case of measurement of a component of angular momentum for a qudit (a $D$-dimensional system, with $D=2J+1$). For weak or continuous-in-time (i.e. repeated weak) measurements, we show that the complementarity condition ensures an average improvement, in the rate of purification, of only 2. However, we show that by choosing the optimal control protocol of this type, one can attain the best possible scaling, $O(D^{2})$, for the average improvement. For this protocol the acquisition of information is nearly deterministic. Finally we contrast these results with those for complementarity-based protocols in a register of qbits.Comment: 21 pages, 21 figures. V2 published versio

Optical coherence and teleportation: Why a laser is a clock, not a quantum channel

It has been argued [T. Rudolph and B.C. Sanders, Phys. Rev. Lett. {\bf 87}, 077903 (2001)] that continuous-variable quantum teleportation at optical frequencies has not been achieved because the source used (a laser) was not `truly coherent'. Van Enk, and Fuchs [Phys. Rev. Lett, {\bf 88}, 027902 (2002)], while arguing against Rudolph and Sanders, also accept that an `absolute phase' is achievable, even if it has not been achieved yet. I will argue to the contrary that `true coherence' or `absolute phase' is always illusory, as the concept of absolute time on a scale beyond direct human experience is meaningless. All we can ever do is to use an agreed time standard. In this context, a laser beam is fundamentally as good a `clock' as any other. I explain in detail why this claim is true, and defend my argument against various objections. In the process I discuss super-selection rules, quantum channels, and the ultimate limits to the performance of a laser as a clock. For this last topic I use some earlier work by myself [Phys. Rev. A {\bf 60}, 4083 (1999)] and Berry and myself [Phys. Rev. A {\bf 65}, 043803 (2002)] to show that a Heisenberg-limited laser with a mean photon number $\mu$ can synchronize $M$ independent clocks each with a mean-square error of $\sqrt{M}/4\mu$ radians$^2$.Comment: 14 pages, no figures, some equations this time. For proceedings of SPIE conference "Fluctuations and Noise 2003

Reply to Norsen's paper "Are there really two different Bell's theorems?"

Yes. That is my polemical reply to the titular question in Travis Norsen's self-styled "polemical response to Howard Wiseman's recent paper." Less polemically, I am pleased to see that on two of my positions --- that Bell's 1964 theorem is different from Bell's 1976 theorem, and that the former does not include Bell's one-paragraph heuristic presentation of the EPR argument --- Norsen has made significant concessions. In his response, Norsen admits that "Bell's recapitulation of the EPR argument in [the relevant] paragraph leaves something to be desired," that it "disappoints" and is "problematic". Moreover, Norsen makes other statements that imply, on the face of it, that he should have no objections to the title of my recent paper ("The Two Bell's Theorems of John Bell"). My principle aim in writing that paper was to try to bridge the gap between two interpretational camps, whom I call 'operationalists' and 'realists', by pointing out that they use the phrase "Bell's theorem" to mean different things: his 1964 theorem (assuming locality and determinism) and his 1976 theorem (assuming local causality), respectively. Thus, it is heartening that at least one person from one side has taken one step on my bridge. That said, there are several issues of contention with Norsen, which we (the two authors) address after discussing the extent of our agreement with Norsen. The most significant issues are: the indefiniteness of the word 'locality' prior to 1964; and the assumptions Einstein made in the paper quoted by Bell in 1964 and their relation to Bell's theorem.Comment: 13 pages (arXiv version) in http://www.ijqf.org/archives/209

Discord in relation to resource states for measurement-based quantum computation

We consider the issue of what should count as a resource for measurement-based quantum computation (MBQC). While a state that supports universal quantum computation clearly should be considered a resource, universality should not be necessary given the existence of interesting, but less computationally-powerful, classes of MBQCs. Here, we propose minimal criteria for a state to be considered a resource state for MBQC. Using these criteria, we explain why discord-free states cannot be resources for MBQC, contrary to recent claims [Hoban et al., arXiv:1304.2667v1]. Independently of our criteria, we also show that the arguments of Hoban et al., if correct, would imply that Shor's algorithm (for example) can be implemented by measuring discord-free states.Comment: 7 pages. Title changed again at request of editors. Significant expository changes. Technical content the same as befor

Quantum feedback for rapid state preparation in the presence of control imperfections

Quantum feedback control protocols can improve the operation of quantum devices. Here we examine the performance of a purification protocol when there are imperfections in the controls. The ideal feedback protocol produces an $x$ eigenstate from a mixed state in the minimum time, and is known as rapid state preparation. The imperfections we examine include time delays in the feedback loop, finite strength feedback, calibration errors, and inefficient detection. We analyse these imperfections using the Wiseman-Milburn feedback master equation and related formalism. We find that the protocol is most sensitive to time delays in the feedback loop. For systems with slow dynamics, however, our analysis suggests that inefficient detection would be the bigger problem. We also show how system imperfections, such as dephasing and damping, can be included in model via the feedback master equation.Comment: 15 pages, 6 figures and 2 tables. V2 the published version, fig. 1 corrected and some minor changes to the tex