The dynamics for an open quantum system can be `unravelled in infinitely many ways, depending on how the environment is monitored, yielding different sorts of conditioned states, evolving stochastically. In the case of ideal monitoring these states a
re pure, and the set of states for a given monitoring forms a basis (which is overcomplete in general) for the system. It has been argued elsewhere [D. Atkins et al., Europhys. Lett. 69, 163 (2005)] that the `pointer basis as introduced by Zurek and Paz [Phys. Rev. Lett 70, 1187(1993)], should be identified with the unravelling-induced basis which decoheres most slowly. Here we show the applicability of this concept of pointer basis to the problem of state stabilization for quantum systems. In particular we prove that for linear Gaussian quantum systems, if the feedback control is assumed to be strong compared to the decoherence of the pointer basis, then the system can be stabilized in one of the pointer basis states with a fidelity close to one (the infidelity varies inversely with the control strength). Moreover, if the aim of the feedback is to maximize the fidelity of the unconditioned system state with a pure state that is one of its conditioned states, then the optimal unravelling for stabilizing the system in this way is that which induces the pointer basis for the conditioned states. We illustrate these results with a model system: quantum Brownian motion. We show that even if the feedback control strength is comparable to the decoherence, the optimal unravelling still induces a basis very close to the pointer basis. However if the feedback control is weak compared to the decoherence, this is not the case.
It has been shown in earlier works that the vertices of Platonic solids are good measurement choices for tests of EPR-steering using isotropically entangled pairs of qubits. Such measurements are regularly spaced, and measurement diversity is a good
feature for making EPR-steering inequalities easier to violate in the presence of experimental imperfections. However, such measurements are provably suboptimal. Here, we develop a method for devising optimal strategies for tests of EPR-steering, in the sense of being most robust to mixture and inefficiency (while still closing the detection loophole of course), for a given number $n$ of measurement settings. We allow for arbitrary measurement directions, and arbitrary weightings of the outcomes in the EPR-steering inequality. This is a difficult optimization problem for large $n$, so we also consider more practical ways of constructing near-optimal EPR-steering inequalities in this limit.
A single photon incident on a beam splitter produces an entangled field state, and in principle could be used to violate a Bell-inequality, but such an experiment (without post-selection) is beyond the reach of current experiments. Here we consider t
he somewhat simpler task of demonstrating EPR-steering with a single photon (also without post-selection). That is, of demonstrating that Alices choice of measurement on her half of a single photon can affect the other half of the photon in Bobs lab, in a sense rigorously defined by us and Doherty [Phys. Rev. Lett. 98, 140402 (2007)]. Previous work by Lvovsky and co-workers [Phys. Rev. Lett. 92, 047903 (2004)] has addressed this phenomenon (which they called remote preparation) experimentally using homodyne measurements on a single photon. Here we show that, unfortunately, their experimental parameters do not meet the bounds necessary for a rigorous demonstration of EPR-steering with a single photon. However, we also show that modest improvements in the experimental parameters, and the addition of photon counting to the arsenal of Alices measurements, would be sufficient to allow such a demonstration.
Feedback control engineers have been interested in MIMO (multiple-input multiple-output) extensions of SISO (single-input single-output) results of various kinds due to its rich mathematical structure and practical applications. An outstanding proble
m in quantum feedback control is the extension of the SISO theory of Markovian feedback by Wiseman and Milburn [Phys. Rev. Lett. {bf 70}, 548 (1993)] to multiple inputs and multiple outputs. Here we generalize the SISO homodyne-mediated feedback theory to allow for multiple inputs, multiple outputs, and emph{arbitrary} diffusive quantum measurements. We thus obtain a MIMO framework which resembles the SISO theory and whose additional mathematical structure is highlighted by the extensive use of vector-operator algebra.
The master equation for the state of an open quantum system can be unravelled into stochastic trajectories described by a stochastic master equation. Such stochastic differential equations can be interpreted as an update formula for the system state
conditioned on results obtained from monitoring the bath. So far only one parameterization (mathematical representation) for arbitrary diffusive unravellings (quantum trajectories arising from monitorings with Gaussian white noise) of a system described by a master equation with $L$ Lindblad terms has been found [H. M. Wiseman and A. C. Doherty, Phys. Rev. Lett. {bf 94}, 070405 (2005)]. This parameterization, which we call the U-rep, parameterizes diffusive unravellings by $L^2+2L$ real numbers, arranged in a matrix ${sf U}$ subject to three constraints. In this paper we investigate alternative parameterizations of diffusive measurements. We find, rather surprisingly, the description of diffusive unravellings can be unified by a single equation for a non-square complex matrix if one is willing to allow for some redundancy by lifting the number of real parameters necessary from $L^2+2L$ to $3L^2+L$. We call this parameterization the M-rep. Both the M-rep and U-rep lack a physical picture of what the measurement should look like. We thus propose another parameterization, the B-rep, that details how the measurement is implemented in terms of beam-splitters, phase shifters, and homodyne detectors. Relations between the different representations are derived.
A $D$-dimensional Markovian open quantum system will undergo quantum jumps between pure states, if we can monitor the bath to which it is coupled with sufficient precision. In general these jumps, plus the between-jump evolution, create a trajectory
which passes through infinitely many different pure states. Here we show that, for any ergodic master equation, one can expect to find an {em adaptive} monitoring scheme on the bath that can confine the system state to jumping between only $K$ states, for some $K geq (D-1)^2+1$. For $D=2$ we explicitly construct a 2-state ensemble for any ergodic master equation, showing that one bit is always sufficient to track a qubit.
Adaptive techniques make practical many quantum measurements that would otherwise be beyond current laboratory capabilities. For example: they allow discrimination of nonorthogonal states with a probability of error equal to the Helstrom bound; they
allow measurement of the phase of a quantum oscillator with accuracy approaching (or in some cases attaining) the Heisenberg limit; and they allow estimation of phase in interferometry with a variance scaling at the Heisenberg limit, using only single qubit measurement and control. Each of these examples has close links with quantum information, in particular experimental optical quantum information: the first is a basic quantum communication protocol; the second has potential application in linear optical quantum computing; the third uses an adaptive protocol inspired by the quantum phase estimation algorithm. We discuss each of these examples, and their implementation in the laboratory, but concentrate upon the last, which was published most recently [Higgins {em et al.}, Nature vol. 450, p. 393, 2007].
