Conference

A number of problems in quantum estimation can be formulated as a convex optimization [1]. Applications include: maximum likelihood estimation, optimal experiment design, quantum state detection, and quantum metrology under instrumentation constraints. This talk will draw on the work I have been involved with, e.g., [2], [3], [4]. Our work in optimal quantum error correction [5, 6] is also relevant. Great benefit is derived using an error model which is specific to the system. Obtaining the errors from tomography is a logical route. How to do this, however, is an open question. The constraint is the form required by the standard error-correction model upon which the optimization is constructed. I will present some ideas on how to do the tomography in this context. • Maximum Likelihood (ML) quantum estimation problems are easily formed as log-convex optimization problems [1]. These include estimation of the state (density), estimation of the distribution of known input states, estimation of the OSR elements for quantum process tomography, and estimation of the coefficients of a preselected basis set of OSR elements. Estimation of Hamiltonian parameters, unfortunately, is not a convex optimization. Associated with these estimation problems, including Hamiltonian parameter estimation, is an optimal experiment design (OED), which is convex, and which can determine the system configurations to maximize the estimation accuracy [2]. Experiments have been performed In Ian Walmsley’s Group at Oxford using these methods [7, 8]. • Quantum state detection can be formulated as a convex optimization problem in the matrices of the POVM which characterize the measurement apparatus. Minimizing the error probability is a semidefinite program (SDP) [9]. Maximizing the posterior probability of detection is a quasiconvex optimization problem [3]. • Quantum metrology subject to instrumentation constraints can be cast as a convex optimization problem [4]. Focusing on the single parameter case, the optimization problem is a linear program (LP). The Fisher information from the LP solution for the constrained problem can be compared to what is possible with no constraints, the Quantum Fisher Information. This approach is easily extended to the multi-parameter case. • Quantum Error Correction (QEC) that is optimized with respect to the specific system at hand can reduce ancilla overhead while raising error thresholds for fault-tolerant operation [5, 6, 10, 11]. The problem is cast as a bi-convex optimization problem, iterating between encoding and recovery, each being an SDP. In [5] we introduced two new aspects of this approach: (i) we modified the objective functions to account for robustness, and (ii) posed the problem in an indirect form which can be solved via a sequence of constrained least-squares problems. This opens the way for solving extremely large problems in a reasonable time period both from offline models and online from measured data, i.e., tomography.

When one makes squeezed light by downconversion of a pulsed pump laser, many temporal / spectral modes are simultaneously squeezed by different amounts. There is no guarantee that any of these modes matches the pump or the local oscillator used to measure the squeezing in homodyne detection. Therefore the state observed in homodyne detection is not pure, and many photons are present in the beam path that do not lie in the local oscillator\'s mode. These problems limit the fidelity of quantum information processing tasks with pulsed squeezed light. I will describe our attempts to make coherent state superpositions (sometimes called \'cat states\') using photon subtraction from squeezed light, the problems caused by multimode squeezing, and methods to characterize the contents of the many squeezed modes.

We propose and demonstrate experimentally a technique for estimating quantum-optical processes in the continuous-variable domain. The process data is determined by applying the process to a set of coherent states and measuring the output. The process output for an arbitrary input state can then be obtained from its Glauber-Sudarshan expansion. Although such expansion is generally singular, it can be arbitrarily well approximated with a regular function.

I will discuss our ongoing attempts to construct Symmetric Informationally Complete Positive Operator Valued Measures, or (minimal) two designs, out of Gaussian states. This poses difficulties both in principle, such as how to introduce a measure on the noncompact group of symplectic transformations, as well as in practice, such as how to truncate the space of states in an experimentally useful way. Joint work with Robin Blume-Kohout.

I\'ll survey recent results from quantum computing theory showing that,if one just wishes to learn enough about a quantum state to predictthe outcomes of most measurements that will actually be made, then itoften suffices to perform exponentially fewer measurements than wouldbe needed in quantum state tomography. I\'ll then describe the resultsof a numerical simulation of the new quantum state learning approach.The latter is joint work with Eyal Dechter.

We introduce the concept of tight POVMs. In analogy with tight frames, these are POVMs that are as close as possible to orthonormal bases for the space they span. We show that tight rank-one POVMs define the exact class of optimal measurements for linear tomography of quantum states. In this setting they are equivalent to complex projective 2-designs. We also show that tight POVMs define the optimal class of measurements on the probe state for ancilla-assisted process tomography of unital channels. In this setting they are equivalent to unitary 2-designs.

The basic principles of quantum state tomography were first outlined by Stokes for the context of light polarisation more than 150 years ago. For an experimentalist the goal is clear: to use a series of measurement outcomes to make the best possible estimate of a system\'s quantum state, including phase information, with the least amount of measurement (and analysis) time and, if possible, also the least expensive and complicated apparatus. However, although the general ideas are straightforward, there is still much flexibility of choice in the practical details of how to implement tomography in the laboratory - details which may substantially influence the tomographic performance. For example, how do I make my measurements? How do I analyse my data? How confident can I be of my reconstruction anyway? Say, for example, I\'ve made my measurements and taken my data and I\'m now faced with two different ways, essentially equivalent at first glance, of analysing my data, where one way gives me \'better numbers\', e.g., a state with higher entanglement. Which do I choose? One may lead to a higher chance of a successful publication, but this is hardly a satisfactory way of making the decision. The perspective of an experimentalist motivates a fairly pragmatic approach to choosing the best technique for performing tomography. Does it work? How well? Can it work better? In this talk, I will use this approach and discuss issues which arise at each stage of the tomography process, using both numerical and real lab data to characterise the performance quality.

We present a novel quantum tomographic reconstruction method based on Bayesian inference via the Kalman filter update equations. The method not only yields the maximum likelihood/optimal Bayesian reconstruction, but also a covariance matrix expressing the measurement uncertainties in a complete way. From this covariance matrix the error bars on any derived quantity can be easily calculated. This is a first step towards the broader goal of devising an omnibus reconstruction method that could be adapted to any tomographic setup with little effort and that treats measurement uncertainties in a statistically well-founded way. We restrict ourselves to the important subclass of tomography based on measurements with discrete outcomes (as opposed to continuous ones), and we also ignore any measurement imperfections (dark counts, less than unit detector efficiency, etc.), which will be treated in further work. We illustrate the general theory on two real tomography experiments of quantum optical information processing elements.

The experimental realization of entangled states requires tools for characterizing the produced states as well as the processes used for creating the entanglement. In my talk, I will present examples of quantum measurements occuring in trapped ion experiments aiming at creating high-fidelity quantum gates.

A new approach to Quantum Estimation Theory will be introduced, based on the novel notions of \'quantum comb\' and \'quantum tester\', which generalize the customary notions of \'channel\' and \'POVM\' [PRL 101 060401 (2008)]. The new approach opens completely new possibilities of optimization in Quantum Estimation, beyond the classic approach of Helstrom and Holevo. Using comb theory it is possible to optimize the input-output arrangement of the black boxes for estimation with many uses. In this way it is possible to prove equivalence of arrangements for optimal estimation of unitaries, and the need of memory assisted protocols for for optimal discrimination of memory channels [arXive:0806.1172]. This also leads to a new notion of distance for channels with memory. Using the theory of quantum testers the optimal tomography schemes are derived---both state and for channel tomography---for arbitrary prior ensemble and arbitrary representation [arXive:0803.3237]. Finally, using the method of generalized pseudo-inverse for optimal data-processing [PRL. 98 020403 (2007)], we derived two improved data-processing for quantum tomography: Adaptive Bayesian and Frequentist [arXive:0807.5058].