Quantum Estimation: Theory and Practice - 2008

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.

Projections onto mutually unbiased bases (MUBs) have the ability to maximize information extraction per measurement and to minimize redundancy. I present an experimental demonstration of quantum state tomography of two-qubit polarization states that takes advantage of MUBs. Estimates of the state taken with this method have a measurably higher fidelity to the true state than estimates taken using standard measurement strategies. I explain how this advantage can be understood from the structure of the measurements we use.

Quantum tomography and fidelity estimation of multi-partite systems is generally a time-consuming task. Nevertheless, this complexity can be reduced if the desired state can be characterized by certain symmetries measurable with the corresponding experimental setup. In this talk I could explain an efficient way (i.e., in polylog(d) time, with d the dimension of the Hilbert space) to perform tomography and estimate the fidelity of generalized coherent state (GCS) preparation. GCSs differ from the well known coherent states in that the associated Hilbert space is finite dimensional. In particular, the class of GCSs is very important in condensed matter applications. These results are useful to experimentalists seeking the simulations of some quantum systems, such as the Ising model in a transverse field. I\'d prefer to give a 30\' + talk late in the week, maybe on Thursday afternoon. Part of this work has been done in collaboration with ion-trap experimentalists J. Chiaverini and D. Berkeland, at Los Alamos National Laboratory. Rolando Somma.

The field of linear optics quantum computing (LOQC) allows the construction of conditional gates using only linear optics and measurement. This quantum computing paradigm bypasses a seemingly serious problem in optical quantum computing: it appears to be very hard to produce a meaningful interaction between two single photons. But what if this obstacle were instead an advantage? By assuming that none of the physical components that make up an LOQC gate produce a direct photon-photon interaction, we dramatically reduce the space of gates which are possible for a given number of input and output qubits. In fact, by parametrizing a gate according to it\'s action on single photons, instead of on multiple photons, it is possible to exponentially reduce the number of measurements necessary to fully characterize an LOQC gate. In addition, this approach to LOQC process tomography may have additional experimental advantages when non-ideal input states are used for this characterization.

We study two related estimation problems involving phase- covariant quantum states. We first address the problem of phase estimation. We give optimal bounds for pure and mixed Gaussian states and find that for a fixed squeezing parameter a larger temperature can enhance the estimation fidelity. In addition we use state estimation concepts to give a benchmark that asses whether experimental implementations of quantum storage and teleportation protocols could be reproduced by classical means, i.e., by a measure and prepare strategy.

Assume one laboratory designed a technique to produce quantum states in a given state $ ho$. The other lab wants to generate exactly the same state and they produce states $sigma$. If we want to know how well the second lab is doing we need to characterize the distance between $sigma$ and $ ho$ by some means,e.g. by trying to measure their fidelity, which allows us to find the Bures distance between them. The task is simple if the given state is pure, $ ho=|psi angle langle psi|$, since then fidelity reduces to the expectation value, $F=langlepsi| sigma| psi angle$. If $ ho$ is mixed the explicit formula for fidelity contains the trace of an absolute value of an operator which is not simple to compute nor to measure. Therefore we provide lower and upper bounds for fidelity and propose schemes to measure them. These experimental schemes require much less effort than the full quantum tomography of both states in question. The bounds for fidelity are called {sl sub-} and {sl super-fidelity}, respectively, due to their properties: as fidelity is multiplicative with respect to the tensor product, the sub-fidelity is sub-multiplicative, while super-fidelity is shown to be super-multiplicative. In the case of any two states of a one qubit system the bounds are strict and all three quantities coincide. The super-fidelity allowes us to define a modified Bures distance which for larger systems induces an alternative geometry of the space of quantum states.

Let us assume a following scenario: In a state of a quantum system one qubit is encoded. The first observer has no prior knowledge about the state of the qubit. He performs an optimal measurement on the system and based on the measured data he estimates the state on the qubit. After performing the measurement the first observer leaves the measured quantum system in a lab. I will study the question whether the second observer who has no knowledge about the measurement setup and the measurement outcome of the first observation can learn anything about the original preparation of the qubit.

After working on this for the past week, I\'m pretty excited about his topic. The method allows easy visualization of single qubit rotations and separable projections, much like the Poincare sphere for one qubit states.

I will briefly describe our recent progress in solving some optimization problems involving metrology with multipath entangled photon states and optimization of quantum operations on such states. We found that in the problem of super-resolution phase measurement in the presence of a loss one can single out two distinct regimes: i) low-loss regime favoring purely quantum states akin the N00N states and ii) high-loss regime where generalized coherent states become the optimal ones. Next I will describe how to optimize photon-entangling operations beyond the Knill-Laflamme-Milburn scheme and, in particular, how to exploit hyperentangled states for entanglement-assisted error correction.If time allows I will briefly review our results on generalization of the Bloch Sphere for the case of two qubits exploiting the SU(4)/Z2-SO(6) group isomorphism. References 1. D. Uskov & Jonathan P. Dowling. Quantum Optical Metrology in the Presence of a Loss (in preparation); Sean D. Huver et al, Entangled Fock States for Robust Quantum Optical Metrology, Imaging, and Sensing, arXiv:0808.1926. 2. D. Uskov et al, Maximal Success Probabilities of Linear-Optical Quantum Gates, arXiv:0808.1926. 3. M. Wilde and D. Uskov, Linear-Optical Hyperentanglement-Assisted Quantum Error-Correcting Code, arXiv:0807.4906. 4. D. Uskov and R. Rau Geometric phases and Bloch sphere constructions for SU(N), with a complete description of SU(4), Phys. Rev. A 78, 022331 (2008).