Conference

Quantum information technologies have recorded enormous progress within the recent fifteen years. They have developed from the early stage of thought experiments into nowadays almost ready-to-use technology. In view of many possible applications the question of efficient analysis and diagnostics of quantum systems appears to be crucial. The quantum state is not an observable and as such it cannot be measured in the traditional sense of thisword. Information encoded in a quantum state may be portrayed by various ways yielding the most complete and detailed picture of the quantum object available. Due to the formal similarities between the quantum estimation and medical non-invasive 3D imaging, this method is also called quantum tomography. Many different methods of quantum tomography have been proposed and implemented for various physical systems. Experiments are being permanently improved in order to increase our ability to unravel even the most exquisite and fragile non-classical effects. Progress has been made not only on the detection side of tomography schemes. Mathematical algorithms too have been improved. The original linear methods based on the inverse Radon transformation are prone to producing artifacts and have other serious drawbacks. For example, the positivity of the reconstructed state required by quantum theory is not guaranteed. This may obviously lead to inconsistent statistical predictions about future events. For such reasons, the simple linear methods are gradually being replaced by statistically motivated methods, for example by Bayesian or maximum-likelihood (ML) [1,2] tomography methods.The quantification of all relevant errors is an indispensable but often neglected part of any tomographic scheme used for quantum diagnostic purposes. The result of quantum tomography cannot be reduced merely to finding the most likely state. What also matters is how much the other states, those being less likely ones, would be consistent with the registered data. In this sense, also states lying in the neighborhood of the most likely state should be taken into account for making future statistical predictions. For this purpose we introduce a novel resolution measure, which provides ``error bars\'\' for any inferred quantity of interest. This is illustrated with an example of the diagnostics of non-classical states based on the value of the reconstructed Wigner function at the origin of the phase space. We show that such diagnostics is meaningful only when some prior information on the measured quantum state is available. In this sense quantum tomography based on homodyne detection is more noisy and more uncertain than widely accepted nowadays. Since the error scales with the dimension, the choice of a proper dimension of the reconstruction space is vital for successful diagnostics of non-classical states. There are two concurring tendencies for the choice of this dimension. When the reconstruction space is low-dimensional, the reconstruction noise is kept low, however there may not be enough free parameters left for fitting of a possibly high-dimensional true state. In the case of high-dimensional reconstruction space, the danger of missing important components of the true state is smaller, however the reconstruction errors may easily exceed acceptable levels. These issues will be discussed in the context of penalization and constraints for maximizing the likelihood [3]. The steps described above are the necessary prerequisites for the programme of objective tomography, where all the conclusions should be derived on the basis of registered data without any additional assumptions. New resolution measure based on the Fisher information matrix may be adopted for designing optimized tomography schemes with resolution tuned to a particular purpose. Quantum state tomography may serve as a paradigm for estimating of more complex objects, for example process tomography. [1] Z. Hradil, Phys. Rev. A 55, R1561 (1997). [2] Z. Hradil, D. Mogilevtsev, and J.Rehacek, Phys. Rev. Lett. 96, 230401 (2006). [3] J.Rehacek, D. Mogilevtsev and Z. Hradil, New J. Phys 8. April, 043022 (2008)

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).

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.

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.

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.

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.

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.

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.

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.