June  8, 2023

August 10-14, 2009
Mathematics in Experimental Quantum Information Processing Workshop


Monday August 10

Robin Blume-Kohout
Perimeter Institute

"Tomography: What is it good for?"

State tomography is a basic primitive in experimental quantum information processing. You prepare a bunch of systems in the same way, measure them, and estimate a state (density matrix) from the results. There is an ongoing debate about the "best" way to infer a density matrix from data. The only way to resolve this confusion is to figure out exactly what task[s] we are trying to accomplish -- which in turn determines an operational metric of success, which we can use to conclusively rank different state estimation procedures. In this talk, I'll show why several common tasks don't actually require state estimation, then argue that data compression is a suitable paradigmatic task. I'll conclude by exploring the consequences of this choice, and showing that it implies (among other things) a radically new conception of state estimation.

A.M. Steinberg, R.B.A. Adamson, and L.K. Shalm
Centre for Quantum Information & Quantum Control and Institute for Optical Sciences
Department of Physics, University of Toronto

Measuring Quantum States in the Presence of Fundamental Symmetries

Quantum State Tomography is the science (and sometimes art) of measuring a quantum state. In this talk, I will discuss some of our recent work on characterizing the state of 2- and 3-photon systems, and the broader implications for tomography of multi-dimensional quantum systems.

Typical approaches to quantum tomography are all, to a greater or lesser extent, brute force attempts to measure enough data to be able to extract a density matrix from or fit one to observations. I will argue that these approaches neglect the underlying symmetries of the state space, in ways that come at a cost.

In particular, I will present our theoretical and experimental work on the characterization of states of "partially distinguishable" photons, and Wigner-function tomography of multi-photon states on the Poincaré sphere.matrix. I will present our experimental results on squeezing and oversqueezing of the triphoton, and tomographical reconstructions of the resulting Wigner function.

Peter Turner
University of Tokyo

Comparison of maximum-likelihood and linear reconstruction schemes in
quantum measurement tomography

The effects on quantum states caused by measurement apparatuses can be described in general by sets of completely positive maps called instruments. There exists a linear reconstruction scheme for the instrument describing a given measurement apparatus from experimental data, but the scheme has the disadvantage that it can give unphysical reconstructions. In this poster we propose a maximum-likelihood reconstruction scheme that addresses this disadvantage. We show that our scheme always gives a physical reconstruction, and that it does so more efficiently than the linear scheme.


Colm Ryan
Institute for Quantum Computing

Randomized benchmarking in liquid-state NMR.

Being able to quantify the level of coherent control in a proposed device implementing a quantum information processor (QIP) is an important task for both comparing different devices and assessing a device's prospects with regards to achieving fault-tolerant quantum control. I will review the motivation behind randomized benchmarking as a solution to this problem. I will show results from our experiments in a liquid-state nuclear magnetic resonance QIP using the randomized benchmarking protocol presented by Knill et al (PRA 77: 012307 (2008)). We report an error per randomized p/2 pulse of (1.3±0.1)10-4 with a single qubit QIP and with a generalization to multiple qubits an average error rate for one and two qubit gates of (4.7±0.3)10-3. We estimate that these error rates are still not decoherence limited and thus can be improved with modifications to the control hardware and software. I will show where this fits with other recent results from other implementations where randomized benchmarking error rates have also been measured.

David Cory

Efficient and Robust Decoupling

I will outline the challenges of using decoupling to remove interactions with an environment, show a variety of approaches from magnetic resonance and introduce a new approach that is robust to the expected experimental errors.

Tuesday August 11

John Holbrook
University of Guelph

Introduction to Numerical Ranges

We survey some of the properties and applications of the classical numerical range of a matrix. We make connections with the higher-rank numerical ranges recently studied in the context of quantum error correction.

Marcus Silva
Université de Sherbrooke

Numeric ranges and minimal fidelity guarantees in the physical realization of unitaries

In some physical realizations of quantum information devices it is highly advantageous to consider the implementation of more complex unitaries through continuously varying Hamiltonians, instead of decomposition such unitaries in terms of a discrete gate set. The control parameters of such Hamiltonians are determined by numerical search, where a particular evolution is compared against the desired unitary by considering the fidelity of their outputs, averaged over a uniform distrubution of all input states. In principle this can be improved upon by looking at the minimal fidelity between the outputs of the two unitaries. Casting this problem in terms of numeric ranges, we demonstrate that the minimum fidelity between two unitaries has a simple geometric interpretation, and that it can be readily computed. We conclude by discussing some of the obstacles in generalizing this approach to the comparison between general quantum maps and unitaries.

[Work done in collaboration with C. Ryan (U.Waterloo), M. Laforest (T. U. Delft) and D. W. Kribs (U. Guelph)]

Yiu Tung Poon
Iowa State University

Generalized numerical ranges and quantum error correction

In this talk, geometric properties of the joint higher rank numerical ranges are obtained and their implications to quantum computing are discussed. It is shown that if the dimension of the underlying Hilbert space of the quantum states is sufficiently large, the joint higher rank numerical range of operators is always star-shaped and contains a non empty convex subset. In case the operators are infinite dimensional, the joint infinite rank numerical range of the operators is a convex set lying in the core of all joint higher rank numerical ranges, and is closely related to the joint essential numerical ranges of the operators. In addition, equivalent formulations of the join infinite rank numerical range are obtained. As by products, previous results on essential numerical range of operators are extended.


Cedric Beny
National University of Singapore

Inverting a channel with near-optimal worst-case entanglement fidelity

Avoiding decoherence is the major challenge of any quantum information experiment. Here we consider the possibility of reversing the effect of the noise on a subspace (code) after it happened. Exactly correctable codes are characterized by the Knill-Laflamme conditions. It was shown however that good codes exist which cannot be found under the assumption of exact correctability. Here we give necessary and sufficient conditions for a code, or a subsystem code, to be approximately correctable in terms of the worst-case entanglement fidelity of the noise channel. We also show how to build a family of near-optimal recovery channels.

Wednesday August 12

Man-Duen Choi
University of Toronto

Hard results of the soft mathematics in quantum information

The beautiful setting of completely positive linear maps serves the most vivid description of quantum information. Here, we look into some profound natures of the simple structure of completely positive linear maps on matrix algebras.

Chi-Kwong Li
College of William and Mary

Completely positive linear maps, unitary orbits, and quantum operations

We will some recent results on completely positive linear maps and unitary orbits in connection to the study of quantum operations.

Claudio Altafini
SISSA-ISAS International School for Advanced Studies

Feedback schemes for radiation damping suppression in NMR: a control-theoretical perspective

In NMR spectroscopy, the collective measurement is weakly invasive and its back-action is called radiation damping. The aim of this study is to provide a control-theoretical analysis of the problem of suppressing this radiation damping. It is shown that two of the feedback schemes commonly used in the NMR practice correspond one to a high gain oputput feedback and the other to a 2-degree of freedom control design with a prefeedback that exactly cancels the radiation damping field. A general high gain feedback stabilization design not requiring the knowledge of the radiation damping time constant is also investigated.


Thursday August 13

Thomas Schulte-Herbruggen
Technical University of Munich, Germany

Matching Lie and Markov properties in open quantum systems

On a general scale, Markovian quantum channels can be defined by their Lie-semigroup properties [1] with the GKS-Lindblad generators as Lie wedge or tangent cone. These differential properties provide powerful tools not only when addressing reachability and controllability, but also for deciding whether effective Liouvillians are physical or whether time-dependent Markovian channels simplify to time-independent ones [1].

Applications of optimal control of Markovian and non-Markovian open quantum systems are shown for realistic examples: they typically cut errors by one order of magnitude [2,3]. These dramatic improvements come at the cost of algorithms that in open systems [1,2,3] are far more intricate than in closed ones [4]. -- Implications in view of quantum CISC-compilation [5] for large systems (with say 100 qubits) are given.

Finally we sketch new controllability criteria based on absence of dynamic symmetries [6].

This work contains collaborations mainly with Gunther Dirr, Indra Kurniawan, and Uwe Helmke from the Institute of Mathematics, University of Wuerzburg, Germany.

[1] Dirr, Helmke, Kurniawan, Schulte-Herbrueggen, arXiv:0811.3906, to appear in Rep. Math. Phys. (2009)
[2] Schulte-Herbrueggen, Spoerl, Khaneja, Glaser, quant-ph/0609037
[3] Rebentrost, Serban, Schulte-Herbrueggen, Wilhelm, Phys. Rev. Lett. 102, 090401 (2009)
[4] Schulte-Herbrueggen, Glaser, Dirr, Helmke, arXiv:0802.4195
[5] Schulte-Herbrueggen, Spoerl, Glaser, arXiv:0712.3227
[6] Sander, Schulte-Herbrueggen, arXiv:0904.4654

Masoud Mohseni

Environment-Assisted Quantum Processes

In this talk, we discuss how decoherence can be helpful in guiding an open system to a particular target quantum operation/process. We first develop a general approach for monitoring and controlling evolution of open quantum systems by introducing a dynamical equation for the evolution of the process matrix operating on the system. This equation is applicable to non-Markovian and/or strong coupling regimes. We show how one can identify quantum Hamiltonian systems via partial tomography/estimation, and discuss its efficiency and limitations for certain classes of sparse Hamiltonians. Additionally, we introduce a novel optimal control setting in order to drive quantum dynamics of Hamiltonian systems to a desired target process matrix, specifically to suppress or manipulate decoherence.

In the second part of this talk, we present certain examples of how environmental-induced dynamics can in fact enhance quantum transport (e.g., energy transfer) in biological and nano-scale systems via an effective interplay of free Hamiltonian and dynamical decoherence. Finally, we discuss how one may develop similar techniques for performing quantum information processing with experimentally available open quantum systems.

Bei-Lok Hu
University of Maryland

Entanglement Dynamics between Two Qubits in a Quantum Field: Birth, Death and Revivals



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