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June 17, 2024

July 6-10, 2009
Workshop on Operator Structures in Quantum Information

Contributed Talks/Posters

Quantum error correction on infinite-dimensional Hilbert spaces
Cedric Beny Centre for Quantum Technologies, National University of Singapore
Coauthors: Achim Kempf, David W. Kribs

We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. The generalization yields new classes of quantum error correcting codes that have no finite-dimensional counterparts. The error correction theory we develop begins with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. Our generalization allows for the correction of codes characterized by any von Neumann algebra, depending on that nature of the noise.

Paper reference: arXiv:0811.0421


The Stinespring Theorem made difficult
Man-Duen Choi Math Dpartment , University of Toronto

For more than three decades, we thought we had known everything about completely positive linear maps. Because of the sudden arrival of the era of quantum computers, we were awakened of many unknown aspects of the the standard structure theorems. Here, I will show off the magic of the difficult Stinespring Theorem in the finite dimensinal case.

Multi-particle decoherence free subspaces and incoherently generated coherences
by Raisa Karasik Applied Science & Technology and Berkeley Quantum Information Center, University of California, Berkeley

Coauthors: K.-P. Marzlin, B. C. Sanders and K. B. Whaley

Decoherence often arises from the interaction between a quantum system and its environment and constitutes a major obstacle for quantum computation and information as it leads to destruction of fragile quantum states. Encoding into decoherence-free subspaces is one strategy to protect quantum states. The bad news is that we have found that decoherence-free subspaces do not exist for extended systems in more than one dimension for a broad class of realistic reservoirs, but the good news is that we have discovered that in some cases the dynamics of a quantum system can connive together with the environmental interactions to reduce decoherence. This property relies on the counterintuitive phenomenon of “incoherently generated coherences” and allows for identification of a new class of decoherence-free states. Examples of such systems are given from cavity quantum electrodynamics and squeezed light.

Paper reference: Phys. Rev. A 77(5): 052301 ( 2008), Phys. Rev. A 76(1): 012331 (2007)


Dilation properties of a bipartite quantum state and quantum analogs of Bell-type inequalities
Elena R. Loubenets. Moscow State Institute of Electronics and Mathematics

We introduce the specifically constructed dilations of a bipartite quantum state; prove the existence of such dilations for any bipartite state, possibly infinitely dimensional, and present, in terms of these dilations, the quantum analogs of bipartite Bell-type inequalities.

Paper reference: JPA 41_ 445303 (2008), JPA 41_ 445304 (2008), Banach Center Publ 73_ 325 (2006), JPA 38_ L653 (2005)


Random quantum channels: graphical calculus
by Ion Nechita, University of Ottawa and Université Lyon 1
Coauthors: Benoît Collins (University of Ottawa and CNRS)

With the aim of studying random constructions arising in quantum information theory, we introduce a diagrammatic notation for tensors, inspired by ideas of Penrose and Coecke. Then, interpreting Weingarten calculus in our formalism, we describe a method for computing expectation values of diagrams which contain Haar-distributed random unitary matrices. This is done by the means of a graph-expansion of the original diagram.

As a first set of applications of the above methods, we compute eigenvalue statistics for outputs of tensor products of independent and conjugate random quantum channels. We obtain the almost sure behavior of the eigenvalues. In the case of conjugate channels, our results improve on known bounds for the largest eigenvalue obtained by Hayden and Winter.

Paper reference: arXiv:0905.2313


Upper bounds on the rate of nondegenerate p-ary stabilizer codes.
Yingkai Ouyang, University of Waterloo

For prime p and q = p2, by viewing the normalizer of a stabilizer code as a q-ary classical code, we use upper bounds on the rate of q-ary codes to obtain upper bounds on the rate of nondegenerate stabilizer codes for fixed stabilizer code distance d and code length n. This is because the distance of any nondegenerate stabilizer code is the distance of its normalizer seen as a classical q-ary code. This substantially improves the quantum Hamming bound for nondegenerate stabilizer codes. The best upper bounds obtained via this method follow from Aaltonen's generalization of the McEcliece, Rumsey, Rodemich and Welch bounds to q-ary codes and the Elias-Bassalygo bound. Previously Rains showed that the maximum distance of an arbitrary quantum code is [(3-v3)/4]n. Using Aaltonen's bound, we show that as n goes goes to infinity, the maximum distance of a nondegenerate stabilizer code is 0.3161n, beating Rain's bound.

All entangled states are useful for channel discrimination
Marco Piani, Institute for Quantum Computing, University of Waterloo
Coauthors: John Watrous

We prove that every entangled state is useful as a resource for the problem of minimum-error channel discrimination. More specifically, given a single copy of an arbitrary bipartite entangled state, it holds that there is an instance of a quantum channel discrimination task for which this state allows for a correct discrimination with strictly higher probability than every separable state.

Paper reference: arXiv:0901.2118


Tsirelson's problem
Volkher Scholz, LUH
Coauthors: Reinhard F. Werner

The situation of two independent observers conducting measurements on a joint quantum system is usually modelled using a Hilbert space of tensor product form, each factor associated to one observer. Correspondingly, the operators describing the observables are then acting non-trivially only on one of the tensor factors. However, the same situation can also be modelled by just using one joint Hilbert space, and requiring that all operators associated to different observers commute, i.e. are jointly measurable without causing disturbance. The problem of Tsirelson is now to decide the question whether all quantum correlation functions between two independent observers derived from commuting observables can also be expressed using observables defined on a Hilbert space of tensor product form. Tsirelson showed already that the distinction is irrelevant in the case that the ambient Hilbert space is of finite dimension. We show here that the problem is equivalent to the question whether all quantum correlation functions can be approximated by correlation function derived from finite-dimensional systems. We also discuss some physical examples which fulfill this requirement.

Paper reference: arXiv:0812.4305

Two-sided bounds on minimum-error quantum measurement, quantum conditional min-entropy, and on the reversibility of quantum dynamics using the first Jezek-Rehacek-Fiurasek-Hradil iterate
Jon Tyson Harvard University

Using a unified framework, we obtain two-sided bounds on the following quantities of interest in quantum information theory:

1. The minimum-error quantum distinguishability of arbitrary ensembles of mixed quantum states.

2. The approximate reversibility of quantum dynamics in terms of entanglement fidelity.

3. The conditional min-entropy of arbitrary bipartite quantum states.

Our primary tool is an abstract generalization of Jezek, Rehacek, and Fiurasek's [Phys. Rev. A 65, 060301] successive approximation scheme for computing optimal measurements and Jezek, Fiurasek, and Hradil's [Phys. Rev. A 68, 012305] scheme for maximum-likelihood reconstruction of quantum channels.

In the case of measurements, we prove a concise factor-of-two estimate for the failure-rate of optimally distinguishing an arbitrary ensemble of mixed quantum states, generalizing work of Holevo [Theor. Probab. Appl. 23, 411 (1978)] and Curlander [Ph.D. Thesis, MIT, 1979]. A modification of the minimal principle of Concha and Poor [Proceedings of the 6th International Conference on Quantum Communication, Measurement, and Computing (Rinton, Princeton, NJ, 2003)] is used to derive a sub-optimal measurement which has an error rate within a factor of two of the optimal by construction. This measurement is quadratically weighted, and has appeared as the first iterate of a sequence of measurements proposed by Jezek, Rehacek, and Fiurasek [Phys. Rev. A 65, 060301]. Unlike the so-called "pretty good" measurement, it coincides with Holevo's asymptotically-optimal measurement in the case of non-equiprobable pure states. A quadratically-weighted version of the measurement bound by Barnum and Knill [J. Math. Phys. 43, 2097 (2002)] is proven. Bounds on the distinguishability of syndromes in the sense of Schumacher and Westmoreland [Phys. Rev. A 56, 131 (1997)] appear as a corollary, allowing one to convert pure-state distinguishability bounds into mixed-state bounds with only a factor-of-two degradation in the failure rate.

Our reversibility bounds for quantum channels are obtained using a quadratically-weighted version of Barnum and Knill's reversal channel. The quadratic weighting of our map is interpreted using a state-dependent functional calculus for quantum channels. Some advantages of our reversal are:

1. We obtain relatively simple reversibility (and distinguishability) estimates, without negative matrix powers.

2. Our channel reversal reduces to Holevo's asymptotically-optimal measurement when used to reverse a pure-state-ensemble-preparation channel.

3. One can replace the output target state with any other state for free.

These advantages are obtained at no cost in tightness of our bounds.

If time permits, I will show how to use matrix monotonicity to prove minimum-error distinguishability bounds.


Paper reference: J. Math. Phys 50, 023106 (2009); Phys Rev A 79, 032343 (2009); to appear

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