We study mathematical aspects of Quantum Information Theory, mainly concerning the issues of data compression, transmission of classical information using quantum channels, teleportation and entanglement.

• Prof. Yuri Suhov
• Dr. Nilanjana Datta, Lecturer in Mathematics & Fellow of Pembroke College
• Dr. Oliver Johnson, Junior Research Fellow of Christ's College
• Petra Scudo, graduate student
• Andrew Skeen, graduate student
• Thomas Voice, graduate student
• Ye Yeo, graduate student

We list below brief summaries of our research.

Nilanjana Datta and Yuri Suhov
A system of interacting qubits can be viewed as a non-i.i.d quantum information source. A possible model of such a source is provided by a quantum spin system, in which spin-1/2 particles located at sites of a lattice interact with each other. We establish the limit for the compression of information from such a source and show that asymptotically it is given by the von Neumann entropy rate. Our result can be viewed as a quantum analogue of Shannon's noiseless coding theorem, and as an extension of Schumacher's coding theorem, for a class of non - i.i.d. quantum information sources. From the probabilistic point of view it is an analogue of the Shannon-McMillan-Breiman theorem, which is considered as a cornerstone of modern Information Theory. We are interested in extending the above work, to exploit the entanglement properties of the quantum system for data compression.

References:
Data compression limit for an information source of interacting qubits; quant-ph/0207069. To appear in the journal Quantum Information Processing.

Oliver Johnson and Yuri Suhov
We use ideas from classical information theory, such as Lempel--Ziv encoding, to study exactly solvable models of quantum statistical mechanics. In the context of quantum information theory, this allows us to assess the entropy of the infinite-volume limit of the Gibbs ensemble. The basic model that we study is a lattice system of of ideal (free) particles (bosons or fermions), though we also discuss the Calogero model.

References:
The von Neumann entropy and information rate for ideal quantum Gibbs ensembles; math-ph/0109023. To appear in the journal Quantum Computers and Computing.

Petra Scudo
In the past two years my research was mainly focused on the study of the use of quantum systems for reliable transmission of classical information. This work was done in collaboration with Asher Peres, at the Technion- Israel Institute for Technology. In particular we considered the use of group theoretical methods in connection with spin systems for the transmission of geometrical coordinates. Later, an extension to relativity was analysed by using the representations of the Poincaré group.
More recently, my research interests moved towards foundational aspects of quantum information in connection with classical probability theory. A collaboration has started with Christopher Fuchs (Bell Laboratories) and Rudiger Schack (University of London) on a new representation of quantum process tomography in relation with the classical De Finetti theorem (work to be published).

References:
A. Peres and P. F. Scudo, "Entangled Quantum States as Direction Indicators", Phys. Rev. Lett. 86, 4160, (2001);
A. Peres and P. F. Scudo, "Transmission of a Cartesian Frame by a Quantum System", Phys. Rev. Lett. 87, 167901, (2001);
A. Peres, P. F. Scudo and D.R. Terno, "Quantum Entropy and Special Relativity", Phys. Rev.Lett. 88, 230402, (2002).
A. Fuchs, P. F. Scudo and R. Schack, "De Finetti representation for quantum operations", (2002), to be published.

Andrew Skeen
I am a second year PhD student supervised by Professor Yuri Suhov in the Statistical Laboratory. We work together on the issue of encoding classical information by using eigenvectors of a density matrix of a system of interacting qubits (the XY model, the one--dimensional hard core model and others). Here we use the classical Lempel--Ziv algorithm, elaborating on a parallel work by O.Johnson and Y.Suhov.
I am also interested in the use of quantum channels to transmit information. This involves a number of questions such as the capacities of quantum channels for transmitting classical or quantum information. Moreover, I am interested in looking at channels with correlated noise (here is an interesting paper). I have also been looking at some ideas in quantum data compression.

Thomas Voice and Prof. Yuri Suhov
We are currently researching into the theoretical properties of "large" quantum systems. Specifically, we consider the limit as the size of a system tends to infinity, and the number of particles in the system is set to be a specific proportion of the size of the system. The aim of this is to try and find examples of theoretical systems which have a large number of subsystems, but each subsystem is greatly entangled with the rest.
Hopefully this research will help give us a better understanding of the nature of entanglement itself, for many particle systems. As yet, much is understood about entanglement between two particle systems, and little is understood about the wider case. In a sense this is analogous to the study of mutual information in classical information theory. Indeed, the entanglement of two states is intuitively very similar to the concept of the mutual information of two random variables.
Entanglement is an important property in the study of quantum information theory. Maximally entangled states have provided many exciting and interesting possibilities, such as quantum information teleportation, superdense coding and transmitting quantum information over channels too noisy for non-entangled transmition.
Also, quantum error correcting codes exhibit entanglement, and perhaps could even be characterised by their entanglement properties. This is another reason for the exploration of entanglement in large systems. Although many good quantum error correcting codes exist in theory, they prove difficult to utilise in practise. Thus the search for a large system that naturally exhibits interesting entanglement properties is also a search for a more natural way of producing quantum error correcting codes.
Initial results have been promising. Our first paper will consider a fermionic system with a natural hamiltonian, under periodic boundary conditions, and calculates entanglement between the system and it's subsystems. The results are considered in the limit for a large system and it is found that in thermal equilibrium non-trivial entanglement occurs and maximal entanglement may be achieved. The paper is currently under preparation and will be available online soon.

Yeo Ye
I am interested in various aspects of quantum information theory. I have studied how thermally entangled spin states of the Heisenberg XX model could be utilized for quantum teleportation of entangled states. The problem of quantum teleportation can be recast into one of reversal of quantum operation. I have explored how this connection could throw light on optimal approximate reversal of quantum operations, and how correlated quantum channels could yield an entanglement teleportation scheme, which allows one to learn about the entanglement associated with the input state without decreasing the amount of entanglement. Currently, Andrew Skeen and I are looking at quantum channels with correlated noise and the role entanglement plays in their capacities.

References:
Quantum channels with correlated noise and entanglement teleportation: quant-ph/0211084;
Optimal approximate reversal of quantum operations on a single qubit: quant-ph/0210071;
Entanglement teleportation via thermally entangled states of two-qubit Heisenberg XX chain: quant-ph/0205088;
Teleportation via thermally entangled state of a two-qubit Heisenberg XX chain : quant-ph/0205014.