Operator structures in quantum information theory (12w5084)
Patrick Hayden (McGill University)
Marius Junge (University of Illinois, Urbana-Champaign)
David Kribs (University of Guelph)
Mary Beth Ruskai (Tufts University)
Andreas Winter (University of Bristol)
Quantum information science is a rapidly developing field whose significance ranges from fundamental issues in the foundations of quantum theory to new state-of-the-art methods for secure transmission of information. The potential for powerful new methods of computation, data transmission and encryption has led to new perspectives on such entire fields as computation complexity and Shannon information theory. Work in this highly interdisciplinary area overlaps many different fields of mathematics.
In this workshop we will concentrate on the role of operator structures, i.e., operator algebras, operator spaces, and operator systems, in quantum information theory and, conversely, the impact of quantum information science on these fields. Operator algebras have a long history as a framework for quantum theory, including quantum statistical mechanics. In quantum information science, interactions with the environment play a major role, giving a natural interpretation to the auxiliary space which is an essential component of operator spaces.
Operator spaces have been used implicitly in quantum information theory (QIT) since its emergence as a distinct fields in the 1990's, most notably by the use of completely positive, trace-preserving maps to model noise, and the use of a completely bounded norm known as the "diamond" norm in QIT. In the last few years operator spaces have begun to play a more prominent role, leading to new developments in mathematics as well as quantum information science. Some of these were direct outgrowths of earlier workshops as described below.
Experiments showing violations of classical correlation inequalities known as "Bell inequalities" have been used to demonstrate the non-classical nature of probability in quantum theory. In 1993, Tsirelson made a connection between Bell inequalities and operator algebras in his proof that violations for bipartite systems were bounded by Grothendieck's constant. In the ground-breaking paper*, Perez-Garcia et al used operator spaces and tensor norms to solve a long-standing question about violations of Bell inequalities for tripartite physical systems. Their results led to a reformulation of an open question (for over 30 years) about Q-algebras and its subsequent resolution. This was the first of a series of papers using operator space methods to prove new results about Bell inequalities, the most recent of which is the counter-intuitive result that large violations can be obtained with small entanglement.
(* The core members of the the Bell inequality group are D. Perez-Garcia, M.M Wolf, C. Palazuelos, and M. Junge. Additional authors have also been involved in some papers and not all members of this group contributed to all papers.)
This work was done, in part, at the first workshop which brought together operator space theorists and quantum information scientists at BIRS in 2007. This featured expository talks to introduce these two communities to their different languages, as well as reports on recent work on quantum error correction, quantum Shannon theory, and quantum state discrimination. A major feature of the workshop was the presentation and vigorous discussion of open questions in quantum information, several of which were subsequently solved. Two of these, "additivity" and the "quantum Birkhoff conjecture" merit a detailed discussion.
A major focus of discussion was a long-standing group of equivalent conjectures known as "additivity". It was then believed that a closely related conjecture about multiplicativity of the maximal output p-norm held for $1 < p leq 2$, and participants were challenged to find explicit counter-examples for all $p > 2$. In July, 2007, A. Winter did so. But instead of providing additional evidence for the conjecture for $1 < p < 2$, this led P. Hayden to extend the counter-examples to this region, leaving only the original question of additivity of minimal output entropy unresolved. (Entropy can be considered the $p = 1$ case when multiplicativity is reformulated in terms of additivity of Renyi entropy.) The minimal entropy p = 1 conjecture was shown to be false a year later by M. Hastings.
Because BIRS could not accommodate another workshop on this theme in 2009, a second workshop was held at the Fields Institute in July, 2009. As a direct result of lectures and discussions about Hastings work during that event, S. Szarek made a critical observation relating additivity counter-examples to Dvoretzky's theorem, a result little-known outside the field of high-dimensional convex geometry. Soon after, Aubrun, Szarek and E. Werner showed that the existence of additivity violations for $p > 2$ were a direct consequence of Dvoretzky's theorem. However, extending this to p = 1 required a sharper version of Dvoretzky's theorem, which has now been achieved by the same group.
By August 2010, Patrick Hayden had used Dvoretzky's theorem to simplify several earlier results about generic entanglement and prove new results in quantum cryptography. As a direct consequence of the 2009 Fields workshop an important (and somewhat obscure) mathematical tool previously unknown to quantum information theorists has begun to play an important role.
Also inspired by Hastings' work, Collins and Nechita took a completely different direction. Working in the framework of free probability (a subject motivated by basic questions about the classification of von Neumann algebras) they developed powerful new tools for analyzing spectral behavior. Their approach is able to provide much better bounds on both the size of additivity violations, and the dimensions at which violations appear, than other approaches. Their methods will be important in other areas i which free probability and random matrices have applications.
Another open question raised in 2007 was called the "asymptotic quantum Birkhoff conjecture". The resolution of this question was announced by Haagerup and Musat in a July, 2010 BIRS workshop on "Noncommutative Lp spaces, Operator spaces and Applications" to which a few quantum information theorists were also invited. This result generated a great deal of excitement and vigorous discussions between the two groups. This led to further discussions in September, 2010 at Mittag-Leffler in which additional connections were made between properties of quantum channels and the notion of factorization for maps on von Neumann algebras. Further work in this direction can be expected to lead to new developments in both fields. It is worth emphasizing that Paulsen's participation in the 2007 BIRS workshop and open problems discussion, made him aware of the "asymptotic quantum Birkhoff conjecture" which he later brought to the attention of Haagerup and Musat.
Quantum information theory has also led to several reformulations of what is known as the "Connes' embedding problem" or Kirchberg's "QWEP conjecture" which is known to be equivalent. Indeed, the Haagerup and Musat work discussed above has implications for this problem. Recently, Scholz and R. Werner worked with the "Bell group" to show that a question of Tsirelson about quantum correlations can stated in a form equivalent to Connes' embedding problem.
Another major open question in QIT, called the "NPT problem" asks about the conditions under which entanglement can be "distilled" using LOCC (local operations and classical communication). As described at the 2009 Fields workshop and the 2010 BIRS workshop, LOCC can be stated in purely operator algebraic terms involving tensor products of algebras for two quantum systems with a third classical algebra. LOCC can then be considered as a sequence of maps non-trivial on only one quantum algebra, but interacting with the classical algebra. Although it was recognized in 1999 that the NPT problem can be reformulated as a question about 2-positivity of maps on operator algebras, this question has only recently come to the attention of operator algebraists.
There are other open questions in which operator structures may play a role. The quantum analogue of Shannon's information theory is much richer than its classical counterpart because of the many different scenarios for transmitting information. Shannon's noisy coding theorem showed that the asymptotic capacity is given by the mutual information associated with a single use of the channel. Although there has been tremendous progress in this field, with capacity expressions known for many protocols, most of these can not be expressed in a simple "one-shot" form from a single use of the channel. Instead, they require a regularization, which is hard to evaluate explicitly. Therefore, good bounds on the capacity are extremely desirable. The possibility of using operator structure toprovide such bounds has just begun to be explored.
A few other topics are worth a brief mention. Some results on continuity of channel capacity were obtained by Leung and G. Smith using CB norms; however, not all cases were settled. The use of positive, but not completely positive, maps as entanglement witnesses has generated interest in cones of k-positive maps and their duals. Non-commutative versions of such classical quantities as Chernoff distance and Hoeffding bounds have been obtained and applied to questions about state discrimination. However, here is still much to be done in the complex field of non-commutative statistics needed for hypothesis testing and state discrimination in quantum theory.
Both QIT and studies of operator structures are extremely active fields which have begun to have a strong influence on each other. It is extremely important to have opportunities for people from these different areas to meet and exchange ideas. The three workshops mentioned above at BIRS in 2007, Fields in 2009 and BIRS in 2010 have had a major impact on both mathematics and quantum information science. With the rapid development and increased cross-fertilization of these areas, it is important to have another workshop in 2012. Although there are many conferences and workshops in QIT, most of them are focussed on other areas, such as computational complexity, algorithm development, cryptography, etc. We are not aware of any other plans for workshops which duplicate our proposal to bring together mathematicians working on operator algebras, spaces and systems with quantum information scientists.