# EQUS

## EQUS (Entanglement in Quantum Systems)

QIT and its importance

In the 80’s, Feynman, Bennett, Brassard and others started to realize that the unusual behavior of Quantum Mechanics (QM) could be used to dramatically improve the way in which we transmit and process information, with the consequent unpredictable applications in essentially all sciences. The amount and importance of the potential consequences of the development of QIT, together with the achievements got so far, have motivated the most prestigious research institutions and companies to invest in this revolutionary field.

Nowadays the field grows so rapidly that it is maybe worthwhile to present it divided in three (interconnected) subfields:

1.- Quantum computation, or how to use QM to improve the way we compute, regarding both hardware and software.

2.- Quantum cryptography, or how to use QM to transmit information in a more secure way.

3.- Theory of entanglement, or how to understand the purely quantum behavior of many body systems.

The role of mathematics

QIT is nowadays an interdisciplinary area, involving mainly Physics, Mathematics and Computer Science. Hence, any development in QIT will necessarily use these three fields. Having this into consideration, the results that our group has obtained so far, as well as those we intend to produce in the future, are very much mathematically oriented and put the emphasis in the mathematical aspects of QIT.

Mathematical theories like Cryptography, Information Theory or Numerical Calculus appear naturally in this context. Moreover, QM is formalized based on the Theory of operators in Hilbert spaces and, therefore, Functional Analysis (which is one of our areas of expertise) also enters the picture. Indeed, fields inside Functional Analysis like Local Banach Space Theory, Tensor Norms, Operator Space Theory, Probability in Banach Spaces, Convex Analysis, … have recently found important applications inside QIT [Au, Au2, Be, Bu, Ha, Ha2, Ho, Pe3].

This interconnection also provides new challenges for mathematicians, proposing new interesting problems, or areas that require completely new mathematics (as it seems to be the case for PEPS theory [Pe2]).

**Concrete projects to be developed**

1.- Understanding quantum correlations.

Bell inequalities [We] are the tool to distinguish quantum from classical correlations using experimental data. However, appart from very particular cases (like CHSH inequality) very few is known in this direction. This is of great importance from three different point of views: from a fundamental point of view it would be desirable to characterize which experimental data are (or not) purely quantum. From a practical point of view, since Bell violations are the only criterion valid to produce unconditionally secure quantum cryptography [Ac], it is desirable to know what is (and not) possible in this context. In addition, recent work [Ke] has shown a closed connection with game theory and complexity theory, that should be explored in more depth.

In our recent paper [Pe3] we established a new connection between Bell inequalities and the theories of Operator Spaces, Tensor Norms and Banach Algebras. Moreover, we showed its power by solving a long standing open problem of Tsirelson. The idea is to follow this direction trying to fully characterize the set of purely quantum correlations.

2.- New resources for Quantum Cryptography.

As commented above, most of the effort in quantum crypto has been devoted to the protocol of key exchange. However, in reality, there are many more situations (specially in the multi-partite case) in which different protocols are requiered (voting, surveying, secret sharing, …). Our aim is to isolate the “primitives” (in the sense of quantum states) needed for different multipartite protocols. This could indirectly lead to a (practical) classification of multipartite entanglement.

3.- MPS and PEPS.

This is the main objective of this project. Our aim is to continue the mathematical study of MPS and PEPS already started in [Pe, Pe2, Pe4]. In particular, we will concentrate on:

- Characterizing the existence of an energy gap (or equivalently criticality) in the Hamiltonians associated to PEPS. Appart from the inherent interest (since there are almost no criterion to decide the presence of gap in 2D) it could have applications in adiabatic quantum computation and in complexity theory.

- Classifying PEPS by renormalization flows (RF), that is, by properties that are scale-independent. Very few is know concerning RF in 2D. One of the few examples known is the recent work of Aguado and Vidal [Ag] in which they show that the toric code is a fix point in each step of the RF given by the MERA ansatz.

- Characterizing topology in PEPS. The first thing to do in this direction would be to look for a proper definition of topological order (at least in the context of PEPS).

- Understanding the existence symmetries in the context of PEPS.

References [Au, Au2, Be, Bu, Ha, Ha2, Ho, Pe3].We Ac Ke Pe Ag

[Ac] A. Acín et al., From Bell's Theorem to Secure Quantum Key Distribution, Phys. Rev. Lett. 97, 120405 (2006).

[Ag] M. Aguado, G. Vidal, Entanglement renormalization and topological order, Phys. Rev. Lett. 100, 070404 (2008)

[Au] G. Aubrun, S.J. Szarek, Tensor products of convex sets and the volume of separable states on N qudits, Phys. Rev. A. 73, 022109 (2006).

[Au2] G. Aubrun, I. Nechita, Catalytic Majorization and lp Norms, Comm. Math. Phys. 278, 133-144 (2008).

[Be] C. H. Bennet et al, Remote preparation of quantum states, IEEE Trans. Inform. Theory, vol. 51, no. 1, pp 56-74, 2005.

[Bu] H. Buhrman er al, Security of quantum bit string commitment depends on the information measure, Phys. Rev. Lett., 97, 250501 (2006) .

[Ha] P. Hayden et al., Randomizing quantum states: Constructions and applications, Commun. Math. Phys. 250(2):371-391, 2004.

[Ha2] P. Hayden, A. Winter, Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Comm. Math. Phys. 284(1):263-280, 2008.

[Ho] M. Horodecjki et al, Partial quantum information, Nature 436:673-676 (2005).

[Pe] D. Pérez-García, M.M. Wolf, F. Verstraete, J.I. Cirac, Matrix Product State Representations, Quant. Inf. Comp. 7, 401 (2007)

[Pe2] D. Pérez-García, M.M. Wolf , F. Verstraete, J.I. Cirac, PEPS as unique ground states of local Hamiltonians, Quant. Inf. Comp. 8, 0650-0663 (2008).

[Pe3] D. Pérez-García, M.M. Wolf, C. Palazuelos, I. Villanueva, M. Junge, Unbounded violation of tripartite Bell inequalities, Comm. Math. Phys. 279, 455 (2008).

[Pe4] D. Pérez-García, M.M. Wolf , M. Sanz, F. Verstraete, J.I. Cirac, String order and symmetries in quantum spin lattices, Phys. Rev. Lett. 100, 167202 (2008).

[Sho] P.W. Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM J.Sci.Statist.Comput. 26 (1997) 1484.

[We] R.F. Werner, M.M. Wolf, Bell inequalities and Entanglement, Quant. Inf. Comp 1 No.3, 1.25 (2001)