**EQUS (Entanglement in Quantum
Systems)**

1. Brief
motivation
2. Programme proposal.

**Brief motivation**

__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)