## Q-Math Seminar

### Fabio Di Cosmo (UC3M)

#### Evolution of Quantum States, a Lagrangian description. Part II

##### Wednesday the 12th of June, 2019, 13:00, UC3M, Seminar Room 2.2D08

The dynamical evolution of a quantum system is assumed to be linear. However, the restriction of linear maps to subsystems, whenever such an operation is meaningful, can result in a non-linear dynamical systems. Therefore, a similar procedure could be considered also in a quantum setting, in order to investigate possible sources of non-linearities. In particular, in the Schrodinger picture we can start with a Hilbert space associated to a given quantum system, and consider a non-linear subset of this vector space. Even if it is not always possible to restrict a vector field to a submanifold, it is always possible to restrict covariant objects. Therefore, in this setting, it seems more natural to consider a Lagrangian description, since it involves differential forms which can be pulled-back to submanifolds. In other words, starting with a Lagrangian description for the quantum evolution, one induces a Lagrangian description on the submanifold of selected states, and the associated Euler-Lagrange equations provide a set of equations of the motion for the restricted system. In general, this evolution will be different from the quantum one, since the submanifold could not be preserved under the evolution. However, in this talk I will present some situations in which the two procedures coincide, but the induced dynamics is not linear.

In continuation of the first talk, in this second part I will focus on some applications involving other types of action at the quantum level, the starting point being the unitary evolution of density matrices for finite dimensional quantum systems. Some remarks on more general actions and separability will be considered in the conlusions.

### Fredy E. Sosa Núñez (UC3M)

#### Structured perturbation theory for eigenvalues of symplectic matrices

##### Wednesday the 29th of May, 2019, 13:00, UC3M, Seminar Room 2.2D08

In this talk, a first order perturbation theory for eigenvalues of real or complex $J$-symplectic matrices under structure-preserving perturbations is presented. Since the class of symplectic matrices has an underlying multiplicative structure, Lidskii's classical formulas for small additive perturbations of the form $\widehat{A} = A + \varepsilon B$, cannot be applied directly, so a new multiplicative perturbation theory is first developed: given an arbitrary square matrix $A$, we obtain the leading terms of the asymptotic expansions in the small, real parameter $\varepsilon$ of multiplicative perturbations $\widehat{A}(\varepsilon)=(I+\varepsilon\,B+\cdots\,)A(I+\varepsilon\,C+\cdots)$ of $A$ for arbitrary matrices $B$ and $C$. After showing that any small structured perturbation $\widehat{S}$ of a symplectic matrix $S$ can be written as $\widehat{S}=\widehat{S}(\varepsilon)=\left( I+\varepsilon B + \cdots \right)S$ with Hamiltonian first-order coefficient $B$, we apply the previously obtained Lidskii-like formulas for multiplicative perturbations to the symplectic case by exploiting the particular connections that symplectic structure induces in the Jordan form between normalized left and right eigenvectors. Special attention is given to eigenvalues on the unit circle, particularly to the exceptional eigenvalues $\pm 1$, whose behavior under structure-preserving perturbations is known to differ significantly from the behavior under arbitrary ones. Also, several numerical examples are generated in order to illustrate the asymptotic expansions and confirm our findings.

### Fabio Di Cosmo (UC3M)

#### Evolution of Quantum States, a Lagrangian description

##### Wednesday the 10th of April, 2019, 13:00, UC3M, Seminar Room 2.2D08

The dynamical evolution of a quantum system is assumed to be linear. However, the restriction of linear dynamical maps to submanifolds of states, whenever such an operation is meaningful, can result in a non-linear dynamical system. Such a procedure will be considered in a quantum setting, in order to investigate the possibility of quantum non-linear equations of motion. In particular, in the Schrödinger picture we can start with a Hilbert space associated to a given quantum system, and consider a non-linear submanifold of this vector space. Even if it is not always possible to restrict a vector field to a submanifold (unless it is tangent to it), it is always possible to restrict possible covariant differential forms associated with the vector field. Therefore, having in mind the restriction to a submanifold of trial states, it seems more natural to consider a Lagrangian description, since it involves differential forms which can be pulled-back to submanifolds. In other words, starting with a Lagrangian description for the quantum evolution, one induces a Lagrangian description on the submanifold of selected states, and the associated Euler-Lagrange equations provide a set of equations of the motion for a “restricted system”. In general, this evolution will be different from the initial quantum one, since the submanifold need not be preserved under the evolution. The proposal is a “dynamical version” of the text-books' “variational method”, used to find approximate solutions for eigenvalues of stationary states by means of trial wave functions. In this talk I will present some cases in which the two procedures coincide, and the induced dynamics is not linear.

The first part of this talk, therefore, will be devoted to the Lagrangian formulation of the unitary evolution for quantum systems. The second part, instead, will be devoted to some applications. The main focus will be on the subspace of squeezed and correlated Gaussian states in $L^2 ( \mathbb{R} )$. However, some remarks on more general situations involving also other types of action at the quantum level will be provided in the conclusions.

### Martin Hebenstreit (Universität Innsbruck)

#### Classically simulable quantum computation & matchgate circuits

##### Wednesday the 27th of March, 2019, 13:00, UC3M, Seminar Room 2.2D08

Although it is believed that quantum computation cannot be classically efficiently simulated in general, there exist certain restricted classes of quantum circuits for which classical simulation is indeed possible. The most prominent example are the Clifford circuits. Here, we consider another such class, the so-called matchgate circuits (MGCs) [1,2]. MGCs can be classically efficiently simulated and moreover, performed as a compressed quantum computation, i.e., the computation can be performed on an quantum computer using exponentially fewer qubits and only polynomial overhead in runtime [3]. We elaborate on and extend recent results [4] on classical simulability of MGCs. To this end, we discuss the notion of magic states in this context.

[1] L. Valiant, SIAM J. Computing **31**, 1229 (2002), B. Terhal and D. DiVincenzo, Phys. Rev. A 65, 032325 (2002)

[2] R. Jozsa and A. Miyake, Proc. R. Soc. A **464**, 3089 (2008)

[3] R. Jozsa, B. Kraus, A. Miyake, J. Watrous, Proc. R. Soc. A **466**, 809 (2010)

[4] D. J. Brod, Phys. Rev. A **93**, 062332 (2016)

### Giuseppe Marmo (Università Federico II, Naples)

#### Quantum Evolution, Contact Manifolds and Dissipation

##### Wednesday the 6th of March, 2019, 13:00, UC3M, Seminar Room 2.2D08

Quantum evolution described on the Hilbert space should preserve the normalization of wave vectors to respect the probabilistic interpretation. The space of normalized vectors in a Hilbert space defines a contact manifold of co-dimension one. A simple generalization of contact-dynamics allows to deal with evolution described by a one-parameter subgroup of the special complex linear subgroup (the complexification of the compact unitary subgroup). The projection on the space of pure states (the complex projective space) represents a Kossakowski-Lindblad vector field up to a 'jumpx' vector field. A 'classical limit' may represent a dissipative system with a 'Rayleigh dissipation'.

References:

[1] Contact Manifolds and Dissipation,Classical and Quantum. F.M.Ciaglia, H.Cruz, G.Marmo. Annals of Physics **398** (2018) 159-179.

[2] Stratified Manifold of Quantum States, Actions of the Complex Special Linear Group. D.Chruscinski, F.M.Ciaglia, A.Ibort, G.Marmo, F.Ventriglia. Annals of Physics **400** (2019) 221-245.

### Patricia Contreras-Tejada (ICMAT)

#### A resource theory of entanglement with a unique multipartite maximally entangled state

##### Wednesday the 27th of February, 2019, 13:00, UC3M, Seminar Room 2.2D08

Entanglement theory is formulated as a quantum resource theory in which the free operations are local operations and classical communication (LOCC). This defines a partial order among bipartite pure states that makes it possible to identify a maximally entangled state, which turns out to be the most relevant state in applications. However, the situation changes drastically in the multipartite regime. Not only do there exist inequivalent forms of entanglement forbidding the existence of a unique maximally entangled state, but recent results have shown that LOCC induces a trivial ordering: almost all pure entangled multipartite states are incomparable (i.e. LOCC transformations among them are almost never possible). In order to cope with this problem we consider alternative resource theories in which we relax the class of LOCC to operations that do not create entanglement. We consider two possible theories depending on whether resources correspond to multipartite entangled or genuinely multipartite entangled (GME) states and we show that they are both non-trivial: no inequivalent forms of entanglement exist in them and they induce a meaningful partial order (i.e. every pure state is transformable to more weakly entangled pure states). Moreover, we prove that the resource theory of GME that we formulate here has a unique maximally entangled state, the generalized GHZ state, which can be transformed to any other state by the allowed free operations.

### J.M. Pérez-Pardo (UC3M)

#### On a proof of monotonicity of quantum relative entropy by A. Uhlmann (part II)

##### Wednesday the 13th of February, 2019, 13:00, UC3M, Seminar Room 2.2D08

Monotonicity is one of the most important properties of the quantum relative entropy. For instance, out of it one can derive the strong sub-additivity property of the entropy. The most general proof was given by A. Uhlmann in 1977 and is now known as Uhlmann's Theorem. This proof relies on a representation result for positive quadratic forms defined on a $*$-Algebra given by W. Pusz and S.L. Woronowicz and that is similar to the GNS construction. The aim of this talk is to present this interesting construction and review the proof of Uhlmann's Theorem.

### J.M. Pérez-Pardo (UC3M)

#### On a proof of monotonicity of quantum relative entropy by A. Uhlmann

##### Wednesday the 6th of February, 2019, 13:00, UC3M, Seminar Room 2.2D08

Monotonicity is one of the most important properties of the quantum relative entropy. For instance, out of it one can derive the strong sub-additivity property of the entropy. The most general proof was given by A. Uhlmann in 1977 and is now known as Uhlmann's Theorem. This proof relies on a representation result for positive quadratic forms defined on a $*$-Algebra given by W. Pusz and S.L. Woronowicz and that is similar to the GNS construction. The aim of this talk is to present this interesting construction and review the proof of Uhlmann's Theorem.

### A. Balmaseda (UC3M)

#### Quantum Control at the Boundary: Quantum Circuits

##### Thursday the 13th of December, 2018, 13:00, UC3M, Seminar Room 2.2D08

The development of Quantum Information Theory and the aim for building quantum computers has increased the relevance of controlling quantum systems. For that reason, some quantum control paradigms have been studied. The controllability of finite dimensional quantum systems is a well understood problem where one can apply the classical theory of control. However, applying such ideas to the infinite dimensional setting is not straightforward, but despite technical difficulties some results on controllability of bilinear systems are known (see for instance Chambrion et. al. [1]).

The quantum control at the boundary (QCB) method is a radically different approach to the problem of controlling the state of a qubit. Instead of seeking the control of the quantum state by directly interacting with it using external magnetic or electric fields, the control of the state will be achieved by manipulating the boundary conditions of the system. The spectrum of a quantum system, for instance an electron moving in a box, depends on the boundary conditions imposed on it, either Dirichlet or Neumann in most cases. A modification of such boundary conditions modifies the state of the system allowing for its manipulation and, eventually, its control.

The QCB paradigm has been used to show how to generate entangled states in composite systems by suitable modifications of the boundary conditions [2], but in spite of its intrinsic interest some basic issues such as the QCB controllability of simple systems has never been addressed. This talk's aim is to explore the (approximate) controllability of such a simple system, a quantum circuit (that is, a free quantum system on a graph), by using QCB.

References:

[1] T. Chambrion, P. Mason, M. Sigalotti, U. Boscain. *Controllability of the discrete-spectrum Schrödinger equation driven by an external field*. Ann. I. H. Poincaré, **26**, 329-349 (2009).

[2] A. Ibort, G. Marmo, J.M. Pérez-Pardo. *Boundary dynamics driven entanglement.* J. Phys. A: Math. Gen. **47**(38), 385301.

### Diego Martínez (UC3M)

#### Amenability in inverse semigroups and C*-algebras

##### Thursday the 29th of November, 2018, 13:00, UC3M, Seminar Room 2.2D08

Amenability, since von Neumann defined it back in 1929, has been an active field of study. By results of various hands, it is well known the amenability (or non-paradoxicality) of a discrete group is equivalent to the existence of a trace in the associated Roe algebra. In this talk we will continue this line of research and prove a similar result for inverse semigroups. We shall prove, for instance, that every amenable inverse semigroup has a tracial Roe algebra, while every properly infinite Roe algebra must come from a paradoxical semigroup. Furthermore, should time allow, we will discuss how these results translate into the groupoid framework. This is joint work with Pere Ara and Fernando Lledó.

### Erik Torrontegui (Instituto de Física Fundamental - CSIC)

#### Implementation and applications of a quantum neuron

##### Thursday the 15th of November, 2018, 13:00, UC3M, Seminar Room 2.2D08

We demonstrate that it is possible to implement a neural network with a sigmoid activation function as an efficient, many-body unitary operation. This unitary operation can be optimally implemented using an Ising model with a transverse field and fast quasi-adiabatic passage. The resulting operation is fully reversible and may have applications also in the realms of quantum sensing or entangled states generation.

### Alberto Ibort (UC3M & ICMAT)

#### On Atiyah’s ‘proof’ of Riemann’s hypothesis

##### Thursday the 18th of October, 2018, 13:00, UC3M, Seminar Room 2.2D08

Quite recently M. Atiyah claimed to have proved the Riemann hypothesis.

In this talk, using Atiyah’s scientific bio as a guide, we will try to understand some of the reasons that could have led him to make such an extraordinary claim.