Research Statement. Dr. Anna Vershynina. August 2017

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1 August 2017 My research interests lie in functional analysis and mathematical quantum physics. In particular, quantum spin systems, open quantum systems and quantum information theory. Quantum information theory is an actively developing area at the intersection of many research area, in particular, mathematics and physics, with an ultimate goal of developing new technologies, in particular quantum computer. In recent years quantum information theory has become a highly attractive and captivating field not only mathematicians, but also physicists, electrical engineers and computer scientists. Mathematical side of quantum information theory involves almost every area of mathematics such as analysis, algebra, geometry, probability, functional analysis, partial differential equations and information theory. In the next few sections I will describe some of the work done by myself and with collaborators in more detail. Some work will be completely omitted; please, find complete references in the list of publications. My long-term research goals include a comprehensive understanding of the limits of quantum computation in which information-theoretic inequalities play an essential role, as discussed in section 1 and 4; an extensive investigation of the dynamics of entanglement in multipartite open systems and its applications, see section 2; the achievement of the universal quantum computation, several methods of which are described in sections 3. 1 Geometric inequalities in quantum information theory Geometric inequalities, such as Brunn-Minkowski inequality and the isoperimetric inequality can be generalized to classical information theory, but present a challenge to do so in quantum information theory. Brunn-Minkowski inequality relates the volumes of individual sets with the volume of their Minkowski sum. For A, B R d their Minkowski sum is defined as A + B := {a + b; a A, b B}. Brunn-Minkowski inequality gives a lower bound on the volume of a sum-set: vol(a + B) 1/d vol(a) 1/d + vol(b) 1/d. In classical information theory one may consider a random variable as an analogue of a set in geometry. Identifying the entropy power of a random variable with a volume of a set, one ends up with a Shannon s entropy power inequality e 2H(X+Y )/d e 2H(X)/d + e 2H(Y )/d, here X and Y are random variables on R d with probability density functions f X and f Y. The entropy of a random variable X is given by H(X) = R d f X (x) log f X (x)dx. The density function of a sum of two random variables X + Y is given by a convolution f X+Y (x) = R d f X (x z)f Y (z)dz. 1

2 In quantum information theory a state (positive operator with unit trace) is a natural analogue of a random variable in classical theory. The entropy of a state ρ is given by S(ρ) = Tr(ρ log ρ) and the entropy power of a d-mode state is given by N(ρ) = exp{s(ρ)/d}. The problem to relate the entropy power of the sum of states with entropy powers of each state arises from the unknown meaning of the sum of states on two different systems. A proposition to consider a beamsplitter produced a meaningful analogue of the entropy power inequality, but failed to generalize the isoperimetric inequality [1], [2]. In my recent work with collaborators [3] we conquer this challenge by considering a special kind of convolution between a classical random variable X and a quantum state ρ f X t ρ = f X (z)w ( tz)ρw ( tz)dz, R d where W (z) is a Weyl displacement operator. When X is taken to be a Gaussian r.v. Z, this convolution describes a heat diffusion semigroup generated by a quantum analogue of a Laplacian, i.e. f Z t ρ = e tl heat(ρ) for t 0. This convolution between a classical probability function and a quantum Wigner function complements the existing two types of convolution operations: between classical probability density functions and between quantum Wigner functions, making a progress towards a unified view on all three of them. For this convolution operation we establish numerous inequalities, some of which we prove to be tight. In particular, we establish the entropy power inequality and the isoperimetric inequality. Consequently, we apply the isoperimetric inequality to show the quantum analogue of a Log-Sobolev inequality. It in turn can be used to show that for a Gaussian initial state a quantum Ornstein-Uhlenbeck semigroup converges to a fixed point with an exponential rate. This work is contained in the article Geometric inequalities from phase space translations [3] submitted for publication to the Journal of Mathematical Physics. Future work: To go from geometric point of view to the information theoretic one we have identified volumes with the entropy power. Instead of the entropy power one may consider a mean photon number of a Gaussian state with identical entropy as the original state. While we prove tightness for many inequalities, for isoperimetric inequality in particular, necessary and sufficient condition for equality are currently unknown for all inequalities. 2 Entanglement rates in closed and open systems One of the governing phenomenons in quantum theory is the entanglement. The question we consider in this work is: given an interaction within a system and/or with an environment, how can we entangle particles in the most efficient way? Consider two parties Alice and Bob, who have access to systems A and B respectively, together with ancilla (additional side ) systems a and b respectively. Starting in a pure state Ψ on all systems+ancillas the entanglement entropy E(Ψ) quantifies the amount of entanglement in this system. In 2007 [4] S. Bravyi conjectured that the rate of change Γ(H) = d dt t=0 E(Ψ(t)) of the entanglement entropy under a unitary evolution governed by a Hamiltonian H is upper bounded by a quantity with a logarithmic dependence on a dimension of the smallest system A or B. Being open for six years, E. H. Lieb and I [6] made a step toward proving the conjecture by providing the upper bound Γ(H) (4/ln2) H d with linear dependence on the smallest dimension d. The logarithmic dependence was later proved by Acoleyen et al. in [7]. Page 2

3 In my subsequent work I have generalized the conjecture to include open bipartite quantum systems with a generator of the irreversible dynamics L given in a Lindblad form. I have provided an upper bound on the entangling rate for a relative entropy of entanglement in open ancilla-free systems which has a logarithmic dependence on a dimension of a smaller system in a bipartite cut, Γ R (L) f(l) ln d, here f(l) is a function that depends on norms of the generator of the irreversible quantum dynamics. I have also investigated the rate of change of quantum mutual information in an ancilla-assisted open system and provided an upper bound independent of dimension of ancillas. This work is contained in the article Entanglement rates in bipartite open systems published in Physical Review A [5]. Future work: My latest results motivate a question of finding a class of entanglement measures for which the entanglement rate is tightly upper bounded by a quantity dependent of the smallest dimension of a system and independent of ancillas. Having considered only bipartite systems, it would be intriguing to generalize the problem to multipartite systems and investigate the dynamics of entanglement there. The entangling rate problem was applied to show the stability of the area law in the closed system [7]. It remains to be seen whether one may find a similar behavior in open systems. 3 Universal quantum computation For the last twenty years it has been known that quantum computers offer exponential computational speedup over classical computers. One of the present goals of the quantum information theory is to achieve the universal quantum computation in order to build quantum computer. In 2004 it was shown that quantum adiabatic computation offers a way to realize it [8]. Adiabatic Hamiltonian depends on a control parameter and the state of the computation is represented by the ground state of this Hamiltonian. By slowly changing a control parameter, one can execute a gate (a unitary transformation of data qubits) and therefore, in time, any quantum computation by moving from one ground state to another. The original class of universal Hamiltonians, considered in [8], was not practical, which motivated the search for a simple Hamiltonian viable for universal adiabatic quantum computer. A conceptually distinct idea to assign clock register to every qubit was first explored for universal adiabatic quantum computation by Mizel et. al. in 2007 [9], but the analysis presented there failed to establish the desired results. In joint work with B. Terhal and D. Gosset, I investigated the model that utilized local clock register, called space-time circuit-to-hamiltonian construction. We showed how one can achieve efficient universal quantum adiabatic computation using this construction. The Hamiltonian we use describes a system of interacting particles which live on the edges of a two dimensional grid. Space-time circuit-to-hamiltonian construction can also be used for universal quantum walk. In 2007 Janzing [10] proposed a scheme for universal computation with a time-independent Hamiltonian. We extended those results and proved that the proposed quantum walk efficiently simulates a quantum circuit. We also presented a new analysis based on the quantum walk on Young s lattice. This work is contained in the article Universal adiabatic quantum computation via the space-time circuit-to-hamiltonian construction published in Physical Review Letters, [11]. Future work: A system that we investigated is two-dimensional, which rises a problem of generalization of this construction to the higher-dimensional systems. A search for a simple practical construction which can be applied to quantum walks or adiabatic quantum computation is also underway. Page 3

4 4 Lieb-Robinson bounds Quantum entanglement shows that local operations can be felt immediately throughout the whole system of interacting particles. A revolutionary idea that there is a finite speed of propagation of information in quantum spin systems with finite (or exponentially decaying) interactions was proved mathematically by Lieb and Robinson in 1972 [12]. The bound they proved showed that a main chunk of a support of a local observable grows with a finite speed under unitary evolution. The bound is now known as the Lieb-Robinson bound and the speed is known as the Lieb-Robinson velocity. The bound turns the locality properties of physical systems into the existence of, and upper bound for this speed. Lieb-Robinson bounds can be applied to prove Lieb-Schultz-Mattis theorem, the existence of the thermodynamic limit and exponential decay of correlations in quantum lattice systems, which arises in quantum information theory and quantum computation. Introduction to Lieb-Robinson bounds can be found online in my article written jointly with Elliott Lieb for Scholarpedia: bounds. In collaboration with B. Nachtergaele and V. Zagrebnov, I generalized the existing Lieb-Robinson bounds to the case of an irreversible quantum dynamics, where the dynamics has a Hamiltonian part as well as a dissipative part. The dissipative part is described by terms of Lindblad form. Moreover, the dynamics generated by both Hamiltonian and dissipative interactions, that have suitably fast decay in space, may depend on time. We used this result to prove the existence of the infinite dynamics as a strongly continuous cocycle of unit preserving completely positive maps. This work has been published in the article Lieb-Robinson bound and the existence of the thermodynamic limit for the class if irreversible dynamics published in AMS Contemporary Mathematics, [13]. Future work: Lieb-Robinson bounds show that the support of an observable in a quantum spin system grows at most linearly in time, up to exponentially small errors. The challenge is to sharpen the bound by showing a faster growth of the support. References [1] R. König, G. Smith, The Entropy Power Inequality for Quantum Systems, Information Theory, IEEE Transactions on, 60(3):1536, (2014) [2] G. De Palma, A. Mari, S. Lloyd, V. Giovannetti, Multimode quantum entropy power inequality, Physical Review A, 91(3):032320, (2015) [3] S. Huber, R. König, A. Vershynina, Geometric inequalities from phase space translations, arxiv: , (submitted to Journal of Mathematical Physics), (2016) [4] S. Bravyi, Upper bounds on entangling rates of bipartite Hamiltonians, Physical Review A, 76(5):052319, (2007) [5] A. Vershynina, Entanglement rates for bipartite open systems, arxiv: , Physical Review A, 92(2):022311, (2015) [6] E. H. Lieb, A. Vershynina, Upper bound on mixing rates, arxiv: , Quantum Information and Computation, 13(11&12):0986, (2013) [7] K. Van Acoleyen, M. Mariën, F. Verstraete, Physical Review Letters, 111:170501, (2013) Page 4

5 [8] D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, O. Regev, Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation, SIAM Journal of Computing, 37:166, (2007) [9] A. Mizel, D. A. Lidar, M. Mitchell, Simple proof of equivalence between adiabatic quantum computation and the circuit model, Physical Review Letters, 99: (2007) [10] D. Janzing, Spin-1/2 particles moving on a 2D lattice with nearest-neighbor interactions can realize an autonomous quantum computer, Physical Review A, 75(1):012307, (2007) [11] D. Gosset, B. M. Terhal, A. Vershynina, Universal adiabatic quantum computation via the spacetime circuit-to-hamiltonian construction, arxiv: , Physical Review Letters, 114: (2015) [12] E. H. Lieb, D. Robinson, The finite group velocity of quantum spin systems, Communications in Mathical Physics, 28:251, (1972) [13] B. Nachtergaele, A. Vershynina, V. A. Zagrebnov, Lieb-Robinson bound and the existence of the thermodynamic limit for the class of irreversible dynamics, arxiv: , AMS Contemporary Mathematics, 552:161, (2011) Page 5