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Session 5
Quantum Information and Representation Theory
Chairs:
Giulio Chiribella (Tsinghua University, China)
Matthias Christandl (ETH Zurich, Switzerland)
Aram Harrow (University of Washington, USA)
Luc Vinet (University of Montreal, Canada)
Titles and Abstracts
Gorjan Alagic (California Institute of Technology, USA)
Title: Quantum Computation and Representations of Mapping Class Groups
Abstract: Ten years ago, Freedman, Kitaev, Larsen and Wang established that
simulating certain Topological Quantum Field Theories is universal for quantum
computation. Their pioneering work has led us to many wonderful connections
between quantum computation, complexity theory, and topology. We now know that
the power of quantum computation is exactly captured by the problem of
approximating certain topological invariants, such as the Jones Polynomial and the
Turaev-Viro invariant. It is natural to view these results as consequences of the
universality of simulating certain representations of mapping class groups on a
quantum computer. In this talk, we will review these results from this point of view,
and then discuss a new potential application: obfuscating quantum and classical
circuits.
Marcus Appleby (Perimeter Institute for Theoretical Physics, Canada)
Title: Galois Theory of Symmetric Measurements
Abstract: The problem of proving (or disproving) the existence of symmetric
informationally complete positive operator valued measures (SICs) has been the focus
of much effort in the quantum information community during the last 12 years. In
this talk we describe the Galois invariances of Weyl-Heisenberg covariant SICs (the
class which has been most intensively studied). It is a striking fact that the published
exact solutions (in dimensions 2--16, 19, 24, 35 and 48) are all expressible in terms of
radicals, implying that the associated Galois groups must be solvable. Building on the
work of Scott and Grassl (J. Math. Phys. 51 042203 (2010)) we investigate the Galois
group in more detail. We show that there is an intriguing interplay between the
Galois and Clifford group symmetries. We also show that there are a number of
interesting regularities in the Galois group structure for the cases we have examined.
We conclude with some speculations about the bearing this may have on the SIC
existence problem.
Alessandro Bisio (Pavia University, Italy)
Title: Optimal covariant processing of quantum gates
Abstract: We address the general problem of transforming a quantum gate given by a
unitary representation of a symmetry group into a gate given by a different
representation of the same group. We investigate which are the optimal performances
that quantum mechanics allows in achieving this general task which includes cloning
and learning of unitary transformations as special cases. We will prove that if many
copies of the unknown unitary are provided, the optimal strategy involves a parallel
call of the available uses and we will provide a bound on the dimension of the
ancillary systems required to realize the optimal strategy. Finally we will reduce the
general problem to a set of quadratic equations and we will provide the explicit
solution for the cloning of a phase gate, and for the conversion between gates given
by irreducible representations.
Giulio Chiribella (Tsinghua University, China)
Title: Can we teleport a quantum clock? Fundamental limits to the use of
entanglement to simulate quantum communication
Abstract: In the standard teleportation protocol, the direct transfer of a quantum
system from a sender to a receiver can be perfectly simulated by the transfer of a
finite amount of classical bits, provided that the sender and the receiver share a
sufficient amount of entanglement.
Extending this idea to a different scenario, one can take a quantum version of
Eddington's clock synchronization---where a quantum clock is transferred from the
sender to the receiver---and try to convert it into a new protocol where the direct
transfer of the clock is replaced by the use of entanglement along with the transfer of
classical bits.
In this talk I will show that such a protocol is impossible: in the absence of previous
synchronization, a quantum clock cannot be transferred perfectly using any finite
amount of quantum entanglement and any finite amount of rounds of classical
communication. In general, the equivalence between entanglement and the direct
transfer of quantum systems breaks down in the absence of a shared reference frames
associated to the action of arbitrary compact Lie groups.
After establishing this no-go result for perfect teleportation in the absence of shared
reference frames, I will discuss some optimal approximate teleportation protocols and
compare them with measure-and-prepare protocols based on the estimation of the
unknown group element connecting the reference frame of the sender with the
reference frame of the receiver.
Related paper: G. Chiribella, V. Giovannetti, L. Maccone, and P. Perinotti, Phys. Rev.
A 86, 010304(R) (2012).
Matthias Christandl (ETH Zurich, Switzerland)
Title: 6j-Symbols Determine Eigenvalues of Tripartite Quantum States
Abstract: We reveal a correspondence between the eigenvalues of tripartite quantum
states and the asymptotic behaviour of symmetric group 6j-symbols of F-matrices.
Strong subadditivity of von Neumann entropy then follows from symmetry properties
of the symbols apparent from a graphical calculus used. Encoding the addition of
three Hermitian matrices into tripartite quantum states, we obtain new
concavity-of-entropy-like inequalities for this generalisation of Horn's problem. Our
work be viewed as an extension of work by Wigner, Ponzano and Regge, and Roberts
who connected the asymptotics of the ordinary SU(2) 6j symbols to the existence of
Euclidean tetrahedra.
Alberto Grunbaum (University of California, Berkeley, USA)
Title: Studying recurrence and localization for Quantum walks by means of the
spectral method
Abstract: I will discuss ongoing joint work with L. Velazquez, R. Werner and A.
Werner in the case of discrete time unitary evolutions. I hope to present some results
involving higher dimensional Quantum walks.
Masahito Hayashi (Nagoya University, Japan)
Title: Fourier Analytic Approach to estimation of Group Action
Abstract: In the estimation of group action in quantum system, we can choose input
state as well as the measurement. In this talk, we describe this problem using the term
of Fourier transform in group representation. As an example, we treat the case of
SU(2) and Weyl-Heisenberg representation.
Iman Marvian (Perimeter Institute for Theoretical Physics, Canada)
Title: A generalization of Schur-Weyl duality with applications in quantum estimation
Abstract: Schur-Weyl duality is a powerful tool in representation theory which has
many applications to quantum information theory. We provide a generalization of this
duality and demonstrate some of its applications. In particular, we use it to develop a
general framework for the study of a family of quantum estimation problems wherein
one is given n copies of an unknown quantum state according to some prior and the
goal is to estimate certain parameters of the given state. In particular, we are
interested to know whether collective measurements are useful and if so to find an
upper bound on the amount of entanglement which is required to achieve the optimal
estimation. In the case of pure states, we show that commutativity of the set of
observables that define the estimation problem implies the sufficiency of unentangled
measurements. Joint work with Robert Spekkens.
Rob Spekkens (Perimeter Institute for Theoretical Physics, Canada)
Title: The resource theory of quantum states that break symmetry
Abstract: The asymmetry properties of a quantum state specify how and to what
extent a given symmetry is broken by the state. If implementing symmetric dynamics
is easy while implementing dynamics that break the symmetry is hard or impossible,
then asymmetry becomes a resource (analogous to the resource of entanglement).
The resource theory of quantum asymmetry is important for the field of quantum
metrology because synchronizing clocks and aligning gyroscopes is a problem of
distributing information about a group and the measure of how well a state can
achieve such an alignment is a measure of its asymmetry. It is also significant for
understanding the consequences of symmetric dynamics. For physical problems
where one does not know all the details of the dynamics or one cannot solve it exactly,
measures of the asymmetry of a state provide new constraints on the possible state
evolutions, constraints that are often independent from those imposed by Noether's
theorem. This talk will provide an overview of the main results in the resource
theory of asymmetry and will highlight some of the important open questions. Joint
work with Iman Marvian.
Michael Walter (ETH Zurich, Switzerland)
Title: Eigenvalue Distributions and the Branching Problem
Abstract: Given a random quantum state of multiple (distinguishable or
indistinguishable) particles, we provide an algorithm to compute the joint probability
distribution of the eigenvalues of its one-body reduced density matrices. The
probability distribution is obtained from the corresponding distribution of diagonal
entries, which is given by the volume function of a parametric polytope. As a
corollary, by taking the support of this probability distribution we recover a complete
solution to the corresponding one-body quantum marginal problem. One way of
establishing the above results is by relating them to the asymptotic growth of the
Kronecker coefficients, an asymptotic branching problem. I will also sketch how
similar ideas lead to an efficient algorithm for the general branching problem of
compact, connected Lie groups.
Reinhard Werner (Leibniz University Hannover, Germany)
Title: The complete boundedness norm for symmetric channels (joint work with A.
Ahlbrecht, T. Mahmood, A. Werner)
Abstract: We show how to simplify the explicit evaluation of the cb-norm (also
known as the diamond norm) of a difference of quantum channels, which intertwine
two unitary actions of a symmetry group. An especially simple formula results, when
the group action is irreducible on the input system.