28 research outputs found
Unbounded entanglement in nonlocal games
Quantum entanglement is known to provide a strong advantage in many two-party
distributed tasks. We investigate the question of how much entanglement is
needed to reach optimal performance. For the first time we show that there
exists a purely classical scenario for which no finite amount of entanglement
suffices. To this end we introduce a simple two-party nonlocal game ,
inspired by Lucien Hardy's paradox. In our game each player has only two
possible questions and can provide bit strings of any finite length as answer.
We exhibit a sequence of strategies which use entangled states in increasing
dimension and succeed with probability for some .
On the other hand, we show that any strategy using an entangled state of local
dimension has success probability at most . In addition,
we show that any strategy restricted to producing answers in a set of
cardinality at most has success probability at most .
Finally, we generalize our construction to derive similar results starting from
any game with two questions per player and finite answers sets in which
quantum strategies have an advantage.Comment: We have removed the inaccurate discussion of infinite-dimensional
strategies in Section 5. Other minor correction
Complexity classification of two-qubit commuting hamiltonians
We classify two-qubit commuting Hamiltonians in terms of their computational
complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can
apply to any pair of qubits, starting in a computational basis state. We prove
a dichotomy theorem: either this model is efficiently classically simulable or
it allows one to sample from probability distributions which cannot be sampled
from classically unless the polynomial hierarchy collapses. Furthermore, the
only simulable Hamiltonians are those which fail to generate entanglement. This
shows that generic two-qubit commuting Hamiltonians can be used to perform
computational tasks which are intractable for classical computers under
plausible assumptions. Our proof makes use of new postselection gadgets and Lie
theory.Comment: 34 page
Nonlocal Games and Quantum Permutation Groups
We present a strong connection between quantum information and quantum
permutation groups. Specifically, we define a notion of quantum isomorphisms of
graphs based on quantum automorphisms from the theory of quantum groups, and
then show that this is equivalent to the previously defined notion of quantum
isomorphism corresponding to perfect quantum strategies to the isomorphism
game. Moreover, we show that two connected graphs and are quantum
isomorphic if and only if there exists and that are
in the same orbit of the quantum automorphism group of the disjoint union of
and . This connection links quantum groups to the more concrete notion
of nonlocal games and physically observable quantum behaviours. We exploit this
link by using ideas and results from quantum information in order to prove new
results about quantum automorphism groups, and about quantum permutation groups
more generally. In particular, we show that asymptotically almost surely all
graphs have trivial quantum automorphism group. Furthermore, we use examples of
quantum isomorphic graphs from previous work to construct an infinite family of
graphs which are quantum vertex transitive but fail to be vertex transitive,
answering a question from the quantum group literature.
Our main tool for proving these results is the introduction of orbits and
orbitals (orbits on ordered pairs) of quantum permutation groups. We show that
the orbitals of a quantum permutation group form a coherent
configuration/algebra, a notion from the field of algebraic graph theory. We
then prove that the elements of this quantum orbital algebra are exactly the
matrices that commute with the magic unitary defining the quantum group. We
furthermore show that quantum isomorphic graphs admit an isomorphism of their
quantum orbital algebras which maps the adjacency matrix of one graph to that
of the other.Comment: 39 page
Counterexamples in self-testing
In the recent years self-testing has grown into a rich and active area of
study with applications ranging from practical verification of quantum devices
to deep complexity theoretic results. Self-testing allows a classical verifier
to deduce which quantum measurements and on what state are used, for example,
by provers Alice and Bob in a nonlocal game. Hence, self-testing as well as its
noise-tolerant cousin -- robust self-testing -- are desirable features for a
nonlocal game to have.
Contrary to what one might expect, we have a rather incomplete understanding
of if and how self-testing could fail to hold. In particular, could it be that
every 2-party nonlocal game or Bell inequality with a quantum advantage
certifies the presence of a specific quantum state? Also, is it the case that
every self-testing result can be turned robust with enough ingeniuty and
effort? We answer these questions in the negative by providing simple and fully
explicit counterexamples. To this end, given two nonlocal games
and , we introduce the -game, in which the players get pairs of questions and choose
which game they want to play. The players win if they choose the same game and
win it with the answers they have given. Our counterexamples are based on this
game.Comment: Added references, changed titl
Graph-theoretical Bounds on the Entangled Value of Non-local Games
We introduce a novel technique to give bounds to the entangled value of
non-local games. The technique is based on a class of graphs used by Cabello,
Severini and Winter in 2010. The upper bound uses the famous Lov\'asz theta
number and is efficiently computable; the lower one is based on the quantum
independence number, which is a quantity used in the study of
entanglement-assisted channel capacities and graph homomorphism games.Comment: 10 pages, submission to the 9th Conference on the Theory of Quantum
Computation, Communication, and Cryptography (TQC 2014