127 research outputs found
Coherent Communication of Classical Messages
We define "coherent communication" in terms of a simple primitive, show it is
equivalent to the ability to send a classical message with a unitary or
isometric operation, and use it to relate other resources in quantum
information theory. Using coherent communication, we are able to generalize
super-dense coding to prepare arbitrary quantum states instead of only
classical messages. We also derive single-letter formulae for the classical and
quantum capacities of a bipartite unitary gate assisted by an arbitrary fixed
amount of entanglement per use.Comment: 5 pages, revtex, v2: updated references, v3: changed title, fixed
error in eq (10
How many copies are needed for state discrimination?
Given a collection of states (rho_1, ..., rho_N) with pairwise fidelities
F(rho_i, rho_j) <= F < 1, we show the existence of a POVM that, given
rho_i^{otimes n}, will identify i with probability >= 1-epsilon, as long as
n>=2(log N/eps)/log (1/F). This improves on previous results which were either
dimension-dependent or required that i be drawn from a known distribution.Comment: 1 page, submitted to QCMC'06, answer is O(log # of states
Extremal eigenvalues of local Hamiltonians
We apply classical algorithms for approximately solving constraint
satisfaction problems to find bounds on extremal eigenvalues of local
Hamiltonians. We consider spin Hamiltonians for which we have an upper bound on
the number of terms in which each spin participates, and find extensive bounds
for the operator norm and ground-state energy of such Hamiltonians under this
constraint. In each case the bound is achieved by a product state which can be
found efficiently using a classical algorithm.Comment: 5 pages; v4: uses standard journal styl
Testing product states, quantum Merlin-Arthur games and tensor optimisation
We give a test that can distinguish efficiently between product states of n
quantum systems and states which are far from product. If applied to a state
psi whose maximum overlap with a product state is 1-epsilon, the test passes
with probability 1-Theta(epsilon), regardless of n or the local dimensions of
the individual systems. The test uses two copies of psi. We prove correctness
of this test as a special case of a more general result regarding stability of
maximum output purity of the depolarising channel. A key application of the
test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain
several structural results that had been previously conjectured, including the
fact that efficient soundness amplification is possible and that two Merlins
can simulate many Merlins: QMA(k)=QMA(2) for k>=2. Building on a previous
result of Aaronson et al, this implies that there is an efficient quantum
algorithm to verify 3-SAT with constant soundness, given two unentangled proofs
of O(sqrt(n) polylog(n)) qubits. We also show how QMA(2) with log-sized proofs
is equivalent to a large number of problems, some related to quantum
information (such as testing separability of mixed states) as well as problems
without any apparent connection to quantum mechanics (such as computing
injective tensor norms of 3-index tensors). As a consequence, we obtain many
hardness-of-approximation results, as well as potential algorithmic
applications of methods for approximating QMA(2) acceptance probabilities.
Finally, our test can also be used to construct an efficient test for
determining whether a unitary operator is a tensor product, which is a
generalisation of classical linearity testing.Comment: 44 pages, 1 figure, 7 appendices; v6: added references, rearranged
sections, added discussion of connections to classical CS. Final version to
appear in J of the AC
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