Research Articles:
Perfect sampling for quantum Gibbs states (pp0361-0388)
Daniel
S. Franca
We show how to obtain perfect
samples from a quantum Gibbs state on a quantum computer. To do so, we
adapt one of the ``Coupling from the Past''-algorithms proposed by
Propp
and Wilson. The algorithm has a probabilistic run-time and produces
perfect samples without any previous knowledge of the mixing time of a
quantum Markov chain. To implement it, we assume we are able to perform
the phase estimation algorithm for the underlying Hamiltonian and
implement a quantum Markov chain such that the transition probabilities
between
eigenstates
only depend on their energy. We provide some examples of quantum Markov
chains that satisfy these conditions and analyze the expected run-time
of the algorithm, which depends strongly on the degeneracy of the
underlying Hamiltonian. For
Hamiltonians
with highly degenerate spectrum, it is efficient, as it is
polylogarithmic in the dimension
and linear in the mixing time. For non-degenerate spectra, its
runtime
is essentially the same as its classical counterpart, which is linear in
the mixing time and quadratic in the dimension, up to a logarithmic
factor in the dimension. We analyze the circuit depth necessary to
implement it, which is proportional to the sum of the depth
necessary to implement one step of the quantum Markov chain and one
phase estimation. This algorithm is stable under noise in the
implementation of different steps. We also briefly discuss how to adapt
different ``Coupling from the Past''-algorithms to the quantum setting.
Ent: A Multipartite
entanglement measure, and parameterization of entangled states (pp0389-0442)
Samuel
R. Hedemann
A multipartite entanglement
measure called the ent is presented and shown to be an entanglement
monotone, with the special property of automatic normalization.
Necessary and sufficient conditions are developed for constructing
maximally entangled states in every multipartite system such that they
are true-generalized X states (TGX) states, a generalization of the Bell
states, and are extended to general nonTGX states as well. These
results are then used to prove the existence of maximally entangled
basis (MEB) sets in all systems. A parameterization of general pure
states of all ent values is given, and proposed as a multipartite
Schmidt decomposition. Finally, we develop an ent vector and ent
array to handle more general definitions of multipartite entanglement,
and the ent is extended to general mixed states, providing a general
multipartite entanglement measure.
Candidates for universal measures of multipartite entanglement (pp0443-0471)
Samuel
R. Hedemann
We propose and examine several
candidates for universal multipartite entanglement measures. The
most promising candidate for applications needing entanglement in the
full Hilbert space is the ent-concurrence, which detects all
entanglement correlations while distinguishing between different types
of distinctly multipartite entanglement, and simplifies to the
concurrence for two-qubit mixed states. For applications where
subsystems need internal entanglement, we develop the absolute ent-concurrence
which detects the entanglement in the reduced states as well as the full
state.
Composition of PPT maps
(pp0472-0480)
Mathew
Kennedy, Nicholas A. Manor, and Vern I. Paulsen
M.
Christandl
conjectured that the composition of any trace preserving
PPT
map with itself is entanglement breaking. We prove that
Christandl's
conjecture holds asymptotically by showing that the distance between the
iterates of any
unital
or trace preserving
PPT
map and the set of entanglement breaking maps tends to zero. Finally,
for every graph we define a one-parameter family of maps on matrices and
determine the least value of the parameter such that the map is
variously, positive, completely positive,
PPT
and entanglement breaking in terms of properties of the graph. Our
estimates are sharp enough to conclude that
Christandl's
conjecture holds for these families.
Linear bosonic
quantum channels defined by superpositions
(pp0481-0496)
T.J. Volkoff
A minimal energy quantum
superposition of two maximally distinguishable,
isoenergetic
single mode Gaussian states is used to construct the system-environment
representation of a class of linear
bosonic
quantum channels acting on a single
bosonic
mode. The quantum channels are further defined by unitary dynamics of
the system and environment corresponding to either a passive linear
optical element $U_{\mathrm{BS}}$
or two-mode squeezing $U_{\mathrm{TM}}$.
The notion of
nonclassicality distance is used to
show that the initial environment superposition state becomes maximally
nonclassical as the constraint
energy is increased. When the system is initially prepared in a coherent
state, application of the quantum channel defined by
$U_{\mathrm{BS}}$
results in a
nonclassical
state for all values of the environment energy constraint. We also
discuss the following properties of the quantum channels: 1) the maximal
noise that a coherent system can tolerate, beyond which the linear
bosonic
attenuator channel defined by
$U_{\mathrm{BS}}$
cannot impart
nonclassical
correlations to the system, 2) the noise added to a coherent system by
the phase-preserving linear amplification channel defined by
$U_{\mathrm{TM}}$,
and 3) a generic lower bound for the trace norm contraction coefficient
on the closed, convex hull of energy-constrained Gaussian states.