976 research outputs found
Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries
The operator-Schmidt decomposition is useful in quantum information theory
for quantifying the nonlocality of bipartite unitary operations. We construct a
family of unitary operators on C^n tensor C^n whose operator-Schmidt
decompositions are computed using the discrete Fourier transform. As a
corollary, we produce unitaries on C^3 tensor C^3 with operator-Schmidt number
S for every S in {1,...,9}. This corollary was unexpected, since it
contradicted reasonable conjectures of Nielsen et al [Phys. Rev. A 67 (2003)
052301] based on intuition from a striking result in the two-qubit case. By the
results of Dur, Vidal, and Cirac [Phys. Rev. Lett. 89 (2002) 057901
quant-ph/0112124], who also considered the two-qubit case, our result implies
that there are nine equivalence classes of unitaries on C^3 tensor C^3 which
are probabilistically interconvertible by (stochastic) local operations and
classical communication. As another corollary, a prescription is produced for
constructing maximally-entangled operators from biunimodular functions.
Reversing tact, we state a generalized operator-Schmidt decomposition of the
quantum Fourier transform considered as an operator C^M_1 tensor C^M_2 -->
C^N_1 tensor C^N_2, with M_1 x M_2 = N_1 x N_2. This decomposition shows (by
Nielsen's bound) that the communication cost of the QFT remains maximal when a
net transfer of qudits is permitted. In an appendix, a canonical procedure is
given for removing basis-dependence for results and proofs depending on the
"magic basis" introduced in [S. Hill and W. Wootters, "Entanglement of a pair
of quantum bits," Phys Rev. Lett 78 (1997) 5022-5025, quant-ph/9703041 (and
quant-ph/9709029)].Comment: More formal version of my talk at the Simons Conference on Quantum
and Reversible Computation at Stony Brook May 31, 2003. The talk slides and
audio are available at
http://www.physics.sunysb.edu/itp/conf/simons-qcomputation.html. Fixed typos
and minor cosmetic
Protected gates for topological quantum field theories
We study restrictions on locality-preserving unitary logical gates for
topological quantum codes in two spatial dimensions. A locality-preserving
operation is one which maps local operators to local operators --- for example,
a constant-depth quantum circuit of geometrically local gates, or evolution for
a constant time governed by a geometrically-local bounded-strength Hamiltonian.
Locality-preserving logical gates of topological codes are intrinsically fault
tolerant because spatially localized errors remain localized, and hence
sufficiently dilute errors remain correctable. By invoking general properties
of two-dimensional topological field theories, we find that the
locality-preserving logical gates are severely limited for codes which admit
non-abelian anyons; in particular, there are no locality-preserving logical
gates on the torus or the sphere with M punctures if the braiding of anyons is
computationally universal. Furthermore, for Ising anyons on the M-punctured
sphere, locality-preserving gates must be elements of the logical Pauli group.
We derive these results by relating logical gates of a topological code to
automorphisms of the Verlinde algebra of the corresponding anyon model, and by
requiring the logical gates to be compatible with basis changes in the logical
Hilbert space arising from local F-moves and the mapping class group.Comment: 50 pages, many figures, v3: updated to match published versio
Broken Symmetries in the Entanglement of Formation
We compare some recent computations of the entanglement of formation in
quantum information theory and of the entropy of a subalgebra in quantum
ergodic theory. Both notions require optimization over decompositions of
quantum states. We show that both functionals are strongly related for some
highly symmetric density matrices. We discuss the presence of broken symmetries
in relation with the structure of the optimal decompositions.Comment: 21 pages, LateX, no figure
Canonical Decompositions of n-qubit Quantum Computations and Concurrence
The two-qubit canonical decomposition SU(4) = [SU(2) \otimes SU(2)] Delta
[SU(2) \otimes SU(2)] writes any two-qubit quantum computation as a composition
of a local unitary, a relative phasing of Bell states, and a second local
unitary. Using Lie theory, we generalize this to an n-qubit decomposition, the
concurrence canonical decomposition (C.C.D.) SU(2^n)=KAK. The group K fixes a
bilinear form related to the concurrence, and in particular any computation in
K preserves the tangle ||^2 for n
even. Thus, the C.C.D. shows that any n-qubit quantum computation is a
composition of a computation preserving this n-tangle, a computation in A which
applies relative phases to a set of GHZ states, and a second computation which
preserves it.
As an application, we study the extent to which a large, random unitary may
change concurrence. The result states that for a randomly chosen a in A within
SU(2^{2p}), the probability that a carries a state of tangle 0 to a state of
maximum tangle approaches 1 as the even number of qubits approaches infinity.
Any v=k_1 a k_2 for such an a \in A has the same property. Finally, although
||^2 vanishes identically when the
number of qubits is odd, we show that a more complicated C.C.D. still exists in
which K is a symplectic group.Comment: v2 corrects odd qubit CCD misstatements, reference chapter for KAK v3
notation change to coincide with sequel, typos. 20 pages, 0 figure
Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem
The set of doubly-stochastic quantum channels and its subset of mixtures of
unitaries are investigated. We provide a detailed analysis of their structure
together with computable criteria for the separation of the two sets. When
applied to O(d)-covariant channels this leads to a complete characterization
and reveals a remarkable feature: instances of channels which are not in the
convex hull of unitaries can return to it when either taking finitely many
copies of them or supplementing with a completely depolarizing channel. In
these scenarios this implies that a channel whose noise initially resists any
environment-assisted attempt of correction can become perfectly correctable.Comment: 31 page
On the dynamics of initially correlated open quantum systems: theory and applications
We show that the dynamics of any open quantum system that is initially
correlated with its environment can be described by a set of (or less)
completely positive maps, where d is the dimension of the system. Only one such
map is required for the special case of no initial correlations. The same maps
describe the dynamics of any system-environment state obtained from the initial
state by a local operation on the system. The reduction of the system dynamics
to a set of completely positive maps allows known numerical and analytic tools
for uncorrelated initial states to be applied to the general case of initially
correlated states, which we exemplify by solving the qubit dephasing model for
such states, and provides a natural approach to quantum Markovianity for this
case. We show that this set of completely positive maps can be experimentally
characterised using only local operations on the system, via a generalisation
of noise spectroscopy protocols. As further applications, we first consider the
problem of retrodicting the dynamics of an open quantum system which is in an
arbitrary state when it becomes accessible to the experimenter, and explore the
conditions under which retrodiction is possible. We also introduce a related
one-sided or limited-access tomography protocol for determining an arbitrary
bipartite state, evolving under a sufficiently rich Hamiltonian, via local
operations and measurements on just one component. We simulate this protocol
for a physical model of particular relevance to nitrogen-vacancy centres, and
in particular show how to reconstruct the density matrix of a set of three
qubits, interacting via dipolar coupling and in the presence of local magnetic
fields, by measuring and controlling only one of them.Comment: 19 pages. Comments welcom
Phase space properties of charged fields in theories of local observables
Within the setting of algebraic quantum field theory a relation between
phase-space properties of observables and charged fields is established. These
properties are expressed in terms of compactness and nuclearity conditions
which are the basis for the characterization of theories with physically
reasonable causal and thermal features. Relevant concepts and results of phase
space analysis in algebraic quantum field theory are reviewed and the
underlying ideas are outlined.Comment: 33 pages, no figures, AMSTEX, DESY 94-18
- …