173 research outputs found
Quantum coin tossing and bit-string generation in the presence of noise
We discuss the security implications of noise for quantum coin tossing
protocols. We find that if quantum error correction can be used, so that noise
levels can be made arbitrarily small, then reasonable security conditions for
coin tossing can be framed so that results from the noiseless case will
continue to hold. If, however, error correction is not available (as is the
case with present day technology), and significant noise is present, then
tossing a single coin becomes problematic. In this case, we are led to consider
random n-bit string generation in the presence of noise, rather than
single-shot coin tossing. We introduce precise security criteria for n-bit
string generation and describe an explicit protocol that could be implemented
with present day technology. In general, a cheater can exploit noise in order
to bias coins to their advantage. We derive explicit upper bounds on the
average bias achievable by a cheater for given noise levels.Comment: REVTeX. 6 pages, no figures. Early versions contained errors in
statements of security conditions, although results were correct. v4: PRA
versio
Symplectic approach to quantum constraints
A general prescription for the treatment of constrained quantum motion is
outlined. We consider in particular constraints defined by algebraic
submanifolds of the quantum state space. The resulting formalism is applied to
obtain solutions to the constrained dynamics of systems of multiple spin-1/2
particles. When the motion is constrained to a certain product space containing
all of the energy eigenstates, the dynamics thus obtained are quasi-unitary in
the sense that the equations of motion take a form identical to that of unitary
motion, but with different boundary conditions. When the constrained subspace
is a product space of disentangled states, the associated motion is more
intricate. Nevertheless, the equations of motion satisfied by the dynamical
variables are obtained in closed form.Comment: 11 page
Optimally Conclusive Discrimination of Non-orthogonal Entangled States Locally
We consider one copy of a quantum system prepared with equal prior
probability in one of two non-orthogonal entangled states of multipartite
distributed among separated parties. We demonstrate that these two states can
be optimally distinguished in the sense of conclusive discrimination by local
operations and classical communications(LOCC) alone. And this proves strictly
the conjecture that Virmani et.al. [8] confirmed numerically and analytically.
Generally, the optimal protocol requires local POVM operations which are
explicitly constructed. The result manifests that the distinguishable
information is obtained only and completely at the last operation and all prior
ones give no information about that state.Comment: 4 pages, no figure, revtex. few typos correcte
Unconditionally secure quantum bit commitment is impossible
The claim of quantum cryptography has always been that it can provide
protocols that are unconditionally secure, that is, for which the security does
not depend on any restriction on the time, space or technology available to the
cheaters. We show that this claim does not hold for any quantum bit commitment
protocol. Since many cryptographic tasks use bit commitment as a basic
primitive, this result implies a severe setback for quantum cryptography. The
model used encompasses all reasonable implementations of quantum bit commitment
protocols in which the participants have not met before, including those that
make use of the theory of special relativity.Comment: 4 pages, revtex. Journal version replacing the version published in
the proceedings of PhysComp96. This is a significantly improved version which
emphasis the generality of the resul
Distinguishability of States and von Neumann Entropy
Consider an ensemble of pure quantum states |\psi_j>, j=1,...,n taken with
prior probabilities p_j respectively. We show that it is possible to increase
all of the pairwise overlaps || i.e. make each constituent pair
of the states more parallel (while keeping the prior probabilities the same),
in such a way that the von Neumann entropy S is increased, and dually, make all
pairs more orthogonal while decreasing S. We show that this phenomenon cannot
occur for ensembles in two dimensions but that it is a feature of almost all
ensembles of three states in three dimensions. It is known that the von Neumann
entropy characterises the classical and quantum information capacities of the
ensemble and we argue that information capacity in turn, is a manifestation of
the distinguishability of the signal states. Hence our result shows that the
notion of distinguishability within an ensemble is a global property that
cannot be reduced to considering distinguishability of each constituent pair of
states.Comment: 18 pages, Latex, 2 figure
Martingale Models for Quantum State Reduction
Stochastic models for quantum state reduction give rise to statistical laws
that are in most respects in agreement with those of quantum measurement
theory. Here we examine the correspondence of the two theories in detail,
making a systematic use of the methods of martingale theory. An analysis is
carried out to determine the magnitude of the fluctuations experienced by the
expectation of the observable during the course of the reduction process and an
upper bound is established for the ensemble average of the greatest
fluctuations incurred. We consider the general projection postulate of L\"uders
applicable in the case of a possibly degenerate eigenvalue spectrum, and derive
this result rigorously from the underlying stochastic dynamics for state
reduction in the case of both a pure and a mixed initial state. We also analyse
the associated Lindblad equation for the evolution of the density matrix, and
obtain an exact time-dependent solution for the state reduction that explicitly
exhibits the transition from a general initial density matrix to the L\"uders
density matrix. Finally, we apply Girsanov's theorem to derive a set of simple
formulae for the dynamics of the state in terms of a family of geometric
Brownian motions, thereby constructing an explicit unravelling of the Lindblad
equation.Comment: 30 pages LaTeX. Submitted to Journal of Physics
Interest Rates and Information Geometry
The space of probability distributions on a given sample space possesses
natural geometric properties. For example, in the case of a smooth parametric
family of probability distributions on the real line, the parameter space has a
Riemannian structure induced by the embedding of the family into the Hilbert
space of square-integrable functions, and is characterised by the Fisher-Rao
metric. In the nonparametric case the relevant geometry is determined by the
spherical distance function of Bhattacharyya. In the context of term structure
modelling, we show that minus the derivative of the discount function with
respect to the maturity date gives rise to a probability density. This follows
as a consequence of the positivity of interest rates. Therefore, by mapping the
density functions associated with a given family of term structures to Hilbert
space, the resulting metrical geometry can be used to analyse the relationship
of yield curves to one another. We show that the general arbitrage-free yield
curve dynamics can be represented as a process taking values in the convex
space of smooth density functions on the positive real line. It follows that
the theory of interest rate dynamics can be represented by a class of processes
in Hilbert space. We also derive the dynamics for the central moments
associated with the distribution determined by the yield curve.Comment: 20 pages, 3 figure
Quantum noise and stochastic reduction
In standard nonrelativistic quantum mechanics the expectation of the energy
is a conserved quantity. It is possible to extend the dynamical law associated
with the evolution of a quantum state consistently to include a nonlinear
stochastic component, while respecting the conservation law. According to the
dynamics thus obtained, referred to as the energy-based stochastic Schrodinger
equation, an arbitrary initial state collapses spontaneously to one of the
energy eigenstates, thus describing the phenomenon of quantum state reduction.
In this article, two such models are investigated: one that achieves state
reduction in infinite time, and the other in finite time. The properties of the
associated energy expectation process and the energy variance process are
worked out in detail. By use of a novel application of a nonlinear filtering
method, closed-form solutions--algebraic in character and involving no
integration--are obtained for both these models. In each case, the solution is
expressed in terms of a random variable representing the terminal energy of the
system, and an independent noise process. With these solutions at hand it is
possible to simulate explicitly the dynamics of the quantum states of
complicated physical systems.Comment: 50 page
Purifying and Reversible Physical Processes
Starting from the observation that reversible processes cannot increase the
purity of any input state, we study deterministic physical processes, which map
a set of states to a set of pure states. Such a process must map any state to
the same pure output, if purity is demanded for the input set of all states.
But otherwise, when the input set is restricted, it is possible to find
non-trivial purifying processes. For the most restricted case of only two input
states, we completely characterize the output of any such map. We furthermore
consider maps, which combine the property of purity and reversibility on a set
of states, and we derive necessary and sufficient conditions on sets, which
permit such processes.Comment: 5 pages, no figures, v2: only minimal change
Is Quantum Bit Commitment Really Possible?
We show that all proposed quantum bit commitment schemes are insecure because
the sender, Alice, can almost always cheat successfully by using an
Einstein-Podolsky-Rosen type of attack and delaying her measurement until she
opens her commitment.Comment: Major revisions to include a more extensive introduction and an
example of bit commitment. Overlap with independent work by Mayers
acknowledged. More recent works by Mayers, by Lo and Chau and by Lo are also
noted. Accepted for publication in Phys. Rev. Let
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