50,263 research outputs found
Discriminating quantum states: the multiple Chernoff distance
We consider the problem of testing multiple quantum hypotheses
, where an arbitrary prior
distribution is given and each of the hypotheses is copies of a quantum
state. It is known that the average error probability decays
exponentially to zero, that is, . However, this error
exponent is generally unknown, except for the case that .
In this paper, we solve the long-standing open problem of identifying the
above error exponent, by proving Nussbaum and Szko\l a's conjecture that
. The right-hand side of this equality is
called the multiple quantum Chernoff distance, and
has been previously
identified as the optimal error exponent for testing two hypotheses,
versus .
The main ingredient of our proof is a new upper bound for the average error
probability, for testing an ensemble of finite-dimensional, but otherwise
general, quantum states. This upper bound, up to a states-dependent factor,
matches the multiple-state generalization of Nussbaum and Szko\l a's lower
bound. Specialized to the case , we give an alternative proof to the
achievability of the binary-hypothesis Chernoff distance, which was originally
proved by Audenaert et al.Comment: v2: minor change
Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances
This paper first analyzes the resolution complexity of two random CSP models
(i.e. Model RB/RD) for which we can establish the existence of phase
transitions and identify the threshold points exactly. By encoding CSPs into
CNF formulas, it is proved that almost all instances of Model RB/RD have no
tree-like resolution proofs of less than exponential size. Thus, we not only
introduce new families of CNF formulas hard for resolution, which is a central
task of Proof-Complexity theory, but also propose models with both many hard
instances and exact phase transitions. Then, the implications of such models
are addressed. It is shown both theoretically and experimentally that an
application of Model RB/RD might be in the generation of hard satisfiable
instances, which is not only of practical importance but also related to some
open problems in cryptography such as generating one-way functions.
Subsequently, a further theoretical support for the generation method is shown
by establishing exponential lower bounds on the complexity of solving random
satisfiable and forced satisfiable instances of RB/RD near the threshold.
Finally, conclusions are presented, as well as a detailed comparison of Model
RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively,
exhibit three different kinds of phase transition behavior in NP-complete
problems.Comment: 19 pages, corrected mistakes in Theorems 5 and
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