123 research outputs found
Single-shot discrimination of quantum unitary processes
We formulate minimum-error and unambiguous discrimination problems for
quantum processes in the language of process positive operator valued measures
(PPOVM). In this framework we present the known solution for minimum-error
discrimination of unitary channels. We derive a "fidelity-like" lower bound on
the failure probability of the unambiguous discrimination of arbitrary quantum
processes. This bound is saturated (in a certain range of apriori
probabilities) in the case of unambiguous discrimination of unitary channels.
Surprisingly, the optimal solution for both tasks is based on the optimization
of the same quantity called completely bounded process fidelity.Comment: 11 pages, 1 figur
Structured Near-Optimal Channel-Adapted Quantum Error Correction
We present a class of numerical algorithms which adapt a quantum error
correction scheme to a channel model. Given an encoding and a channel model, it
was previously shown that the quantum operation that maximizes the average
entanglement fidelity may be calculated by a semidefinite program (SDP), which
is a convex optimization. While optimal, this recovery operation is
computationally difficult for long codes. Furthermore, the optimal recovery
operation has no structure beyond the completely positive trace preserving
(CPTP) constraint. We derive methods to generate structured channel-adapted
error recovery operations. Specifically, each recovery operation begins with a
projective error syndrome measurement. The algorithms to compute the structured
recovery operations are more scalable than the SDP and yield recovery
operations with an intuitive physical form. Using Lagrange duality, we derive
performance bounds to certify near-optimality.Comment: 18 pages, 13 figures Update: typos corrected in Appendi
Optimum Quantum Error Recovery using Semidefinite Programming
Quantum error correction (QEC) is an essential element of physical quantum
information processing systems. Most QEC efforts focus on extending classical
error correction schemes to the quantum regime. The input to a noisy system is
embedded in a coded subspace, and error recovery is performed via an operation
designed to perfectly correct for a set of errors, presumably a large subset of
the physical noise process. In this paper, we examine the choice of recovery
operation. Rather than seeking perfect correction on a subset of errors, we
seek a recovery operation to maximize the entanglement fidelity for a given
input state and noise model. In this way, the recovery operation is optimum for
the given encoding and noise process. This optimization is shown to be
calculable via a semidefinite program (SDP), a well-established form of convex
optimization with efficient algorithms for its solution. The error recovery
operation may also be interpreted as a combining operation following a quantum
spreading channel, thus providing a quantum analogy to the classical diversity
combining operation.Comment: 7 pages, 3 figure
Constructing new optimal entanglement witnesses
We provide a new class of indecomposable entanglement witnesses. In 4 x 4
case it reproduces the well know Breuer-Hall witness. We prove that these new
witnesses are optimal and atomic, i.e. they are able to detect the "weakest"
quantum entanglement encoded into states with positive partial transposition
(PPT). Equivalently, we provide a new construction of indecomposable atomic
maps in the algebra of 2k x 2k complex matrices. It is shown that their
structural physical approximations give rise to entanglement breaking channels.
This result supports recent conjecture by Korbicz et. al.Comment: 9 page
Unital quantum operators on the Bloch ball and Bloch region
For one qubit systems, we present a short, elementary argument characterizing
unital quantum operators in terms of their action on Bloch vectors. We then
show how our approach generalizes to multi-qubit systems, obtaining
inequalities that govern when a ``diagonal'' superoperator on the Bloch region
is a quantum operator. These inequalities are the n-qubit analogue of the
Algoet-Fujiwara conditions. Our work is facilitated by an analysis of
operator-sum decompositions in which negative summands are allowed.Comment: Revised and corrected, to appear in Physical Review
All entangled states are useful for channel discrimination
We prove that every entangled state is useful as a resource for the problem
of minimum-error channel discrimination. More specifically, given a single copy
of an arbitrary bipartite entangled state, it holds that there is an instance
of a quantum channel discrimination task for which this state allows for a
correct discrimination with strictly higher probability than every separable
state.Comment: 5 pages, more similar to the published versio
Iterations of nonlinear entanglement witnesses
We describe a generic way to improve a given linear entanglement witness by a
quadratic, nonlinear term. This method can be iterated, leading to a whole
sequence of nonlinear witnesses, which become stronger in each step of the
iteration. We show how to optimize this iteration with respect to a given
state, and prove that in the limit of the iteration the nonlinear witness
detects all states that can be detected by the positive map corresponding to
the original linear witness.Comment: 11 pages, 5 figure
Mathematical Biology at an Undergraduate Liberal Arts College
Since 2002 we have offered an undergraduate major in Mathematical Biology at Harvey Mudd College. The major was developed and is administered jointly by the mathematics and biology faculty. In this paper we describe the major, courses, and faculty and student research and discuss some of the challenges and opportunities we have experienced
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