33 research outputs found
The Clifford group, stabilizer states, and linear and quadratic operations over GF(2)
We describe stabilizer states and Clifford group operations using linear
operations and quadratic forms over binary vector spaces. We show how the
n-qubit Clifford group is isomorphic to a group with an operation that is
defined in terms of a (2n+1)x(2n+1) binary matrix product and binary quadratic
forms. As an application we give two schemes to efficiently decompose Clifford
group operations into one and two-qubit operations. We also show how the
coefficients of stabilizer states and Clifford group operations in a standard
basis expansion can be described by binary quadratic forms. Our results are
useful for quantum error correction, entanglement distillation and possibly
quantum computing.Comment: 9 page
Asymptotic adaptive bipartite entanglement distillation protocol
We present a new asymptotic bipartite entanglement distillation protocol that
outperforms all existing asymptotic schemes. This protocol is based on the
breeding protocol with the incorporation of two-way classical communication.
Like breeding, the protocol starts with an infinite number of copies of a
Bell-diagonal mixed state. Breeding can be carried out as successive stages of
partial information extraction, yielding the same result: one bit of
information is gained at the cost (measurement) of one pure Bell state pair
(ebit). The basic principle of our protocol is at every stage to replace
measurements on ebits by measurements on a finite number of copies, whenever
there are two equiprobable outcomes. In that case, the entropy of the global
state is reduced by more than one bit. Therefore, every such replacement
results in an improvement of the protocol. We explain how our protocol is
organized as to have as many replacements as possible. The yield is then
calculated for Werner states.Comment: 11 pages, 5 figures, RevTeX
On the geometry of entangled states
The basic question that is addressed in this paper is finding the closest
separable state for a given entangled state, measured with the Hilbert Schmidt
distance. While this problem is in general very hard, we show that the
following strongly related problem can be solved: find the Hilbert Schmidt
distance of an entangled state to the set of all partially transposed states.
We prove that this latter distance can be expressed as a function of the
negative eigenvalues of the partial transpose of the entangled state, and show
how it is related to the distance of a state to the set of positive partially
transposed states (PPT-states). We illustrate this by calculating the closest
biseparable state to the W-state, and give a simple and very general proof for
the fact that the set of W-type states is not of measure zero. Next we show
that all surfaces with states whose partial transposes have constant minimal
negative eigenvalue are similar to the boundary of PPT states. We illustrate
this with some examples on bipartite qubit states, where contours of constant
negativity are plotted on two-dimensional intersections of the complete state
space.Comment: submitted to Journal of Modern Optic
Hashing protocol for distilling multipartite CSS states
We present a hashing protocol for distilling multipartite CSS states by means
of local Clifford operations, Pauli measurements and classical communication.
It is shown that this hashing protocol outperforms previous versions by
exploiting information theory to a full extent an not only applying CNOTs as
local Clifford operations. Using the information-theoretical notion of a
strongly typical set, we calculate the asymptotic yield of the protocol as the
solution of a linear programming problem.Comment: 13 pages, 3 figures, RevTeX
Stabilizer states and Clifford operations for systems of arbitrary dimensions, and modular arithmetic
We describe generalizations of the Pauli group, the Clifford group and
stabilizer states for qudits in a Hilbert space of arbitrary dimension d. We
examine a link with modular arithmetic, which yields an efficient way of
representing the Pauli group and the Clifford group with matrices over the
integers modulo d. We further show how a Clifford operation can be efficiently
decomposed into one and two-qudit operations. We also focus in detail on
standard basis expansions of stabilizer states.Comment: 10 pages, RevTe
An efficient algorithm to recognize local Clifford equivalence of graph states
In [Phys. Rev. A 69, 022316 (2004)] we presented a description of the action
of local Clifford operations on graph states in terms of a graph transformation
rule, known in graph theory as \emph{local complementation}. It was shown that
two graph states are equivalent under the local Clifford group if and only if
there exists a sequence of local complementations which relates their
associated graphs. In this short note we report the existence of a polynomial
time algorithm, published in [Combinatorica 11 (4), 315 (1991)], which decides
whether two given graphs are related by a sequence of local complementations.
Hence an efficient algorithm to detect local Clifford equivalence of graph
states is obtained.Comment: 3 pages. Accepted in Phys. Rev.
Stabilizer state breeding
We present a breeding protocol that distills pure copies of any stabilizer
state from noisy copies and a pool of predistilled pure copies of the same
state, by means of local Clifford operations, Pauli measurements and classical
communication.Comment: RevTeX4, 9 pages, 1 figur
Local invariants of stabilizer codes
In [Phys. Rev. A 58, 1833 (1998)] a family of polynomial invariants which
separate the orbits of multi-qubit density operators under the action of
the local unitary group was presented. We consider this family of invariants
for the class of those which are the projection operators describing
stabilizer codes and give a complete translation of these invariants into the
binary framework in which stabilizer codes are usually described. Such an
investigation of local invariants of quantum codes is of natural importance in
quantum coding theory, since locally equivalent codes have the same
error-correcting capabilities and local invariants are powerful tools to
explore their structure. Moreover, the present result is relevant in the
context of multipartite entanglement and the development of the
measurement-based model of quantum computation known as the one-way quantum
computer.Comment: 10 pages, 1 figure. Minor changes. Accepted in Phys. Rev.
A comparison of the entanglement measures negativity and concurrence
In this paper we investigate two different entanglement measures in the case
of mixed states of two qubits. We prove that the negativity of a state can
never exceed its concurrence and is always larger then
where is the concurrence of the state.
Furthermore we derive an explicit expression for the states for which the upper
or lower bound is satisfied. Finally we show that similar results hold if the
relative entropy of entanglement and the entanglement of formation are
compared