Nonlocal Properties of Two-Qubit Gates and Mixed
States, and the Optimization of Quantum Computations
Received April 30, 2002; accepted July 6, 2002
Entanglement of two parts of a quantum system is a nonlocal property unaﬀected by
local manipulations of these parts. It can be described by quantities invariant under
local unitary transformations. Here we present, for a system of two qubits, a set of
invariants which provides a complete description of nonlocal properties. The set
contains 18 real polynomials of the entries of the density matrix. We prove that one of
two mixed states can be transformed into the other by single-qubit operations if and
only if these states have equal values of all 18 invariants. Corresponding local
operations can be found eﬃciently. Without any of these 18 invariants the set is
incomplete. Similarly, nonlocal, entangling properties of two-qubit unitary gates are
invariant under single-qubit operations. We present a complete set of 3 real poly-
nomial invariants of unitary gates. Our results are useful for optimization of quantum
computations since they provide an eﬀective tool to verify if and how a given two-qubit
operation can be performed using exactly one elementary two-qubit gate, imple-
mented by a basic physical manipulation (and arbitrarily many single-qubit gates).
KEY WORDS: quantum information; entanglement; invariants; nonlocality.
PACS: 03.67-a; 03.67.Lx.
Nonlocality is an important ingredient in quantum information processing,
e.g., in quantum computation and quantum communication. Nonlocal
correlations in quantum systems reﬂect entanglement between its parts.
Genuine nonlocal properties should be described in a form invariant under
1570-0755/02/0800-0243/0 # 2003 Plenum Publishing Corporation
r Theoretische Festko
t Karlsruhe, D-76128 Karlsruhe,
Germany and Landau Institute for Theoretical Physics, Kosygin st. 2, 117940 Moscow,
Russia. E-mail: email@example.com
Quantum Information Processing, Vol. 1, No. 4, August 2002 (# 2003)