The Real Density Matrix
Timothy F. Havel
Received February 24, 2003; accepted March 10, 2003
We introduce a nonsymmetric real matrix which contains all the information that
the usual Hermitian density matrix does, and which has exactly the same tensor
product structure. The properties ofthis matrix are analyzed in detail in the case of
multi-qubit (e.g., spin ¼
=2) systems, where the transformation between the real
and Hermitian density matrices is given explicitly as an operator sum, and used to
convert the essential equations ofthe density matrix formalism into the real domain.
KEY WORDS: Density matrix; superoperator; Hadamard product; nuclear
magnetic resonance; product operator formalism; Pauli algebra.
PACS: 03.65.Ca; 03.67a; 33.25.+k; 02.10.Xm.
The density matrix plays a central role in the modern theory of quantum
mechanics, and an equally important role in its applications to optics,
spectroscopy, and condensed matter physics. Viewed abstractly, it is a self-
adjoint operator on system’s Hilbert space, the expectation values
0 h jj i1 of which give the probability of observing the system in the
state j i. As a matrix, however, it is generally represented vs. the operator
(or ‘‘Liouville’’) basis jiih jj induced by a choice of a complete orthonormal
basis fjiiji ¼ 0; 1; ...g in the underlying Hilbert space. These complex-valued
matrices æ ½hijj ji
are necessarily Hermitian and positive semi-deﬁnite.
Their diagonal entries are the probabilities of these mutually exclusive basis
states, whereas their off-diagonal entries prescribe the amounts by which the
probabilities of their coherent superpositions deviate from the correspond-
ing classical mixtures due to interference.
Another option is to use an operator basis the elements of which
have rank exceeding one, so that it is not induced by any Hilbert space
Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139. E-mail: email@example.com
Quantum Information Processing, Vol. 1, No. 6, December 2002 (# 2003)
1570-0755/02/1200–0511/0 # 2003 Plenum Publishing Corporation