Schmidt number for quantum operations
Siendong Huang
*
Department of Applied Mathematics, National Dong Hwa University, Hualien 974, Taiwan
Received 5 January 2006; published 26 May 2006
To understand how entangled states behave under local quantum operations is an open problem in quantum-
information theory. The Jamiolkowski isomorphism provides a natural way to study this problem in terms of
quantum states. We introduce the Schmidt number for quantum operations by this duality and clarify how the
Schmidt number of a quantum state changes under a local quantum operation. Some characterizations of
quantum operations with Schmidt number k are also provided.
DOI: 10.1103/PhysRevA.73.052318 PACS numbers: 03.67.Mn, 03.65.Ud
Entanglement is one of the central concepts in quantum-
information theory. It makes possible many nonintuitive ap-
plications, such as quantum parallelism 1, quantum cryp-
tography 2, quantum teleportation 3, and quantum dense
coding 4. Such applications proceed more effectively if
quantum states are more entangled. One is then interested in
how much a quantum state is entangled. Many entanglement
measures have been suggested, most notably the entangle-
ment of distillation 5 and of formation 5, and the relative
entropy of entanglement 6,7. A basic idea regarding en-
tanglement measures is that the entanglement measures can-
not increase under local quantum operations with classical
communications LOCC. The theory of entanglement mea-
surements is then connected with the behavior of the quan-
tum states under quantum operations. To understand how en-
tanglement behaves when only part of an entangled state is
manipulated becomes a challenging open problem in
quantum-information theory.
By quantum operations we mean general quantum state
manipulations including unitary transformations, positive-
operator-valued measurements, and postselections. In gen-
eral, the trace of a density matrix may not be preserved under
quantum operations. Let S denote the set of positive matrices
with trace less than or equal to 1. In this paper the elements
of S are generally called quantum states. Mathematically,
quantum operations are linear, completely positive, and
trace-nonincreasing mappings from S into itself. A natural
way of describing quantum operations is given by the
Jamiolkowski isomorphism J = between operations
and states , which encodes the dynamical properties of op-
erations with the static properties of states 8,9.
Suppose that H is an n-dimensional Hilbert space and
: BH →BH is a linear bounded mapping from BH
into BH. To each we associate a matrix BH H,
according to
= I P
+
1
where P
+
= 1/ n
i, j=1
n
| ii jj | is the normalized maximally
entangled state. Here P
+
is so chosen that is a quantum
state if is a quantum operation. The mapping = J is
called Jamiolkowski isomorphism. This duality between
and has been discussed in recent works 10–12 and em-
ployed for various purposes 13–15. The inverse mapping is
given by
X = nTr
1
X
T
I 2
for all A BH where A
T
is the transpose of A with respect
to the fixed basis |i and Tr
1
means the partial trace over the
first Hilbert space. Two important relations between and
are that i is Hermitian preserving if and only if is
Hermitian and is completely positive if and only if is
positive; ii is trace preserving if and only if
Tr
1
=
1
n
I . 3
Hence the associated quantum states of quantum operations
are positive matrices whose partial traces are smaller than or
equal to the maximally entangled state.
The physical interpretation of in Eq. 1 corresponding
to the quantum operation is then straightforward if we
identify the first Hilbert space with the system held by Bob
and the second by Alice. Suppose that Alice and Bob share
the maximally entangled normalized state P
+
. Alice performs
the quantum operation on her own subsystem and tells
Bob her outcome. Then is the not necessarily normalized
quantum state shared by Alice and Bob after Alice’s local
operation. It has been shown that the most general strategy of
entanglement manipulations of a pure bipartite is equivalent
to a strategy involving only a single quantum operation by
Alice followed by one-way communication of the result
from Alice to Bob and finally locally unitary transforma-
tions by Bob and Alice16. Thus, if represents the result
of LOCC on P
+
done by Alice and Bob together, is the
total effect of these LOCC on P
+
done only by Alice. In this
way we see that reflects the nonlocal effect of , though
is manipulated locally.
Since contains all the dynamical information of , the
entanglement properties of may reflect how the entangle-
ment of quantum states behaves under . For example, if
is a quantum channel, i.e., is a linear, completely positive,
and trace-preserving mapping and the associated quantum
state is separable, is called an entanglement-breaking
EB channel, which maps every quantum state to a sepa-
rable quantum state 17,18. The entanglement of quantum *Electronic address: sdhuang@mail.ndhu.edu.tw
PHYSICAL REVIEW A 73, 052318 2006
1050-2947/2006/735/0523184 ©2006 The American Physical Society 052318-1