Partial trace

Partial trace

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In linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

 and functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, the partial trace is a generalization of the trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

. Whereas the trace is a scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

 valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum information
Quantum information
In quantum mechanics, quantum information is physical information that is held in the "state" of a quantum system. The most popular unit of quantum information is the qubit, a two-level quantum system...

 and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories
Consistent histories
In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology...

 and the relative state interpretation.


Suppose V, W are finite-dimensional vector spaces over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, with dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

s m and n, respectively. For any space A let L(A) denote the space of linear operators on A. The partial trace over W, TrW, is a mapping

It is defined as follows:


be bases for V and W respectively; then T
has a matrix representation

relative to the basis


Now for indices k, i in the range 1, ..., m, consider the sum

This gives a matrix bk, i. The associated linear operator on V is independent of the choice of bases and is by definition the partial trace.

Among physicists, this is often called "tracing out" or "tracing over" W to leave only an operator on V in the context where W and V are Hilbert spaces associated with quantum systems (see below).

Invariant definition

The partial trace operator can be defined invariantly (that is, without reference to a basis) as follows: it is the unique linear operator
such that
To see that the conditions above determine the partial trace uniquely, let form a basis for , let form a basis for , let be the map that sends to (and all other basis elements to zero), and let be the map that sends to . Since the vectors form a basis for , the maps form a basis for .

From this abstract definition, the following properties follow:

Partial trace for operators on Hilbert spaces

The partial trace generalizes to operators on infinite dimensional Hilbert spaces. Suppose V, W are Hilbert spaces, and

be an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

 for W. Now there is an isometric isomorphism

Under this decomposition, any operator can be regarded as an infinite matrix
of operators on V

where .

First suppose T is a non-negative operator. In this case, all the diagonal entries of the above matrix are non-negative operators on V. If the sum

converges in the strong operator topology
Strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...

 of L(V), it is independent of the chosen basis of W. The partial trace TrW(T) is defined to be this operator. The partial trace of a self-adjoint operator is defined if and only if the partial traces of the positive and negative parts are defined.

Computing the partial trace

Suppose W has an orthonormal basis, which we denote by ket vector notation as . Then

Partial trace and invariant integration

In the case of finite dimensional Hilbert spaces, there is a useful way of looking at partial trace involving integration with respect to a suitably normalized Haar measure μ over the unitary group U(W) of W. Suitably normalized means that μ is taken to be a measure with total mass dim(W).

Theorem. Suppose V, W are finite dimensional Hilbert spaces. Then

commutes with all operators of the form and hence is uniquely of the form . The operator R is the partial trace of T.

Partial trace as a quantum operation

The partial trace can be viewed as a quantum operation
Quantum operation
In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan...

. Consider a quantum mechanical system whose state space is the tensor product of Hilbert spaces. A mixed state is described by a density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

 ρ, that is
a non-negative trace-class operator of trace 1 on the tensor product
The partial trace of ρ with respect to the system B, denoted by , is called the reduced state of ρ on system A. In symbols,

To show that this is indeed a sensible way to assign a state on the A subsystem to ρ, we offer the following justification. Let M be an observable on the subsystem A, then the corresponding observable on the composite system is . However one chooses to define a reduced state , there should be consistency of measurement statistics. The expectation value of M after the subsystem A is prepared in and that of when the composite system is prepared in ρ should be the same, i.e. the following equality should hold:

We see that this is satisfied if is as defined above via the partial trace. Furthermore it is the unique such operation.

Let T(H) be the Banach space of trace-class operators on the Hilbert space H. It can be easily checked that the partial trace, viewed as a map
is completely positive and trace-preserving.

The partial trace map as given above induces a dual map between the C*-algebras of bounded operators on and given by

maps observables to observables and is the Heisenberg picture
Heisenberg picture
In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...

 representation of .

Comparison with classical case

Suppose instead of quantum mechanical systems, the two systems A and B are classical. The space of observables for each system are then abelian C*-algebras. These are of the form C(X) and C(Y) respectively for compact spaces X, Y. The state space of the composite system is simply

A state on the composite system is a positive element ρ of the dual of C(X × Y), which by the Riesz-Markov theorem corresponds to a regular Borel measure on X × Y. The corresponding reduced state is obtained by projecting the measure ρ to X. Thus the partial trace is the quantum mechanical equivalent of this operation.