Positive Semidefinite Operator

linear-algebra quantum-mechanics

Definition

Positive Semidefinite Operator

Let be a complex Hilbert space. A linear operator is positive semidefinite if

In this case one writes

In quantum theory, positive semidefinite operators are the operators that can assign non-negative probabilities through expressions of the form .

Basic Consequences

If , then is Hermitian:

Hence all eigenvalues of are real. In finite dimensions, positivity is equivalent to non-negative eigenvalues:

Spectral Form

In finite dimensions, a Hermitian operator has a spectral decomposition

where each is an orthogonal projection. Then

This gives a direct test: diagonalise the operator and check whether all spectral weights are non-negative.

Factorisation

An operator is positive semidefinite iff it can be written as

for some linear operator .

Indeed,

In particular, every positive semidefinite operator has a positive square root such that

Operator Order

For Hermitian operators and , define

This is the Loewner order. A measurement effect is an operator satisfying

The lower bound ensures non-negative probabilities. The upper bound ensures probabilities at most .

Matrix Example

Let

For ,

Therefore .

By contrast,

is Hermitian but not positive semidefinite, since for ,