Loewner Order

linear-algebra operator-theory quantum-mechanics

Definition

Loewner Order

Let be a complex Hilbert space, and let

be the set of Hermitian operators on . The Loewner order is the partial order on defined by

where means that is a positive semidefinite operator:

Quadratic Form Characterisation

The definition can be read directly through expectation values:

Thus is larger than exactly when every state vector assigns at least as large an expectation value to as to .

Order Properties

On , the Loewner order is a partial order.

Partial order

For Hermitian operators :

1. Reflexivity: .

2. Antisymmetry: If and , then .

3. Transitivity: If and , then .

Proof

Reflexivity follows from .

For antisymmetry, and imply

for all . Hence for all , so .

For transitivity, and imply

The sum of positive semidefinite operators is positive semidefinite, so .

Positive Cone

The positive semidefinite operators form the positive cone of the order:

The interval between and the identity is

This interval is exactly the set of measurement effects.

Relation to Eigenvalues

In finite dimensions,

where are the eigenvalues of the Hermitian operator .

Equivalently, diagonalise the difference and check whether all eigenvalues are non-negative.

Not an Entrywise Order

For matrices, in the Loewner order does not mean

It means instead that the difference has non-negative quadratic form.

Matrix comparison

Let

Then

The eigenvalues of are and , so . Hence

Quantum Measurement Use

For a measurement effect ,

For every normalised quantum state , the Loewner bounds imply

So the Loewner order is the formal reason why effects produce valid Born-rule probabilities.