Quantum Logic - Automorphisms

Automorphisms

An automorphism of Q is a bijective mapping α:Q → Q which preserves the orthocomplemented structure of Q, that is

for any sequence {E_i}_i of pairwise orthogonal self-adjoint projections. Note that this property implies monotonicity of α. If P is a quantum probability measure on Q, then E → α(E) is also a quantum probability measure on Q. By the Gleason theorem characterizing quantum probability measures quoted above, any automorphism α induces a mapping α* on the density operators by the following formula:

The mapping α* is bijective and preserves convex combinations of density operators. This means

whenever 1 = r₁ + r₂ and r₁, r₂ are non-negative real numbers. Now we use a theorem of Richard V. Kadison:

Theorem. Suppose β is a bijective map from density operators to density operators which is convexity preserving. Then there is an operator U on the Hilbert space which is either linear or conjugate-linear, preserves the inner product and is such that

for every density operator S. In the first case we say U is unitary, in the second case U is anti-unitary.

Remark. This note is included for technical accuracy only, and should not concern most readers. The result quoted above is not directly stated in Kadison's paper, but can be reduced to it by noting first that β extends to a positive trace preserving map on the trace class operators, then applying duality and finally applying a result of Kadison's paper.

The operator U is not quite unique; if r is a complex scalar of modulus 1, then r U will be unitary or anti-unitary if U is and will implement the same automorphism. In fact, this is the only ambiguity possible.

It follows that automorphisms of Q are in bijective correspondence to unitary or anti-unitary operators modulo multiplication by scalars of modulus 1. Moreover, we can regard automorphisms in two equivalent ways: as operating on states (represented as density operators) or as operating on Q.