Whether the following two inequalities are correct: Tr(M^2 rho) <= Tr(M rho). norm(Mv) <= Tr(M rho), where norm(⋅) is the 2-norm of a vector.

Aryan Lowery 2022-09-29 Answered
Let ρ be a density matrix (positive semidefinite and trace 1), with its rank being 1 such that
ρ = v v ,
where v is a n × 1 unit vector.
Let M be a positive semidefinite matrix such that all its eigenvalues are between 0 and 1.
I am trying to see whether the following two inequalities are correct:
Tr ( M 2 ρ ) Tr ( M ρ ) .
| | M v | | Tr ( M ρ ) ,
where | | | | is the 2-norm of a vector.
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Answers (1)

Conor Daniel
Answered 2022-09-30 Author has 11 answers
Recall the eigendecomposition M = P Λ P . Since
t r a c e ( M 2 ρ ) = t r a c e ( v M 2 v ) = v M 2 v
and
t r a c e ( M ρ ) = v M v ,
we have that
v ( M M 2 ) v = v ( P Λ P P Λ 2 P ) v = v P ( Λ Λ 2 ) P v 0
as the difference Λ Λ 2 is positive semidefinite (because the eigenvalues are bounded by 0 from below and by 1 from above).
Note that your assumptions about v are optional.
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