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.

Question

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.

Conor Daniel · Accepted Answer

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    vand      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  ≥  0as 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.

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.

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