Physlib

Physlib.QuantumMechanics.DDimensions.Operators.Covariance

Covariance

i. Overview

In this module we define the covariance of two partial linear maps `A` and `B` in a common state `ψ` as the real part of the inner product of their centered vectors.

ii. Key results

- `covariance` : the real part of the centered inner product. - `covariance_comm` : covariance is symmetric in the two observables. - `covariance_eq_re_symm_centered` : covariance as the real part of the symmetrized centered inner product. - `covariance_self_eq_variance` : the covariance of an observable with itself is its variance.

iii. Table of contents

  • A. Covariance

iv. References

  • [B. C. Hall, *Quantum Theory for Mathematicians*, Chapter 12][hall2013quantum].

A. Covariance

5 declarations

definition

Covariance of AA and BB in state ψ\psi

For two partial linear maps AA and BB on a complex inner product space and a state vector ψ\psi in the domain of both, the covariance is defined as the real part of the inner product of their centered vectors. Mathematically, it is given by Re⟨ψA,ψB⟩\text{Re} \langle \psi_A, \psi_B \rangle, where ψA=Aψ−⟨A⟩ψψ\psi_A = A\psi - \langle A \rangle_\psi \psi and ψB=Bψ−⟨B⟩ψψ\psi_B = B\psi - \langle B \rangle_\psi \psi are the centered vectors of AA and BB with respect to ψ\psi, and ⟨T⟩ψ\langle T \rangle_\psi denotes the expectation value of an operator TT.

theorem

Covψ(A,B)=Re⟨ψA,ψB⟩\text{Cov}_\psi(A, B) = \text{Re} \langle \psi_A, \psi_B \rangle

Let AA and BB be partial linear maps on a complex inner product space. For a state vector ψ\psi in the domain of both AA and BB, the covariance of AA and BB in state ψ\psi is equal to the real part of the inner product of their centered vectors: Covψ(A,B)=Re⟨ψA,ψB⟩\text{Cov}_\psi(A, B) = \text{Re} \langle \psi_A, \psi_B \rangle where ψA=Aψ−⟨A⟩ψψ\psi_A = A\psi - \langle A \rangle_\psi \psi and ψB=Bψ−⟨B⟩ψψ\psi_B = B\psi - \langle B \rangle_\psi \psi are the centered vectors, and ⟨T⟩ψ=Re⟨ψ,Tψ⟩\langle T \rangle_\psi = \text{Re} \langle \psi, T\psi \rangle is the expectation value of an operator TT.

theorem

Symmetry of Covariance: Covψ(A,B)=Covψ(B,A)\text{Cov}_\psi(A, B) = \text{Cov}_\psi(B, A)

For any two partial linear maps AA and BB on a complex inner product space HH and a state vector ψ\psi belonging to the domain of both AA and BB, the covariance of AA and BB in the state ψ\psi is equal to the covariance of BB and AA in the state ψ\psi. Mathematically, this is expressed as Covψ(A,B)=Covψ(B,A)\text{Cov}_\psi(A, B) = \text{Cov}_\psi(B, A), where the covariance is defined as the real part of the inner product of the centered vectors of AA and BB.

theorem

Covψ(A,B)=12Re(⟨ψA,ψB⟩+⟨ψB,ψA⟩)\text{Cov}_\psi(A, B) = \frac{1}{2} \text{Re} (\langle \psi_A, \psi_B \rangle + \langle \psi_B, \psi_A \rangle)

Let AA and BB be two partial linear maps on a complex inner product space HH, and let ψ\psi be a vector in the domain of both operators. Let ψA=Aψ−⟨A⟩ψψ\psi_A = A\psi - \langle A \rangle_\psi \psi and ψB=Bψ−⟨B⟩ψψ\psi_B = B\psi - \langle B \rangle_\psi \psi be the centered vectors of AA and BB with respect to ψ\psi. The covariance of AA and BB in the state ψ\psi is equal to the real part of the symmetrized inner product of their centered vectors: Covψ(A,B)=12Re(⟨ψA,ψB⟩+⟨ψB,ψA⟩)\text{Cov}_\psi(A, B) = \frac{1}{2} \text{Re} \left( \langle \psi_A, \psi_B \rangle + \langle \psi_B, \psi_A \rangle \right)

theorem

covψ(A,A)=Varψ(A)\text{cov}_\psi(A, A) = \text{Var}_\psi(A)

Let HH be a complex inner product space and AA be a partial linear map on HH. For any vector ψ\psi in the domain of AA, the covariance of AA with itself in the state ψ\psi is equal to the variance of AA in the state ψ\psi: covψ(A,A)=Varψ(A)\text{cov}_\psi(A, A) = \text{Var}_\psi(A) where the covariance is defined as the real part of the inner product of the centered vectors, and the variance is the squared norm of the centered vector.