Covariance

by pxlogpx

Covariance is defined, for two random variables, as the formula
Con(X,Y)=E[(X-E(X))(Y-E(Y))]=E(XY)-E(X)E(Y)

If we have quantum variable, the definition becomes
V_{XY}=Con(X,Y)=\frac{1}{2}E[\{X-E(X),Y-E(Y)\}]. If we have a set of random variables, say x=\{x_1,x_2...x_n\}, a covariance matrix can be defined, V=(V_{ij}), where V_{ij}=\frac{1}{2}E[\{\hat{x}_i-E(x_i),\hat{x}_j-E(x_j)\}]

Now the interesting thing is that the covariance matrix is positive.

Proof:
Since V=\frac{1}{2}E[(x-E(x))\cdot (x-E(x))+(x-E(x))\cdot (x-E(x))]

Given any column vector u, we have
u^TVu=\frac{1}{2}E[u^T(x-E(x))\cdot (x-E(x))u+u^T(x-E(x))\cdot (x-E(x))u]\\ =\frac{1}{2}E[2A^2]\ge 0 where A=(x-E(x))u.
Since u is any vector, V must be positive.