2
$\begingroup$

Let {$E_{i}$} be a set of 2x2 POVM operators, satisfying $\sum_{i}E_{i}=\mathbb{I_{2x2}}$.

We know that a general 2x2 Hermitian matrix (say, $H$) can be represented by

$$ H = \left[{\begin{array}{cc} a_{0}+a_{3} & a_{1}-ia_{2} \\ a_{1}+ia_{2} & a_{0}-a_{3} \end{array}}\right]=a_{0}\mathbb{I_{2x2}}+a_{1}\sigma_{1}+a_{2}\sigma_{2}+a_{3}\sigma_{3} $$ where the quantities $a_{k}$ are real, and $\sigma_{i}$'s represent the Pauli matrices.

Is there such a compact way to represent $E_{i}$ [basically satisfying (i) Hermitian peroperty and (ii) positive semi-definiteness], possibly with added constraints?

$\endgroup$
1
$\begingroup$

By applying extra constraints on the above generalized 2x2 Hermitian matrix for satisfying positive semi-definiteness criteria, we can arrive at a generalized representation for 2x2 POVM operator, say $E_{i}$.

The constraint is that the Hermitian matrix should have only non-negative eigenvalues [1].

Let $E_{i}$ be $$ E_{i} = \left[{\begin{array}{cc} a_{i0}+a_{i3} & a_{i1}-ia_{i2} \\ a_{i1}+ia_{i2} & a_{i0}-a_{i3} \end{array}}\right]. $$ The characteristic equation for above the matrix would be $$ \left|{\begin{array}{cc} \lambda-(a_{i0}+a_{i3}) & a_{i1}-ia_{i2} \\ a_{i1}+ia_{i2} & \lambda-(a_{i0}-a_{i3}) \end{array}}\right|=0 $$ The roots of the characteristic polynomial should be non-negative, $$ \lambda^{2}-2a_{i0}\lambda+a_{i0}^{2}-a_{i3}^{2}-a_{i1}^{2}-a_{i2}^{2}=0\\ (\lambda-a_{i0})^{2}=a_{i3}^{2}+a_{i1}^{2}+a_{i2}^{2}\\ \lambda=\pm k+a_{i0} $$ where $k=|\sqrt[]{a_{i1}^{2}+a_{i2}^{2}+a_{i3}^{2}}|$. Hence, we have the condition: $$ a_{i0} \ge k $$ because $\lambda_{min}=-k+a_{i0}$.

By definition, $\sum_{i}E_{i}=\mathbb{I_{2x2}}$. In total, we have (4+$i$) constraints:

  • $\sum_{i}a_{i0}=1$
  • $\sum_{i}a_{i1}=0$
  • $\sum_{i}a_{i2}=0$
  • $\sum_{i}a_{i3}=0$
  • $\forall i, a_{i0} \ge |\sqrt[]{a_{i1}^{2}+a_{i2}^{2}+a_{i3}^{2}}|$

[1] Weisstein, Eric W. "Positive Semidefinite Matrix." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.