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?


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

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