[Feature] Computing partial trace outputs for statevectors and density matrices

[rajaiitp]

It would nice to have basic density matrix related functionalities that are part of basic quantum information theory computations (partial trace, entropy, fidelity, purity..)

[RolandMacDoland]

@rajaiitp Could you please be more specific about the mathematical formulation of these ? Thanks.

[rajaiitp]

• Partial trace of qubits marked by index i,j

$PT_{ij}(\rho_{ijkl}) = tr_{ij}(\rho_{ijkl}) =\sum_{i=(0,1),j=(0,1)}\rho_{ijkl}$

• Von-Neuman Entropy of density matrix
  $Entropy = - tr(\rho log \rho)$
  trace over all qubit indices.

• Fidelity between 2 density matrices $\rho, \sigma$:
  $Fidelity(\rho,\sigma) = (tr(\sqrt{\sqrt{\rho}\sigma \sqrt{\sigma}}))^2$

here sqrt is defined as diagonalize, sqrt and undiagonalize

• n_th purity of a density matrix rho,
  $Purity(n,\rho) = tr(\rho^n)$

[RolandMacDoland]

Hey @rajaiitp could you please make sure the formulae above are correct ? Thanks.

[RolandMacDoland]

In my opinion, this could go with the introduction of a Result type.

[rajaiitp]

@Roland-djee yes the formulae above are correct, just edited some brackets, didnt know we had support for latex on here. I dont know what a Result type is

[RolandMacDoland]

@Roland-djee yes the formulae above are correct, just edited some brackets, didnt know we had support for latex on here. I dont know what a Result type is

Yeah no worries, it is just an idea for us internally to handle what you suggested.

[mlahariya]

This should be added to backend utils - and should be resolved in qadence 2. Refinement on implementation details is needed.

@rajaiitp Can you help us define the priority of this issue? Is it needed urgently?

[rajaiitp]

Rarely used. Its just a property of the state that sometimes people care to look into. (probably used once or twice so far at best)