The definition of fidelity for two mixed ensembles is $F=Tr\sqrt{\sqrt{\rho_1}\rho_2\sqrt{\rho_1}}$.

Now I came across a problem in numerical calculation, Systems A,B are identical, but attached to even and odd numbers of electrons in environment respectively, we will have $\langle c_A(i)\rangle=-\langle c_B(i)\rangle$(assuming environments have smaller fermionic numbering), thus $F<1$(1 expected).

This difference in sign is an artifact, because we won't be allowed to measure a fermionic type operator without attaching a string of sign operators to it, like $P_1P_2\ldots P_{i-1}c(i)$.

How can we see whether two systems are identical or not from fidelity calculation? Or, we must put systems block before environment block in fermionic ordering? What is the most widely accepted way to cope with this sign problem?

  • $\begingroup$ Possibly relevant: physics.stackexchange.com/questions/80555/… $\endgroup$
    – Rococo
    Jun 6, 2016 at 14:39
  • $\begingroup$ @Rococo, Thank you for your suggestion, I think they are essentially the same physics. However, obtaining fidelity that do not rely on fermionic ordering between two mixed ensembles using matrix product states(MPS) is not that easy ...(It is unknown whether it can be achieved ). $\endgroup$
    – 刘金国
    Jun 7, 2016 at 5:08


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