Quantum state exclusion

In this tutorial, we are going to cover the problem of quantum state exclusion. We are going to briefly describe the problem setting and then describe how one may use toqito to calculate the optimal probability with which this problem can be solved for a number of different scenarios.

Quantum state exclusion is very closely related to the problem of quantum state distinguishability. It may be useful to consult the following tutorial that covers quantum state distinguishability:

Further information beyond the scope of this tutorial can be found in the text [tPBR12] as well as the course [tBJOP14].

The state exclusion problem

The quantum state exclusion problem is phrased as follows.

  1. Alice possesses an ensemble of \(n\) quantum states:

    \[\begin{equation} \eta = \left( (p_0, \rho_0), \ldots, (p_n, \rho_n) \right), \end{equation}\]

where \(p_i\) is the probability with which state \(\rho_i\) is selected from the ensemble. Alice picks \(\rho_i\) with probability \(p_i\) from her ensemble and sends \(\rho_i\) to Bob.

  1. Bob receives \(\rho_i\). Both Alice and Bob are aware of how the ensemble is defined but he does not know what index \(i\) corresponding to the state \(\rho_i\) he receives from Alice is.
  2. Bob wants to guess which of the states from the ensemble he was not given. In order to do so, he may measure \(\rho_i\) to guess the index \(i\) for which the state in the ensemble corresponds.

This setting is depicted in the following figure.

quantum state exclusion

The quantum state exclusion setting.

Note

The primary difference between the quantum state distinguishability scenario and the quantum state exclusion scenario is that in the former, Bob want to guess which state he was given, and in the latter, Bob wants to guess which state he was not given.

Optimal probability of conclusively excluding a quantum state

Optimal probability of unambiguously excluding a quantum state

References

[tPBR12]Pusey, Matthew, Barret, Jonathan, and Rudolph, Terry “On the reality of the quantum state” Nature Physics 8.6 (2012): 475-478. arXiv:1111.3328
[tBJOP14]Bandyopadhyay, Somshubhro, Jain, Rahul, Oppenheim, Jonathan, Perry, Christopher “Conclusive exclusion of quantum states” Physical Review A 89.2 (2014): 022336. arXiv:1306.4683