# Indefinite causal order aids quantum depolarizing channel identification

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## Abstract

Quantum channel identification is the metrological determination of one or more parameters of a quantum channel. This is accomplished by passing probes in prepared states through the channel and then statistically estimating the parameter(s) from the measured channel outputs. In quantum channel identification, the channel parameters’ quantum Fisher information is a means to assess and compare different probing schemes. We use quantum Fisher information to study a probing scheme in which the channel is put in indefinite causal order (ICO) with copies of itself, focusing our investigation on probing the qudit (*d*-dimensional) depolarizing channel to estimate its state preservation probability. This ICO arrangement is one in which both the eigenvectors and eigenvalues of the channel output depend on the channel’s state preservation probability. We overcome this complication to obtain the quantum Fisher information in analytical form. This result shows that ICO-assisted probing yields greater information than does the comparable probe re-circulation scheme with definite causal order, that the information gained is greater when the channel ordering is more indefinite, and that the information gained is greatest when the channel ordering is maximally indefinite. This leads us to conclude that ICO is acting here in a strong sense as an aid to channel probing. The effectiveness of ICO for probing the depolarizing channel decreases with probe dimension, being most effective for qubits.

## Keywords

Quantum channel identification Quantum metrology Indefinite causal order Depolarizing channel Channel probing Quantum Fisher information Quasi-classical Kraus operators Quantum switch## Notes

## References

- 1.Fujiwara, A.: Quantum channel identification problem. Phys. Rev. A
**63**(4), 042304 (2001)ADSCrossRefGoogle Scholar - 2.Frey, M., Collins, D., Gerlach, K.: Probing the qudit depolarizing channel. J. Phys. A Math. Theor.
**44**(20), 205306 (2011)ADSMathSciNetCrossRefGoogle Scholar - 3.Paris, M.G.: Quantum estimation for quantum technology. Int. J. Quantum Inf.
**7**(supp01), 125–137 (2009)CrossRefGoogle Scholar - 4.Chiribella, G.: Perfect discrimination of no-signalling channels via quantum superposition of causal structures. Phys. Rev. A
**86**(4), 040301 (2012)ADSMathSciNetCrossRefGoogle Scholar - 5.Guerin, P.A., Feix, A., Araujo, M., Brukner, C.: Exponential communication complexity advantage from quantum superposition of the direction of communication. Phys. Rev. Lett.
**117**(10), 100502 (2016)ADSCrossRefGoogle Scholar - 6.Oreshkov, O., Costa, F., Brukner, C.: Quantum correlations with no causal order. Nat. Commun.
**3**, 1092 (2012)ADSCrossRefGoogle Scholar - 7.Ebler, D., Salek, S., Chiribella, G.: Enhanced Communication with the Assistance of Indefinite Causal Order (2017) arXiv preprint. arXiv:1711.10165
- 8.Hardy, L.: Quantum Gravity Computers: On the Theory of Computation with Indefinite Causal Structure. In: Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle. The Western Ontario Series in Philosophy of Science
**73**, 379 (2009)Google Scholar - 9.Chiribella, G., D’Ariano, G.M., Perinotti, P., Valiron, B.: Quantum computations without definite causal structure. Phys. Rev. A
**88**(2), 022318 (2013)ADSCrossRefGoogle Scholar - 10.Helstrom, C.W.: Minimum mean-squared error of estimates in quantum statistics. Phys. Lett. A
**25**(2), 101–102 (1967)ADSCrossRefGoogle Scholar - 11.Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
- 12.Procopio, L.M., Moqanaki, A., Araujo, M., Costa, F., Calafell, I.A., Dowd, E.G., Hamel, D.R., Rozema, L.A., Brukner, C., Walther, P.: Experimental superposition of orders of quantum gates. Nat. Commun.
**6**, 7913 (2015)CrossRefGoogle Scholar - 13.Frey, M., Collins, D.: Quantum Fisher information and the qudit depolarization channel. In: Quantum Information and Computation VII (Vol. 7342, p. 73420N). International Society for Optics and Photonics (2009, April)Google Scholar
- 14.S̆afránek, D.: A simple expression for the quantum Fisher information matrix (2018) arXiv preprint. arXiv:1801.00945
- 15.Coecke, B., Fritz, T., Spekkens, R.W.: A mathematical theory of resources. Inf. Comput.
**250**, 59–86 (2016)MathSciNetCrossRefGoogle Scholar