Using and reusing coherence to realize quantum processes

María García Díaz1, Kun Fang2, Xin Wang2, Matteo Rosati1, Michalis Skotiniotis1, John Calsamiglia1, and Andreas Winter1,3

1Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
2Centre for Quantum Software and Information, University of Technology Sydney, NSW 2007, Australia
3ICREA-Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys, 23, ES-08001 Barcelona, Spain

Coherent superposition is a key feature of quantum mechanics that underlies the advantage of quantum technologies over their classical counterparts. Recently, coherence has been recast as a resource theory in an attempt to identify and quantify it in an operationally well-defined manner. Here we study how the coherence present in a state can be used to implement a quantum channel via incoherent operations and, in turn, to assess its degree of coherence. We introduce the robustness of coherence of a quantum channel-which reduces to the homonymous measure for states when computed on constant-output channels-and prove that: i) it quantifies the minimal rank of a maximally coherent state required to implement the channel; ii) its logarithm quantifies the amortized cost of implementing the channel provided some coherence is recovered at the output; iii) its logarithm also quantifies the zero-error asymptotic cost of implementation of many independent copies of a channel. We also consider the generalized problem of imperfect implementation with arbitrary resource states. Using the robustness of coherence, we find that in general a quantum channel can be implemented without employing a maximally coherent resource state. In fact, we prove that every pure coherent state in dimension larger than $2$, however weakly so, turns out to be a valuable resource to implement some coherent unitary channel. We illustrate our findings for the case of single-qubit unitary channels.

Static coherence, the degree of superposition present in a state, can be thought of as different from dynamic coherence, which is the ability to generate coherence itself. Here we develop a framework that puts these two types of coherence on an equal footing and allows us to study their interconversion thereby shifting the paradigm of coherence theory from states to processes. In particular, we introduce a measure for dynamic coherence which uniquely quantifies the implementation cost of a channel using static coherence as a resource, and show that coherence can be used and reused in a continuous fashion.

► BibTeX data

► References

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arXiv:quant-ph/0612146

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https://doi.org/10.1145/276698.276708

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[4] T. Baumgratz, M. Cramer, and M. B. Plenio. Quantifying coherence. Phys. Rev. Lett., 113: 140401, Sep 2014. 10.1103/​PhysRevLett.113.140401. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.113.140401.
https://doi.org/10.1103/PhysRevLett.113.140401

[5] Khaled Ben Dana, María García Díaz, Mohamed Mejatty, and Andreas Winter. Resource theory of coherence: Beyond states. Phys. Rev. A, 95: 062327, Jun 2017. 10.1103/​PhysRevA.95.062327. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.95.062327.
https://doi.org/10.1103/PhysRevA.95.062327

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https://doi.org/10.1109/TIT.2013.2268533

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