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Quantum channel discrimination without entanglement

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Abstract

We address the problem of unambiguous discrimination of quantum channels (UDQC) without entanglement. As our main result, we show that even in the absence of entanglement, partial UDQC (PUDQC) can still be performed—depending on the unknown given channel. We provide a necessary and sufficient condition for PUDQC and put forth a method to perform the PUDQC once the said condition is met. We propose the performance metrics that capture the expected performance of the PUDQC independent of the specific channels to be distinguished. Finally, we perform PUDQC on several qubit channel pairs as concrete examples and derive the proposed performance metrics for these channel pairs.

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Notes

  1. Unambiguous discrimination refers to the scenario where the margin of error is zero in the announced result. Such methods of unambiguous discrimination in quantum state discrimination are obtained at the expense of allowing a nonzero probability of inconclusive results at the end of the protocol.

  2. This parameterization has a minor difference from standard parameterization on the Bloch sphere and leads to some redundant pairs of \(\phi \) and \(\theta \) values. We have made this choice because later in the article, we fix \(\phi = 0\), then the range of \(0 \le \theta \le 2\pi \) allows us to cover a whole circle in the X–Z plane.

  3. A Bloch vector defines the location of a (possibly mixed) qubit in the Bloch sphere (ball). The density matrix corresponding to a Bloch vector is given by .

  4. We consider when \(\mathcal {N}_1\) is distinguishable from \(\mathcal {N}_2\). The other way round can be obtained using the same reasoning.

  5. In fact we can choose as well. The decisive event, then, will be observing the state \(|0\rangle \) at the output of the channel. This will not change the optimal POVMs and the value of \(\epsilon \).

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Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2B2014462) and ICT R&D program of MSIP/IITP [R0190-15-2030, Reliable crypto-system standards and core technology development for secure quantum key distribution network].

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Authors and Affiliations

  1. Department of Electronic Engineering, Kyung Hee University, Yongin-si, 17104, Korea

    Junaid ur Rehman, Ahmad Farooq, Youngmin Jeong & Hyundong Shin

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  1. Junaid ur Rehman
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  2. Ahmad Farooq
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Correspondence to Youngmin Jeong or Hyundong Shin.

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ur Rehman, J., Farooq, A., Jeong, Y. et al. Quantum channel discrimination without entanglement. Quantum Inf Process 17, 271 (2018). https://doi.org/10.1007/s11128-018-2037-0

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