Ancilla Dimension in Quantum Channel Discrimination
- 95 Downloads
Single-shot quantum channel discrimination is a fundamental task in quantum information theory. It is well known that entanglement with an ancillary system can help in this task, and, furthermore, that an ancilla with the same dimension as the input of the channels is always sufficient for optimal discrimination of two channels. A natural question to ask is whether the same holds true for the output dimension. That is, in cases when the output dimension of the channels is (possibly much) smaller than the input dimension, is an ancilla with dimension equal to the output dimension always sufficient for optimal discrimination? We show that the answer to this question is “no” by construction of a family of counterexamples. This family contains instances with arbitrary finite gap between the input and output dimensions, and still has the property that in every case, for optimal discrimination, it is necessary to use an ancilla with dimension equal to that of the input. The proof relies on a characterization of all operators on the trace norm unit sphere that maximize entanglement negativity. In the case of density operators, we generalize this characterization to a broad class of entanglement measures, which we call weak entanglement measures. This characterization allows us to conclude that a quantum channel is reversible if and only if it preserves entanglement as measured by any weak entanglement measure, with the structure of maximally entangled states being equivalent to the structure of reversible maps via the Choi isomorphism. We also include alternate proofs of other known characterizations of channel reversibility.
Unable to display preview. Download preview PDF.
- 8.Bjelakovic, I., Siegmund-Schultze, R.: Quantum Stein’s lemma revisited, inequalities for quantum entropies, and a concavity theorem of Lieb (2003). arXiv:quant-ph/0307170
- 11.Kitaev, A., Shen, A., Vyalyi, M.: Classical and Quantum Computation. Graduate Studies in Mathematics, vol. 47. American Mathematical Society (2002)Google Scholar
- 12.Aharonov, D., Kitaev, A., Nisan, N.: Quantum circuits with mixed states. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pp 20–30 (1998)Google Scholar
- 17.Rosgen, B., Watrous, J.: On the hardness of distinguishing mixed-state quantum computations. In: Proceedings of the 20th Annual Conference on Computational Complexity, pp. 344–354 (2005)Google Scholar
- 23.Watrous, J.: Theory of Quantum Information (2015). https://cs.uwaterloo.ca/~watrous/TQI
- 24.Haagerup, U.: Injectivity and decomposition of completely bounded maps. In: Araki, H., Moore, C., Stratila, Ş.-V., Voiculescu, D.-V. (eds.) Operator Algebras and their Connections with Topology and Ergodic Theory. Lecture Notes in Mathematics, vol. 1132, pp. 170–222. Springer, Berlin, Heidelberg (1985)Google Scholar
- 36.Tucci, R.: Quantum entanglement and conditional information transmission (1999). arXiv:quant-ph/9909041
- 40.Johnston, N.: QETLAB: a MATLAB toolbox for quantum entanglement, version 0.9 (2016). http://qetlab.com
- 41.Johnston, N.: How to compute hard-to-compute matrix norms [weblog post] (2016). http://www.njohnston.ca/2016/01/
- 42.Puzzuoli, D.: ancilla_dimension (2016). https://github.com/DanPuzzuoli/ancilla_dimension