Programmability of covariant quantum channels

Martina Gschwendtner1,2, Andreas Bluhm3, and Andreas Winter4,5

1Munich Center for Quantum Science and Technology (MCQST), 80799 München, Germany
2Zentrum Mathematik, Technical University of Munich, 85748 Garching, Germany
3QMATH, Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark
4Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluis Companys, 23, 08001 Barcelona, Spain
5Grup d'Informació Quàntica, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's $\textit{No-Programming Theorem}$), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via $\textit{teleportation simulation}$, which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.

The idea of programmable quantum processors resembles very much its classical equivalent to program a machine such that it implements several operations. The quantum version is able to apply quantum channels to an input state specified by the user via a program state from a program register containing all relevant information. Nielsen and Chuang show in their No-Programming Theorem that devices implementing any infinite set of unitaries exactly with a finite-dimensional program register do not exist. This result kicked off the search for approximate processors, optimizing the program size for any given precision. However, we can bypass this no-go theorem by implementing particular families of quantum channels, which is the scope of this article.

Since symmetries are of fundamental importance in physics, we consider covariant quantum channels. To those, the No-Programming Theorem is not necessarily applicable. We show that an exact implementation is possible if the group describing the symmetry acts irreducibly on the channel input. Using the special block-diagonal structure of the Choi-Jamiolkowski states corresponding to the covariant quantum channels allows us to show that the program dimension is at most the sum of the dimensions of the blocks occurring in this structure. Addressing the interest in approximate versions of programmable quantum processors, we first provide upper bounds on the program dimension, which we show for arbitrary representations instead of irreducible ones. These are finally complemented by lower bounds. The approximate bounds are in general worse than the exact ones, but they apply in a more general setting. Moreover, they show that our upper bounds in the exact case are tight.

This article opens up possibilities for future work on the exact implementation in the broader setting with reducible representations.

► BibTeX data

► References

[1] M. Al Nuwairan. SU(2)-Irreducibly covariant and EPOSIC channels. arXiv:1306.5321, art. arXiv:1306.5321, 2013. URL https:/​/​​abs/​1306.5321.

[2] A. Ambainis and J. Emerson. Quantum t-designs: t-wise independence in the quantum world. In Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07), pages 129–140, 2007. 10.1109/​CCC.2007.26.

[3] K. M. R. Audenaert. A sharp continuity estimate for the von Neumann entropy. Journal of Physics A: Mathematical and Theoretical, 40 (28): 8127––8136, 2007. 10.1088/​1751-8113/​40/​28/​s18.

[4] L. Banchi, J. Pereira, S. Lloyd, and S. Pirandola. Convex optimization of programmable quantum computers. npj Quantum Information, 6 (1): 42, 2020. 10.1038/​s41534-020-0268-2.

[5] F. Benatti. Dynamics, Information and Complexity in Quantum Systems. Springer, 2009. 10.1007/​978-1-4020-9306-7.

[6] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters, 70: 1895–1899, 1993. 10.1103/​PhysRevLett.70.1895.

[7] C. H. Bennett, D. P. Divincenzo, J. A. Smolin, and W. K. Wootters. Mixed-state entanglement and quantum error correction. Physical Review A, 54 (5): 3824–3851, 1996. 10.1103/​PhysRevA.54.3824.

[8] B. Blackadar. Operator Algebras: Theory of C*-Algebras and Von Neumann Algebras, volume 13 of Encyclopaedia of Mathematical Sciences. Springer, 2006. 10.1007/​3-540-28517-2.

[9] G. Chiribella, G. M. D'Ariano, and P. Perinotti. Realization schemes for quantum instruments in finite dimensions. Journal of Mathematical Physics, 50 (4): 042101–042101, 2009. 10.1063/​1.3105923.

[10] M.-D. Choi. Completely positive maps on complex matrices. Linear Algebra and Its Applications, 10: 285–290, 1975. 10.1016/​0024-3795(75)90075-0.

[11] C. Cı̂rstoiu, K. Korzekwa, and D. Jennings. Robustness of Noether's principle: Maximal disconnects between conservation laws and symmetries in quantum theory. Physical Review X, 10: 041035, 2020. 10.1103/​PhysRevX.10.041035.

[12] T. Cubitt, A. Montanaro, and A. Winter. On the dimension of subspaces with bounded Schmidt rank. Journal of Mathematical Physics, 49 (2): 022107, 2008. 10.1063/​1.2862998.

[13] G. M. D’Ariano and P. Perinotti. Efficient universal programmable quantum measurements. Physical Review Letters, 94 (9): 090401, 2005. 10.1103/​physrevlett.94.090401.

[14] T. Eggeling and R. F. Werner. Separability properties of tripartite states with $U{\bigotimes}U{\bigotimes}U$ symmetry. Physical Review A, 63: 042111, 2001. 10.1103/​PhysRevA.63.042111.

[15] M. Fannes, B. Haegeman, M. Mosonyi, and D. Vanpeteghem. Additivity of minimal entropy output for a class of covariant channels. arXiv:quant-ph/​0410195, 2004. URL https:/​/​​abs/​quant-ph/​0410195.

[16] W. Fulton and J. Harris. Representation Theory, volume 129 of Graduate Texts in Mathematics. Springer, 2004. 10.1007/​978-1-4612-0979-9.

[17] A. Hayashi, T. Hashimoto, and M. Horibe. Reexamination of optimal quantum state estimation of pure states. Physical Review A, 72: 032325, 2005. 10.1103/​PhysRevA.72.032325.

[18] P. Hayden, D. Leung, and A. Winter. Aspects of generic entanglement. Communications in Mathematical Physics, 265 (1): 95–117, 2006. 10.1007/​s00220-006-1535-6.

[19] T. Heinosaari and M. Ziman. The Mathematical Language of Quantum Theory. Cambridge University Press, 2012. 10.1017/​CBO9781139031103.

[20] M. Hillery, V. Bužek, and M. Ziman. Probabilistic implementation of universal quantum processors. Physical Review A, 65 (2): 022301, 2002a. 10.1103/​PhysRevA.65.022301.

[21] M. Hillery, M. Ziman, and V. Bužek. Implementation of quantum maps by programmable quantum processors. Physical Review A, 66 (4): 042302, 2002b. 10.1103/​PhysRevA.66.042302.

[22] M. Hillery, M. Ziman, and V. Bužek. Approximate programmable quantum processors. Physical Review A, 73 (2): 022345, 2006. 10.1103/​PhysRevA.73.022345.

[23] M. Horodecki, P. Horodecki, and R. Horodecki. General teleportation channel, singlet fraction, and quasidistillation. Physical Review A, 60: 1888–1898, 1999. 10.1103/​PhysRevA.60.1888.

[24] S. Ishizaka and T. Hiroshima. Asymptotic teleportation scheme as a universal programmable quantum processor. Physical Review Letters, 101 (24): 240501, 2008. 10.1103/​physrevlett.101.240501.

[25] A. Jamiołkowski. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3 (4): 275–278, 1972. 10.1016/​0034-4877(72)90011-0.

[26] M. Keyl. Fundamentals of quantum information theory. Physics Reports, 369 (5): 431–548, 2002. 10.1016/​S0370-1573(02)00266-1.

[27] A. M. Kubicki, C. Palazuelos, and D. Pérez-García. Resource quantification for the no-programing theorem. Physical Review Letters, 122 (8): 080505, 2019. 10.1103/​PhysRevLett.122.080505.

[28] M. Ledoux and M. Talagrand. Probability in Banach Spaces: Isoperimetry and Processes, volume 23 of A Series of Modern Surveys in Mathematics Series. Springer, 1991. 10.1007/​978-3-642-20212-4.

[29] I. Marvian and R. W. Spekkens. Extending Noether's theorem by quantifying the asymmetry of quantum states. Nature Communications, 5: 3821, 2014. 10.1038/​ncomms4821.

[30] M. Mozrzymas, M. Studziński, and N. Datta. Structure of irreducibly covariant quantum channels for finite groups. Journal of Mathematical Physics, 58 (5): 052204, 2017. 10.1063/​1.4983710.

[31] A. Müller-Hermes. Transposition in quantum information theory. Master's thesis, Technical University of Munich, 2012. URL https:/​/​​foswiki/​pub/​M5/​CQC/​mth.pdf.

[32] M. A. Nielsen and I. L. Chuang. Programmable quantum gate arrays. Physical Review Letters, 79 (2): 321–324, 1997. 10.1103/​PhysRevLett.79.321.

[33] E. Noether. Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1918: 235–257, 1918. URL https:/​/​​wiki/​Invariante_Variationsprobleme.

[34] D. Petz. Quantum Information Theory and Quantum Statistics. Theoretical and Mathematical Physics. Springer, 2008. 10.1007/​978-3-540-74636-2.

[35] S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi. Fundamental limits of repeaterless quantum communications. Nature Communications, 8 (1): 15043, 2017. 10.1038/​ncomms15043.

[36] S. Pirandola, R. Laurenza, C. Lupo, and J. L. Pereira. Fundamental limits to quantum channel discrimination. npj Quantum Information, 5: 50, 2019. 10.1038/​s41534-019-0162-y.

[37] D. Pérez-García. Optimality of programmable quantum measurements. Physical Review A, 73 (5): 052315, 2006. 10.1103/​physreva.73.052315.

[38] B. Simon. Representations of Finite and Compact Groups, volume 10 of Graduate Studies in Mathematics. American Mathematical Society, 1996. 10.1090/​gsm/​010.

[39] G. Vidal, L. Masanes, and J. I. Cirac. Storing quantum dynamics in quantum states: A stochastic programmable gate. Physical Review Letters, 88 (4): 047905, 2002. 10.1103/​PhysRevLett.88.047905.

[40] K. G. H. Vollbrecht and R. F. Werner. Entanglement measures under symmetry. Physical Review A, 64 (6): 062307, 2001. 10.1103/​PhysRevA.64.062307.

[41] D.-S. Wang. Choi states, symmetry-based quantum gate teleportation, and stored-program quantum computing. Physical Review A, 101 (5): 052311, 2020. 10.1103/​PhysRevA.101.052311.

[42] J. Watrous. The Theory of Quantum Information. Cambridge University Press, Cambridge, 2018. 10.1017/​9781316848142.

[43] R. F. Werner and A. S. Holevo. Counterexample to an additivity conjecture for output purity of quantum channels. ournal of Mathematical Physics, 43 (9): 4353–4357, 2002. 10.1063/​1.1498491.

[44] M. M. Wilde. Quantum Information Theory. Cambridge University Press, 2nd edition, 2017. 10.1017/​9781316809976.001.

[45] M. M. Wilde, M. Tomamichel, and M. Berta. Converse bounds for private communication over quantum channels. IEEE Transactions on Information Theory, 63 (3): 1792–1817, 2017. 10.1109/​tit.2017.2648825.

[46] A. Winter. Tight uniform continuity bounds for quantum entropies: Conditional entropy, relative entropy distance and energy constraints. Communications in Mathematical Physics, 347 (1): 291–313, 2016. 10.1007/​s00220-016-2609-8.

[47] Y. Yang, R. Renner, and G. Chiribella. Optimal universal programming of unitary gates. Physical Review Letters, 125: 210501, 2020. 10.1103/​PhysRevLett.125.210501.

Cited by

[1] Rhea Alexander, Si Gvirtz-Chen, and David Jennings, "Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system", New Journal of Physics 24 5, 053023 (2022).

[2] Dmitry Grinko and Maris Ozols, "Linear programming with unitary-equivariant constraints", arXiv:2207.05713.

The above citations are from Crossref's cited-by service (last updated successfully 2022-10-04 05:02:39) and SAO/NASA ADS (last updated successfully 2022-10-04 05:02:41). The list may be incomplete as not all publishers provide suitable and complete citation data.