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Quantum Codes of Maximal Distance and Highly Entangled Subspaces

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Felix Huber1,2,3 and Markus Grassl4,5

1ICFO – The Institute of Photonic Sciences, 08860 Castelldefels (Barcelona), Spain
2Institut für Theoretische Physik, Universität zu Köln, 50937 Köln, Germany
3Naturwissenschaftlich-Technische Fakultät, Universität Siegen, 57068 Siegen, Germany
4International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308 Gdańsk, Poland
5Max Planck Institute for the Science of Light, 91058 Erlangen, Germany

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Abstract

We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length $n$ of all QMDS codes with local dimension $D$ and distance $d geq 3$ is bounded by $n leq D^2 + d – 2$. We obtain their weight distribution and present additional bounds that arise from Rains’ shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform $r$-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.

Quantum error-correcting codes are essential to protect quantum information against decoherence and interaction with the environment. While this interaction creates undesired entanglement between the system and its environment, it is again entanglement among the individual subsystems composing the code that allows to fight decoherence.

We investigate bounds on the parameters of the code that relate to the entanglement in the code, as manifested by maximally mixed marginals of the logical states. The first bound is the quantum Singleton bound, which has already been known very early in the theory of quantum error-correction. It is independent of the local dimension and can always be reached when the local dimension is sufficiently large. The corresponding codes are known as quantum maximum distance separable (QMDS) codes.

In this paper, we derive additional bounds on the existence of QMDS codes. Crucially, they are valid for all QMDS codes, including codes beyond the stabilizer formalism. We show that another characteristic property, the weight enumerator, is also independent of whether the QMDS code is of the stabilizer type or not.

In many cases the known stabilizer constructions match our upper bounds. It it surprising that these combinatorial, inherently classical constructions yield optimal codes also in the quantum case, dealing with arbitrary subspaces of complex vector spaces. We conclude with the open question whether or not there are QMDS codes which do not arise from classical MDS codes.

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Cited by

[1] Daniel Alsina and Mohsen Razavi, “Absolutely maximally entangled states, quantum maximum distance separable codes, and quantum repeaters”, arXiv:1907.11253.

[2] Paweł Mazurek, Máté Farkas, Andrzej Grudka, Michał Horodecki, and Michał Studziński, “Quantum error-correction codes and absolutely maximally entangled states”, Physical Review A 101 4, 042305 (2020).

[3] Maciej Demianowicz and Remigiusz Augusiak, “Entanglement of genuinely entangled subspaces and states: Exact, approximate, and numerical results”, Physical Review A 100 6, 062318 (2019).

[4] Felix Huber, “Positive Maps and Matrix Contractions from the Symmetric Group”, arXiv:2002.12887.

[5] Zahra Raissi, “Modified-Shortening: Modifying method of constructing quantum codes from highly entangled states”, arXiv:2005.01426.

[6] Maciej Demianowicz and Remigiusz Augusiak, “An approach to constructing genuinely entangled subspaces of maximal dimension”, arXiv:1912.07536.

[7] Sathwik Chadaga, Mridul Agarwal, and Vaneet Aggarwal, “Encoders and Decoders for Quantum Expander Codes Using Machine Learning”, arXiv:1909.02945.

The above citations are from SAO/NASA ADS (last updated successfully 2020-06-19 12:26:57). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2020-06-19 12:26:55).

Source: https://quantum-journal.org/papers/q-2020-06-18-284/

Quantum

Cellular automata in operational probabilistic theories

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Paolo Perinotti

QUIT Group, Dipartimento di Fisica, Università degli studi di Pavia, and INFN sezione di Pavia, via Bassi 6, 27100 Pavia, Italy

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Abstract

The theory of cellular automata in operational probabilistic theories is developed. We start introducing the composition of infinitely many elementary systems, and then use this notion to define update rules for such infinite composite systems. The notion of causal influence is introduced, and its relation with the usual property of signalling is discussed. We then introduce homogeneity, namely the property of an update rule to evolve every system in the same way, and prove that systems evolving by a homogeneous rule always correspond to vertices of a Cayley graph. Next, we define the notion of locality for update rules. Cellular automata are then defined as homogeneous and local update rules. Finally, we prove a general version of the wrapping lemma, that connects CA on different Cayley graphs sharing some small-scale structure of neighbourhoods.

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Cited by

[1] Robin Lorenz and Jonathan Barrett, “Causal and compositional structure of unitary transformations”, arXiv:2001.07774.

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Quantum

The Platonic solids and fundamental tests of quantum mechanics

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Armin Tavakoli and Nicolas Gisin

Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland

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Abstract

The Platonic solids is the name traditionally given to the five regular convex polyhedra, namely the tetrahedron, the octahedron, the cube, the icosahedron and the dodecahedron. Perhaps strongly boosted by the towering historical influence of their namesake, these beautiful solids have, in well over two millennia, transcended traditional boundaries and entered the stage in a range of disciplines. Examples include natural philosophy and mathematics from classical antiquity, scientific modeling during the days of the European scientific revolution and visual arts ranging from the renaissance to modernity. Motivated by mathematical beauty and a rich history, we consider the Platonic solids in the context of modern quantum mechanics. Specifically, we construct Bell inequalities whose maximal violations are achieved with measurements pointing to the vertices of the Platonic solids. These Platonic Bell inequalities are constructed only by inspecting the visible symmetries of the Platonic solids. We also construct Bell inequalities for more general polyhedra and find a Bell inequality that is more robust to noise than the celebrated Clauser-Horne-Shimony-Holt Bell inequality. Finally, we elaborate on the tension between mathematical beauty, which was our initial motivation, and experimental friendliness, which is necessary in all empirical sciences.

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Cited by

[1] Armin Tavakoli, “Semi-device-independent certification of independent quantum state and measurement devices”, arXiv:2003.03859.

[2] H. Chau Nguyen, Sébastien Designolle, Mohamed Barakat, and Otfried Gühne, “Symmetries between measurements in quantum mechanics”, arXiv:2003.12553.

[3] Armin Tavakoli, Ingemar Bengtsson, Nicolas Gisin, and Joseph M. Renes, “Compounds of symmetric informationally complete measurements and their application in quantum key distribution”, arXiv:2007.01007.

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Squeezing metrology: a unified framework

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Lorenzo Maccone and Alberto Riccardi

Dip. Fisica and INFN Sez. Pavia, University of Pavia, via Bassi 6, I-27100 Pavia, Italy

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Abstract

Quantum metrology theory has up to now focused on the resolution gains obtainable thanks to the entanglement among $N$ probes. Typically, a quadratic gain in resolution is achievable, going from the $1/sqrt{N}$ of the central limit theorem to the $1/N$ of the Heisenberg bound. Here we focus instead on quantum squeezing and provide a unified framework for metrology with squeezing, showing that, similarly, one can generally attain a quadratic gain when comparing the resolution achievable by a squeezed probe to the best $N$-probe classical strategy achievable with the same energy. Namely, here we give a quantification of the Heisenberg squeezing bound for arbitrary estimation strategies that employ squeezing. Our theory recovers known results (e.g. in quantum optics and spin squeezing), but it uses the general theory of squeezing and holds for arbitrary quantum systems.

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Cited by

[1] Stella Seah, Stefan Nimmrichter, Daniel Grimmer, Jader P. Santos, Valerio Scarani, and Gabriel T. Landi, “Collisional Quantum Thermometry”, Physical Review Letters 123 18, 180602 (2019).

[2] Lorenzo Maccone and Changliang Ren, “Quantum Radar”, Physical Review Letters 124 20, 200503 (2020).

[3] Emanuele Polino, Mauro Valeri, Nicolò Spagnolo, and Fabio Sciarrino, “Photonic Quantum Metrology”, arXiv:2003.05821.

[4] Dario Gatto, Paolo Facchi, Frank A. Narducci, and Vincenzo Tamma, “Distributed quantum metrology with a single squeezed-vacuum source”, Physical Review Research 1 3, 032024 (2019).

[5] Giovanni Gramegna, Danilo Triggiani, Paolo Facchi, Frank A. Narducci, and Vincenzo Tamma, “Typicality of Heisenberg scaling precision in multi-mode quantum metrology”, arXiv:2003.12551.

[6] Giovanni Gramegna, Danilo Triggiani, Paolo Facchi, Frank A. Narducci, and Vincenzo Tamma, “Heisenberg scaling precision in multi-mode distributed quantum metrology”, arXiv:2003.12550.

[7] Sascha Wald, Saulo V. Moreira, and Fernando L. Semião, “In- and out-of-equilibrium quantum metrology with mean-field quantum criticality”, Physical Review E 101 5, 052107 (2020).

The above citations are from SAO/NASA ADS (last updated successfully 2020-07-09 11:02:12). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2020-07-09 11:02:10: Could not fetch cited-by data for 10.22331/q-2020-07-09-292 from Crossref. This is normal if the DOI was registered recently.

Source: https://quantum-journal.org/papers/q-2020-07-09-292/

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