Orthonormal Bases Of Extreme Quantumness

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Marcin Rudziński^1,2, Adam Burchardt³, and Karol Życzkowski^1,4

¹Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
²Doctoral School of Exact and Natural Sciences, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
³QuSoft, CWI and University of Amsterdam, Science Park 123, 1098 XG Amsterdam, the Netherlands
⁴Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland

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Abstract

Spin anticoherent states acquired recently a lot of attention as the most "quantum" states. Some coherent and anticoherent spin states are known as optimal quantum rotosensors. In this work, we introduce a measure of quantumness for orthonormal bases of spin states, determined by the average anticoherence of individual vectors and the Wehrl entropy. In this way, we identify the most coherent and most quantum states, which lead to orthogonal measurements of extreme quantumness. Their symmetries can be revealed using the Majorana stellar representation, which provides an intuitive geometrical representation of a pure state by points on a sphere. Results obtained lead to maximally (minimally) entangled bases in the $2j+1$ dimensional symmetric subspace of the $2^{2j}$ dimensional space of states of multipartite systems composed of $2j$ qubits. Some bases found are iso-coherent as they consist of all states of the same degree of spin-coherence.

Featured image: In the left image, the most "quantum" basis in $mathcal{H}_4$ is depicted using the stellar representation. On the right, the Husimi function for states in the most coherent ("classical") basis within $mathcal{H}_4$ is presented.

Popular summary

Extremal states, coherent and anticoherent, have practical applications in quantum metrology as optimal rotosensors. This work provides a natural extension of previous studies concerning the search for such states proposing optimal orthogonal measurements of Lüders and von Neumann of the extreme spin coherence. We introduce the measure $mathcal{B}_t$ as the tool to characterize the quantumness of a measurement given by a basis in $mathcal{H}_N$. The search for the most quantum bases for $N=3,4,5$ and $7$ is performed. Numerical results suggest, that the obtained solutions are unique. A set of candidates for the "classical" bases consisting of the most spin-coherent states is indicated for $N=3,4,5,6$. Some of the most quantum bases, analyzed in the stellar representation of Majorana, reveal symmetries of Platonic solids. Most classical bases display symmetric structures too. We also considered other measures of the quantumness of vectors forming a given basis. Optimization of the mean Wehrl entropy of $N$ orthogonal vectors leads to the same bases distinguished by extremal values of the quantities $mathcal{B}_t$, with a single exception of the quantum basis for $N=6$.

► BibTeX data

@article{Rudzinski2024orthonormalbasesof,
  doi = {10.22331/q-2024-01-25-1234},
  url = {https://doi.org/10.22331/q-2024-01-25-1234},
  title = {Orthonormal bases of extreme quantumness},
  author = {Rudzi{'{n}}ski, Marcin and Burchardt, Adam and {.{Z}}yczkowski, Karol},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1234},
  month = jan,
  year = {2024}
}

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Cited by

[1] Michał Piotrak, Marek Kopciuch, Arash Dezhang Fard, Magdalena Smolis, Szymon Pustelny, and Kamil Korzekwa, "Perfect quantum protractors", arXiv:2310.13045, (2023).

[2] Aaron Z. Goldberg, "Correlations for subsets of particles in symmetric states: what photons are doing within a beam of light when the rest are ignored", arXiv:2401.05484, (2024).

The above citations are from SAO/NASA ADS (last updated successfully 2024-01-25 11:53:23). The list may be incomplete as not all publishers provide suitable and complete citation data.

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Source: https://quantum-journal.org/papers/q-2024-01-25-1234/

Time Stamp: January 25, 2024

Orthonormal bases of extreme quantumness

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