Complete order equivalence of spin unitaries

Farenick, Douglas; Huntinghawk, Farrah; Masanika, Adili; Plosker, Sarah

doi:10.1016/j.laa.2020.09.033

Cited by 3 publications

(4 citation statements)

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“…Although the Pauli-Weyl-Brauer matrices are universal for selfadjoint anticommuting unitaries [7], the analogous result fails for p > 2. Proposition 4.1.…”

Section: Weyl-brauer Unitaries Are Not Universalmentioning

confidence: 97%

“…As in [10,Definition 6.63] and by adapting the method of the proof of Theorem 4.3 in [7], we shall invoke an iteration whereby we produce, from m invertible matrices x 1 , . .…”

Section: Weyl-brauer Unitariesmentioning

confidence: 99%

“…As mentioned in [7], it is often the case that spin systems arise in quantum theory; for this reason, the following modification of the construction above is worth a brief mention. Theorem 3.2.…”

Section: Weyl-brauer Unitariesmentioning

confidence: 99%

“…The proofs of Theorems 5.3 and 5.4 extend beyond Weyl pairs to all Weyl-Brauer unitaries, once it has been shown that the Weyl-Brauer matrices generate irreducible operator systems. The details are left to the reader (using, if one wishes, the method of proof in [7] that showed the irreducibility of the operator system generated by the Pauli-Weyl-Brauer unitaries). However, at the very least, we state below the version of this theorem for the three basic Weyl-Brauer unitaries ω a , ω b , and ω c .…”

Section: The Matrix Range Of the Weyl Unitariesmentioning

confidence: 99%

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Universality of Weyl Unitaries

Farenick¹,

Ojo²,

Plosker³

2021

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Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [11] on group theory and quantum mechanics, are p × p unitary matrices given by the diagonal matrix whose entries are the p-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by u and v, satisfy u p = v p = 1 p (the p × p identity matrix) and the commutation relation uv = ζvu, where ζ is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if u and v are any d × d unitary matrices such that u p = v p = 1 d and uv = ζvu, then there exists a unital completely positive linear map φ : M p (C) → M d (C) such that φ(u) = u and φ(v) = v. We also show, moreover, that any two pairs of p-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent.When p = 2, the Weyl matrices are two of the three Pauli matrices from quantum mechanics. It was recently shown in [7] that g-tuples of Pauli-Weyl-Brauer unitaries are universal for all g-tuples of anticommuting selfadjoint unitary matrices; however, we show here that the analogous result fails for positive integers p > 2.Finally, we show that the Weyl matrices are extremal in their matrix range, using recent ideas from noncommutative convexity theory.

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“…Although the Pauli-Weyl-Brauer matrices are universal for selfadjoint anticommuting unitaries [7], the analogous result fails for p > 2. Proposition 4.1.…”

Section: Weyl-brauer Unitaries Are Not Universalmentioning

confidence: 97%

“…As in [10,Definition 6.63] and by adapting the method of the proof of Theorem 4.3 in [7], we shall invoke an iteration whereby we produce, from m invertible matrices x 1 , . .…”

Section: Weyl-brauer Unitariesmentioning

confidence: 99%

Section: Weyl-brauer Unitariesmentioning

confidence: 99%

Section: The Matrix Range Of the Weyl Unitariesmentioning

confidence: 99%

See 2 more Smart Citations