Elementary rings and modules, by Iain T. Adamson. Pp 136. £1·50. 1972 (Oliver and Boyd)

Fletcher, Colin R.

doi:10.1017/s0025557200132267

Cited by 2 publications

(4 citation statements)

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“…14 The orthogonal direct sum is also referred as internal direct sum and it is isomorphic to the external orthogonal direct sum, which constructs a Hilbert space from two or more Hilbert spaces [47,48]. 15 In order to overcome this technicality, when Λ has infinity elements we define + λ∈Λ X λ as follows:…”

Section: A1 Notations For Linear Algebramentioning

confidence: 99%

Consequences of preserving reversibility in quantum superchannels

Yokojima

Quintino

Soeda

et al. 2021

Quantum

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Similarly to quantum states, quantum operations can also be transformed by means of quantum superchannels, also known as process matrices. Quantum superchannels with multiple slots are deterministic transformations which take independent quantum operations as inputs. While they are enforced to respect the laws of quantum mechanics, the use of input operations may lack a definite causal order, and characterizations of general superchannels in terms of quantum objects with a physical implementation have been missing. In this paper, we provide a mathematical characterization for pure superchannels with two slots (also known as bipartite pure processes), which are superchannels preserving the reversibility of quantum operations. We show that the reversibility preserving condition restricts all pure superchannels with two slots to be either a quantum circuit only consisting of unitary operations or a coherent superposition of two unitary quantum circuits where the two input operations are differently ordered. The latter may be seen as a generalization of the quantum switch, allowing a physical interpretation for pure two-slot superchannels. An immediate corollary is that purifiable bipartite processes cannot violate device-independent causal inequalities.

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Section: A1 Notations For Linear Algebramentioning

confidence: 99%

Consequences of preserving reversibility in quantum superchannels

Yokojima

Quintino

Soeda

et al. 2021

Quantum

View full text Add to dashboard Cite

show abstract

“…Particular cases do however occur; significantly, 25 , the very first proposition in Book V to deal with proportions, Proposition 4, proves that (paraphrasing) if a:b::c:d 9 then ma:nb::mc:nd, describing the compound of a ratio of numbers and a proportion between magnitudes. Another case occurs within the combined definition and proposition, V, Definition 17:…”

Section: The Terminology Is Defined In An Interpolated Corrupt Definimentioning

confidence: 99%

“…76 The procedure is described in a forthcoming paper, Book II of Euclid 9 s Elements and a pre-Eudoxan theory of ratio, (to appear).…”

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confidence: 99%