The Drew–Johnson–Loewy conjecture for matrices over max–min semirings

Mohindru, Preeti

doi:10.1080/03081087.2014.908874

Cited by 4 publications

(10 citation statements)

References 14 publications

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“…It has been shown in [21], that the Drew-Johnson-Loewy conjecture over the two element Boolean algebra (the binary Boolean semiring) is equivalent to a well-known result in graph theory by Erdos, Goodman and Posa [10]. Also it has been shown in [7] that the Drew-Johnson-Loewy conjecture holds for completely positive matrices over the binary Boolean algebra.…”

Section: Cp Matrices Over Boolean Algebras and The Djl Conjecturementioning

confidence: 91%

“…In [21], the Drew-Johnson-Loewy conjecture was generalized to completely positive matrices over semirings and was proved for completely positive matrices over max-min semirings. Although the original DrewJohnson-Loewy conjecture was disproved, the generalized Drew-Johnson-Loewy conjecture is still open for many other semirings.…”

Section: Definition 113 Let S Be An Antinegative Commutative Semirimentioning

confidence: 98%

“…In [21], it has been shown that the diagonally dominance condition is necessary and sufficient for symmetric matrices over max-min semirings or totally ordered Boolean algebras to be completely positive. We now generalize this result for completely positive matrices over Boolean algebras in general (finite or infinite).…”

Section: Cp Matrices Over Boolean Algebrasmentioning

confidence: 99%

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Completely positive matrices over Boolean algebras and their CP-rank

Mohindru

2015

Special Matrices

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Drew, Johnson and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [︀ n 2 /4 ]︀ . In this paper, we prove this conjecture for n × n completely positive matrices over Boolean algebras (finite or infinite). In addition, we formulate various CP-rank inequalities of completely positive matrices over special semirings using semiring homomorphisms. Definition 1.4. [11] A semiring is a set S together with two operations ⊕ and ⊗ and two distinguished elements0, 1 in S with 0 ≠ 1, such that 1. (S, ⊕, 0) is a commutative monoid, 2. (S, ⊗, 1) is a monoid, 3. ⊗ is both left and right distributive over ⊕, 4. The additive identity 0 satisfies the property r ⊗ 0 = 0 ⊗ r = 0, for all r ∈ S.

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Section: Cp Matrices Over Boolean Algebras and The Djl Conjecturementioning

confidence: 91%

Section: Definition 113 Let S Be An Antinegative Commutative Semirimentioning

confidence: 98%

Section: Cp Matrices Over Boolean Algebrasmentioning

confidence: 99%

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Completely positive matrices over Boolean algebras and their CP-rank

Mohindru

2015

Special Matrices

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“…Since the Boolean semiring and maxmin semirings are regular inclines with the LI-property, the following result is a generalization the corresponding result for max-min semirings in [21].…”

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confidence: 86%

The DJL Conjecture for CP Matrices over Special Inclines

Mohindru

Pereira

2016

Communications in Algebra

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Drew, Johnson and Loewy conjectured that for $n \geq 4$, the CP-rank of every $n \times n$ completely positive real matrix is at most $\left[n^{2}/4\right]$. While this conjecture is false for completely positive real matrices, we show that this conjecture is true for $n \times n$ completely positive matrices over certain special types of inclines. In addition, we prove an incline version of Markham's theorems which gives sufficient conditions for completely positive matrices over special inclines to have triangular factorizations

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“…For a proof, see Ref [54],. Proposition 3.1.12 This follows directly from the universal CPM construction of Selinger[55].…”

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confidence: 99%

Routed quantum circuits

Vanrietvelde

Kristjánsson

Barrett

2021

Quantum

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We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces. We therefore propose an extension to the framework of quantum circuits, given by routed linear maps and routed quantum circuits. We prove that this new framework allows for a consistent and intuitive diagrammatic representation in terms of circuit diagrams, applicable to both pure and mixed quantum theory, and exemplify its use in several situations, including the superposition of quantum channels and the causal decompositions of unitaries. We show that our framework encompasses the `extended circuit diagrams' of Lorenz and Barrett [arXiv:2001.07774 (2020)], which we derive as a special case, endowing them with a sound semantics.

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The Drew–Johnson–Loewy conjecture for matrices over max–min semirings

Cited by 4 publications

References 14 publications

Completely positive matrices over Boolean algebras and their CP-rank

Completely positive matrices over Boolean algebras and their CP-rank

The DJL Conjecture for CP Matrices over Special Inclines

Routed quantum circuits

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