## Quantum logic in dagger kernel categories

Venue: | Order |

Citations: | 7 - 7 self |

### BibTeX

@ARTICLE{Heunen_quantumlogic,

author = {Chris Heunen and Bart Jacobs},

title = {Quantum logic in dagger kernel categories},

journal = {Order},

year = {},

pages = {2010}

}

### OpenURL

### Abstract

This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/order-theoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and and-then connectives are obtained, as adjoints, via the existential-pullback adjunction between fibres. 1

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Citation Context ...) diagonal · ◦ �� �� · � � � �� · � � � · making both triangles commute. As a result, the factorisation (6) is unique up to isomorphism. Indeed, kernels and zero-epis form a factorisation system (see =-=[4]-=-). Proof Assume the zero-epi e: E → Y and kernel m = ker(h): M ↣ X satisfy m ◦ f = g ◦ e, as below, E ◦e �� �� Y f M � � � m � X Then: h ◦ g ◦ e = h ◦ m ◦ f = 0 ◦ f = 0 and h ◦ g = 0 because e is zero... |

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Citation Context ...tions, where † is induced by the inner product. The use of daggers, mostly with additional assumptions, dates back to [31, 35]. Daggers are currently of interest in the context of quantum computation =-=[1, 40, 7]-=-. The dagger abstractly captures the reversal of a computation. Mostly, dagger categories are used with fairly strong additional assumptions, like compact closure in [1]. Here we wish to follow a diff... |

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Citation Context ...his allows us to show that ̂ B is Boolean: if m ∧ n = 0, them m † ◦ n = m ◦ n = m ∧ n = 0. □ It remains an open question whether a similar construction can be performed for orthomodular lattices (see =-=[13]-=-), instead of Boolean algebras. The straightforward extension of the above construction does not work: in order to get kernels one needs to use the and-then connective (&, see Proposition 6.1) for com... |

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Citation Context ...ure in [1]. Here we wish to follow a different approach and start from minimal assumptions. This paper is a first step to understand quantum logic, from the perspective of categorical logic (see e.g. =-=[32, 28, 41, 21]-=-). It grew from the work of one of the authors [20]. Although that paper enjoys a satisfactory relation to traditional quantum logic [18], this one generalises it, by taking the notion of dagger categ... |

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Citation Context ...tions, where † is induced by the inner product. The use of daggers, mostly with additional assumptions, dates back to [31, 35]. Daggers are currently of interest in the context of quantum computation =-=[1, 40, 7]-=-. The dagger abstractly captures the reversal of a computation. Mostly, dagger categories are used with fairly strong additional assumptions, like compact closure in [1]. Here we wish to follow a diff... |

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Citation Context ...rnels, but without the general perspective given by categorical logic. 1Upon this structure of “dagger kernel categories” the paper constructs pullbacks of kernels and factorisation (both similar to =-=[14]-=-). It thus turns out that the kernels form a “bifibration” (both a fibration and an opfibration, see [21]). This structure can be used as a basis for categorical logic, which captures substitution in ... |

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Citation Context ...ure in [1]. Here we wish to follow a different approach and start from minimal assumptions. This paper is a first step to understand quantum logic, from the perspective of categorical logic (see e.g. =-=[32, 28, 41, 21]-=-). It grew from the work of one of the authors [20]. Although that paper enjoys a satisfactory relation to traditional quantum logic [18], this one generalises it, by taking the notion of dagger categ... |

36 |
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Citation Context ...ure in [1]. Here we wish to follow a different approach and start from minimal assumptions. This paper is a first step to understand quantum logic, from the perspective of categorical logic (see e.g. =-=[32, 28, 41, 21]-=-). It grew from the work of one of the authors [20]. Although that paper enjoys a satisfactory relation to traditional quantum logic [18], this one generalises it, by taking the notion of dagger categ... |

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Citation Context ...ler elements. Consequently, 0 is not an atom. A poset is called atomic if for any x ̸= 0 in it there exists an atom a with a ≤ x. Finally, a lattice is atomistic when every element is a join of atoms =-=[10]-=-. Proposition 36 For an arbitrary object I in a dagger kernel category, the following are equivalent: 1. idI = 1 is an atom in KSub(I); 2. KSub(I) = {0, 1}; 3. each nonzero kernel x: I ↣ X is an atom ... |

27 |
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Citation Context ...first see how the so-called Sasaki hook [26] arises naturally in this setting, and then investigate Booleanness. For a kernel m: M ↣ X we shall write E(m) = m ◦ m † : X → X for the “effect” of m, see =-=[11]-=-. This E(m) is easily seen to be a self-adjoint idempotent: one has E(m) † = E(m) and E(m) ◦ E(m) = E(m). The endomap E(m): X → X associated with a kernel/predicate m on X maps everything in X that is... |

22 | D.: Quantum measurements without sums
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Citation Context ...tions, where † is induced by the inner product. The use of daggers, mostly with additional assumptions, dates back to [31, 35]. Daggers are currently of interest in the context of quantum computation =-=[1, 40, 7]-=-. The dagger abstractly captures the reversal of a computation. Mostly, dagger categories are used with fairly strong additional assumptions, like compact closure in [1]. Here we wish to follow a diff... |

20 |
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Citation Context ... Hilb has equalisers by eq(f, g) = ker(g − f), which takes care of the right equality. As is well-known, the ℓ2 construction forms a functor ℓ2 : PInj → Hilb (but not a functor Sets → Hilb), see e.g. =-=[3, 15]-=-. Since it preserves daggers, zero object and kernels it is a map in the category DagKerCat, and therefore yields a map of kernel fibrations like in (1). It does not form a pullback (change-of-base) b... |

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Citation Context ...as ⊃: m ⊃ n def = E(m) −1 (n) = m ⊥ ∨ (m ∧ n). The associated left adjoint ∃ E(m) ⊣ E(m) −1 yields the “and then” operator: k & m def = ∃ E(m)(k) = m ∧ (m ⊥ ∨ k), so that the “Sasaki adjunction” (see =-=[12]-=-) holds by construction: k & m ≤ n ⇐⇒ k ≤ m ⊃ n. Quantum logic based on this “and-then” & connective is developed in [30], see also [36, 37]. This & connective is in general non-commutative and nonass... |

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Citation Context ...al example is the category Hilb of Hilbert spaces and continuous linear transformations, where † is induced by the inner product. The use of daggers, mostly with additional assumptions, dates back to =-=[31, 35]-=-. Daggers are currently of interest in the context of quantum computation [1, 40, 7]. The dagger abstractly captures the reversal of a computation. Mostly, dagger categories are used with fairly stron... |

8 | The Sasaki Hook Is Not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One That Assigns Causes
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Citation Context ..., and what is perpendicular to m to 0, as expressed by the equations E(m) ◦ m = m and E(m) ◦ m ⊥ = 0. Of interest is the following result. It makes the dynamical aspects of quantum logic described in =-=[8]-=- explicit. Proposition 24 For kernels m: M ↣ X, n: N ↣ X the pullback E(m) −1 (n) is the Sasaki hook, written here as ⊃: m ⊃ n def = E(m) −1 (n) = m ⊥ ∨ (m ∧ n). The associated left adjoint ∃ E(m) ⊣ E... |

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Citation Context ... Hilb has equalisers by eq(f, g) = ker(g − f), which takes care of the right equality. As is well-known, the ℓ2 construction forms a functor ℓ2 : PInj → Hilb (but not a functor Sets → Hilb), see e.g. =-=[3, 15]-=-. Since it preserves daggers, zero object and kernels it is a map in the category DagKerCat, and therefore yields a map of kernel fibrations like in (1). It does not form a pullback (change-of-base) b... |

6 |
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Citation Context ...al example is the category Hilb of Hilbert spaces and continuous linear transformations, where † is induced by the inner product. The use of daggers, mostly with additional assumptions, dates back to =-=[31, 35]-=-. Daggers are currently of interest in the context of quantum computation [1, 40, 7]. The dagger abstractly captures the reversal of a computation. Mostly, dagger categories are used with fairly stron... |

6 |
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Citation Context ...s in general non-commutative and nonassociative 1 . Some basic properties are: m & m = m, 1 & m = m & 1 = m, 1 The “and-then” connective & should not be confused with the multiplication of a quantale =-=[39]-=-, since the latter is always associative. 22�� � �� �� � �� � 0 & m = m & 0 = 0, and both k & m ≤ n, k ⊥ & m ≤ n imply m ≤ n (which easily follows from the Sasaki adjunction). Proof Consider the foll... |

5 |
The Theory of Symmetry Actions in Quantum Mechanics
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Citation Context ... Additionally, the category Hilb is an example—and, interestingly—also the category PHilb of Hilbert spaces modulo phase. The latter category provides the framework in which physicists typically work =-=[6]-=-. It has much weaker categorical structure than Hilb. We also present a construction to turn an arbitrary Boolean algebra into a dagger kernel category. The authors are acutely aware of the fact that ... |

5 | An embedding theorem for Hilbert categories
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(Show Context)
Citation Context ...rom the perspective of categorical logic (see e.g. [32, 28, 41, 21]). It grew from the work of one of the authors [20]. Although that paper enjoys a satisfactory relation to traditional quantum logic =-=[18]-=-, this one generalises it, by taking the notion of dagger category as starting point, and adding kernels, to be used as predicates. The interesting thing is that in the presence of a dagger † much els... |

5 | Categorical quantum models and logics - Heunen - 2009 |

5 | A presentation of quantum logic based on an ”and then” connective
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Citation Context ...r: k & m def = ∃ E(m)(k) = m ∧ (m ⊥ ∨ k), so that the “Sasaki adjunction” (see [12]) holds by construction: k & m ≤ n ⇐⇒ k ≤ m ⊃ n. Quantum logic based on this “and-then” & connective is developed in =-=[30]-=-, see also [36, 37]. This & connective is in general non-commutative and nonassociative 1 . Some basic properties are: m & m = m, 1 & m = m & 1 = m, 1 The “and-then” connective & should not be confuse... |

5 |
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Citation Context ...∃ E(m)(k) = m ∧ (m ⊥ ∨ k), so that the “Sasaki adjunction” (see [12]) holds by construction: k & m ≤ n ⇐⇒ k ≤ m ⊃ n. Quantum logic based on this “and-then” & connective is developed in [30], see also =-=[36, 37]-=-. This & connective is in general non-commutative and nonassociative 1 . Some basic properties are: m & m = m, 1 & m = m & 1 = m, 1 The “and-then” connective & should not be confused with the multipli... |

3 | Quantifiers for quantum logic
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Citation Context ...rt from minimal assumptions. This paper is a first step to understand quantum logic, from the perspective of categorical logic (see e.g. [32, 28, 41, 21]). It grew from the work of one of the authors =-=[20]-=-. Although that paper enjoys a satisfactory relation to traditional quantum logic [18], this one generalises it, by taking the notion of dagger category as starting point, and adding kernels, to be us... |

3 |
matrices and generalized dynamic algebra
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Citation Context ...ned from a Boolean algebra the factorisation of f : x → y is the composite x f −→f f −→y. In particular, for m ≤ x, considered as kernel m: m → x one has ∃f (m) = (m ∧ f : (m ∧ f) → x). Example 22 In =-=[33]-=- the domain Dom(f) of a map f : X → Y is the complement of its kernel, so Dom(f) = ker(f) ⊥ , and hence a kernel itself. It can be described as an image, namely of f † , since: Dom(f) = ker(f) ⊥ = ker... |

3 | On quantic conuclei in orthomodular lattices
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Citation Context ...∃ E(m)(k) = m ∧ (m ⊥ ∨ k), so that the “Sasaki adjunction” (see [12]) holds by construction: k & m ≤ n ⇐⇒ k ≤ m ⊃ n. Quantum logic based on this “and-then” & connective is developed in [30], see also =-=[36, 37]-=-. This & connective is in general non-commutative and nonassociative 1 . Some basic properties are: m & m = m, 1 & m = m & 1 = m, 1 The “and-then” connective & should not be confused with the multipli... |

2 |
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Citation Context ...y Definition 30 there are ϕ, ψ : Im(p) → X with 2 The name “effect” was chosen because of connections to effect algebras [11]. For example, in the so-called standard effect algebra of a Hilbert space =-=[13]-=-, an effect corresponds a positive operator beneath the identity. 29� � ��� � � � �� � � �� �� ψ † = (ip) † , ϕ ◦ mp = ψ, ϕ † = mp ◦ ψ † and ϕ = ip. This yields ψ = ip and mp = id. Hence p = ip ◦ (ip... |

2 | Compactly accessible categories and quantum key distribution
- Heunen
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Citation Context ...ness, by showing that the existence of directed colimits ensures that kernel subobject lattices are complete. This, too, is a natural categorical requirement in the context of infinite-dimensionality =-=[17]-=-. Recall that a directed colimit is a colimit of a directed poset, considered as a diagram. The following result can be obtained abstractly in two steps: directed colimits in D yield direct colimits i... |

2 |
Foundations of quantum physics. Number 19
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Citation Context ...nd thus p ≤ id. 8 Completeness and atomicity of kernel posets In traditional quantum logic, orthomodular lattices are usually considered with additional properties, such as completeness and atomicity =-=[34]-=-. This section considers how these requirements on the lattices KSub(X) translate to categorical properties. For convenience, let us recall the following standard order-theoretical definitions.complet... |

2 | A characterization of quantic quantifiers in orthomodular lattices. Theory and Applications of Categories, v - Román |

2 |
First Order Categorical Logic,” Number 611
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(Show Context)
Citation Context ...ure in [1]. Here we wish to follow a different approach and start from minimal assumptions. This paper is a first step to understand quantum logic, from the perspective of categorical logic (see e.g. =-=[17,14,24,12]-=-). It grew from the work of one of the authors [11]. Although that paper enjoys a satisfactory relation to traditional quantum logic [10], this one generalises it, by taking the notion of dagger categ... |

1 | Orthomodularity of decompositions in a categorical setting - Harding |

1 | Orthomodular lattices, Foulis semigroups and dagger kernel categories
- Jacobs
(Show Context)
Citation Context ...(x) = ↓ x char ∼ = �� B(1, ̂ x) (m ≤ x) � �� (m: 1 → x) As before, f ◦ char(m) = f ∧ m = ∃f (m) = char(∃f (m)). The category OMLatGal of orthomodular lattices and Galois connections between them from =-=[22]-=- also has such an opclassifier. There is no obvious kernel opclassifier for the category Hilb. The category PInj is easily seen not to have a kernel opclassifier. 5 Images and coimages We continue to ... |

1 | Quantifiers and orthomodular lattices - Janowitz - 1963 |

1 |
The theory of symmetry actions
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- 2004
(Show Context)
Citation Context ... Additionally, the category Hilb is an example—and, interestingly—also the category PHilb of Hilbert spaces modulo phase. The latter category provides the framework in which physicists typically work =-=[5]-=-. It has much weaker categorical structure than Hilb. Finally, we present a construction to turn an arbitrary Boolean algebra into a dagger kernel category. We suspect that there is a similar construc... |

1 |
An embedding theorem for Hilbert categories, preprint
- Heunen
- 811
(Show Context)
Citation Context ..., from the perspective of categorical logic (see e.g. [17,14,24,12]). It grew from the work of one of the authors [11]. Although that paper enjoys a satisfactory relation to traditional quantum logic =-=[10]-=-, this one generalises it, by taking the notion of dagger category as starting point, and adding kernels, to be used as predicates. The interesting thing is that in the presence of a dagger functor † ... |

1 |
Quantifiers for quantum logic, preprint
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- 2008
(Show Context)
Citation Context ...start from minimal assumptions. This paper is a first step to understand quantum logic, from the perspective of categorical logic (see e.g. [17,14,24,12]). It grew from the work of one of the authors =-=[11]-=-. Although that paper enjoys a satisfactory relation to traditional quantum logic [10], this one generalises it, by taking the notion of dagger category as starting point, and adding kernels, to be us... |