Bob Coecke and Ross Duncan- Interacting Quantum Observables

8/3/2019 Bob Coecke and Ross Duncan- Interacting Quantum Observables

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Interacting Quantum Observables

Bob Coecke and Ross Duncan

Oxford University Computing Laboratory

Abstract. We formalise the constructive content of an essential fea-ture of quantum mechanics: the interaction of complementary quantumobservables, and information flow mediated by them. Using a generalcategorical formulation, we show that pairs of mutually unbiased quan-tum observables form bialgebra-like structures. We also provide an ab-stract account on the quantum data encoded in complex phases, and

prove a normal form theorem for it. Together these enable us to describeall observables of finite dimensional Hilbert space quantum mechanics.The resulting equations suffice to perform computations with elemen-tary quantum gates, translate between distinct quantum computationalmodels, establish the equivalence of entangled quantum states, and sim-ulate quantum algorithms such as the quantum Fourier transform. Allthese computations moreover happen within an intuitive diagrammaticcalculus.

1 Introduction

Complementary quantum observables such as position and momentum cannotbe assigned sharp values at the same time. This fact constitutes the heart ofquantum physics. That the self-adjoint operators which characterise these dontcommute, motivated the study of non-commutative C-algebras, and that theirpropositional lattices are not distributive resulted in Birkhoff-von Neumannquantum logic. Neither of these axiomatic approaches unveils the true capabilitieswhich these complementary observables provide. They merely involve weakening

the commutativity/distributivity equation, rendering them essentially useless forany quantum informatic purpose. In this paper we provide an axiomatic accountof complementary quantum observables which enables us to tackle problems ofactual interest to quantum informatics: algorithm design, identifying the ca-pabilities of multi-partite entanglement, translation between distinct quantumcomputational models etc. Our starting point is the axiomatisation of quantumobservables proposed by Pavlovic and one of the authors in [5] which substan-tially relied on Carboni and Walters cartesian bicategories [2]. This notion ofquantum observable strongly improves on the one due to Abramsky and one ofthe authors in [1], the paper which initiated categorical quantum axiomatics, inthat it axiomatises quantum observables in terms of dagger symmetric monoidalstructure only, allowing for an operational interpretation, a diagrammatic cal-culus, as well as the necessary higher level of abstraction.1

1 For a detailed discussion of this necessity see [3,12].

L. Aceto et al. (Eds.): ICALP 2008, Part II, LNCS 5126, pp. 298 310, 2008.c Springer-Verlag Berlin Heidelberg 2008


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Interacting Quantum Observables 299

Somewhat ironically, while classical structures were crafted to reason aboutclassical control, this paper shows that considering a pair of interacting classicalstructurescorresponding to complementary quantum observablesare a pow-erful vehicle to specify and reason about pure quantum states and operations,

with many applications. We formalise this notion of complementarity througha set of equations which axiomatise copyability of classical states and the infor-mation flow through incompatible classical structures. Surprisingly the relevantequations are almost exactly those of a bialgebra [13], differing only by scalarfactors. We show that the axioms of this structure, a scaled bialgebra, expressthe essential features of quantum mechanics in very direct yet usable fashion.

2 Categories of Quantum States and Processes

A -symmetric monoidal category(-SMC) [12] is a symmetric monoidal category(C, , I) together with an identity-on-objects contravariant endofunctor : C C which preserves the monoidal structure. An elementary account of -SMCsand their graphical representations is in [12]. Well known examples include Rel,the category of sets and relations, and FdHilb, the category of finite dimensionalHilbert spaces and linear maps.

However quantum states are not vectors in a Hilbert space: they areone-dimensional subspaces. To articulate this fact we will use the word state ex-clusively to refer to such one-dimensional subspaces. Similarly, a non-degenerate

observable does not correspond to a basis, but rather a maximal family of mutu-ally orthogonal states. The move from a linear to a projective setting is formalisedusing a pair of categories, FdHilbp and FdHilbwp. The category FdHilbphas the same objects as FdHilb but its morphisms are equivalence classes ofFdHilb-morphisms for the congruence

f g c C \ {0} s.t. f = c g.This quotient reduces the scalar monoid to a two-element set, hence the capacityfor probabilistic reasoning is lost. The solution consists of enriching FdHilbpwith probabilistic weights i.e. to consider morphisms of the form r f wherer R+ and f a morphism in FdHilbp. Therefore, let FdHilbwp be the categorywhose objects are those ofFdHilb and whose morphisms are equivalence classesof FdHilb-morphisms for the congruence

f g [0, 2) s.t. f = ei g.We regain the absolute values of the inner-product, and thus the probabilisticdistance between states.2 These three categories are related via inclusions:

FdHilbp

rR+-

FdHilbwp

[0,2)-

FdHilb

2 A detailed categorical account on FdHilbwp is in [3]; in particular, neither FdHilbpnor FdHilbwp has biproducts, so the approach to measurements taken in [1] willnot work here.


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300 B. Coecke and R. Duncan

We write | (or rarely ) to denote vectors; || denotes a state spanned bythis vector. Similarly, |i ci| i is the state spanned by vectori ci| i . We takeas given a canonical basis for the Hilbert space Cn, which we write {| i }i. Thisbasis then fixes a canonical observable given by the states {|| i }i. The hom-setFdHilbp(C,C

2

)that is, the space of linear maps C C2

corresponds tothe the points of the Bloch sphere.3 The unitaries in FdHilbp(C,C

2), i.e. thosemaps U satisfying U U = UU = 1 correspond to rotations of the Bloch sphere.

3 Classical Structures and the Spider Theorem

A classical structure [5] in a -SMC is an internal cocommutative comonoid(A, : A A A, : A I) with both isometric and Frobenius, that is,

= 1A and = ( 1A) (1A ),respectively. The unit object I canonically comes with classical structure I : I I I and 1I. An orthonormal base {i}i for Hilbert space H induces

: H H H :: i i i and : H C :: i 1 (1)as a classical structure. Conversely, each classical structure in FdHilb arises inthis way [6]. Hence, classical structure axiomatises the concept of (orthonormal)base in a Hilbert space. Obviously, these classical structures are inherited by

FdHilbwp, and passing to FdHilbwp clarifies what the data which specify aclassical structure represent. A state || is unbiasedfor some observable {||i}iif for all i we have that | | i|2 = 1/ dim(H) whenever and i are unitvectors. Two observables {||i}i and {||i}i are complementary whenever ||iis unbiased for {||j}j for all i.

Proposition 1. InFdHilbwp each pair consisting of an observable {||i}i ona Hilbert space H and another state || of H which is unbiased for {||i}idefines a unique classical structure by setting, for all i,

(||i) = ||i i and |()| = |, |Conversely, all classical structures inFdHilbwp arise in this way.

The crux to this result is the fact that a set of n base vectors {|i}i of aHilbert space, up to a common global phase, is faithfully represented by the n+1states {||i}i {||

i i}. On the Bloch sphere an observable {||0, ||1},

e.g. {||0, ||1}, comprises two antipodal points, while ||, e.g. ||+, lies on thecorresponding equator, together making up a T-shape:

3 In any monoidal category maps of the type I A are called points.


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It is standard to interpret the eigenstates ||i for an observable {||i}i asclassical data. Hence, in FdHilbwp, the operation of a classical structure copiesthe eigenstates ||i of the observable it is associated with. We can interpret|| as the state which uniformly deletes these eigenstates: by unbiasedness theprobabilistic distance of each eigenstate ||i to || is equal. Therefore we willrefer to as (classical) copying and to as (classical) erasing. A crucial pointhere is that given an observable there is a choice involved in picking .

Within graphical calculus for -SMCs (see [12]) we depict the morphisms and by and , and their adjoints, and by and , When takingthe monoidal structure to be strictwhich we will do throughout this paperclassical structures obey the following remarkable theorem [4].4

Theorem 1. Let f, g : An

Am be two morphisms generated from classical

structure (A,,) and the dagger symmetric monoidal structure. If the graphicalrepresentation both of f and g is connected then f = g.

Hence, such a morphism only depends on the object A and the number of inputsand outputs. We represent this morphism as an n + m-legged spider

.

Theorem 1 allows the dots representing , , and to fuse into a singledot, provided all the dots are connected. Note that, conversely, the axioms ofclassical structure are consequences of this fusing principle.

Classical structure refines the -compact structure which was used in [1,12],provided the latter is self-dual. Graphical reasoning in compact structure byyanking is subsumed by reasoning in terms of the above spider theorem. Thiswill become clear in the first example of 6.3. We can define the conjugate f :A B of a morphism f : A B relative to classical structures (A, A, A) and(B, B, B) to be f := (1BA)(1Bf1A)(B1A) where X := X X .In FdHilb the linear function f is obtained by conjugating the entries of thematrix off when expressed in the classical structure bases. The dimension ofAis dim(A) := A A represented graphically by a circle.

4 A Generalised Spider Theorem and Abstract Phase

Data

Let (A,,) be a classical structure in a -SMC. On points , : I A wedefine

= ( ) i.e. .

4 Similar results are known for concrete dagger Frobenius algebras, e.g. 2D topologicalquantum field theories, as well as in more abstract categorical settings [11].


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Since (A, , ) forms a commutative monoid, this operation is immediatelyassociative and commutative, with unit . Now define

: C(I, A) C(A, A) :: ( 1A) i.e. .

From the properties of it immediately follows that is a homomorphism ofmonoids, and that for every

() = (1A ()) = (() 1A) i.e. .Since is commutative, we also have

() () = () () i.e. .

and since () = ( 1A) , the spider theorem yields () = ().Now let n : A An be defined by the recursion 0 = , 1 = 1A andn = (n1 1A).Theorem 2. Letf : An Am be a morphism generated from classical struc-ture (A,,), points i : I A (not necessarily all distinct), and dagger sym-metric monoidal structure. If the graphical representation of f is connected then

f = m

i

i

n . (2)

This is a strict generalisation of Theorem 1: besides the number of inputs andoutputs there is now also the product of all points which distinguishes classes ofequal diagrams. We obtain a decorated spider:

In graphical terms, Theorem 2 allows arbitrary decorated dots of the same colourto fuse together provided we multiply their decorations.

In Cn, consider | = i ci| i ; when written in the basis fixed by (, ), ()consists of the diagonal n n matrix with c1, . . . , cn on the diagonal. Hence,() is unitary, upto a normalisation factor, if and only if || is unbiased for{||1, . . . , ||n}. This fact admits generalisation to arbitrary -SMCs.Definition 1. We call : I A unbiased relative to if () is unitary.

Proposition 2. The set of points which are unbiased relative to a classicalstructure forms a group under with () as the inverse.For unbiased relative to , by Theorem 2 and Proposition 2, we have

=

() () () and = ()


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and again by the generalised spider theorem it then follows that these morphismsdefine a classical structure. We call it a phase shiftof (A,,). In C2, these phasedvariants to the classical structure {||0, ||1, ||+} (cf. Proposition 1) are thoseobtained by varying the choice of || on the equator of the Bloch sphere:

.

The states which are unbiased relative to {||0, ||1} are of the form ||+ :=|0 + ei|1 so form a family parameterised by a phase . In particular, we have||+1 ||+2 = ||+1+2, that is, the operation boils down to adding upphases modulo 2, which is an abelian group with minus as inverse.

5 Complementary Observables as Scaled Bialgebras

The goal of this section is to show that each pair of complementary observablesin FdHilbwp defines a scaled bialgebra. In the next section we will then use thisscaled algebra structure together with the generalised spider theorem for phasedata to reason about quantum informatics. First we define and study an abstractnotion of complementary observables. We then derive a general scaled bialgebra

law in categories with enough points such as FdHilb. This result then carriesover to FdHilbwp where it takes a much simpler form.

5.1 Complementary Classical Structures (CCSs)

In eq.(1) we described classical structures in FdHilb as maps which copy basevectors, and hence also the corresponding states in FdHilbwp. We introduce anabstract counterpart to these copy-able points. We assume as given a classicalstructure (A,,) in a -SMC.

Recall that if a

-SMC has a classical structure on an object A then themonoidal subcategory generated by A is -compact, and hence we can define thedimension of A by dim(A) = .5 For brevity, we define D = dim(A).We will, in addition, assume the existence of a self-adjoint scalar

D, which we

we denote graphically as . As the notation suggests,

D satisfies

D

D = D = dim(A) or, graphically: .

Notation. We represent all the points a : I A which are unbiased withrespect to (A,,) by dots of the same green (light grey) colour used before.Those points which are copied by in the sense of the definition below wemark by a different colour, here red, or darker grey. Any other points are markedin black. In light of the special role played by unbiased points, we will use thespider notation only for these.

5 One can show that dim(A) does not depend on the choice of classical structure.


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Definition 2. We call a point ai : I A classical relative to (A,,) if bothD = ai and

D ( ai) = ai ai hold, that is, graphically,

and .

The classical points for a classical structure in FdHilbwp are of course the states{||i}i of Proposition 1.

The abstract conception of a classical point allows the concrete notion ofunbiasedness to be derived from the abstract formulation of Definition 1:

Lemma 1. If ai : I A is classical and : I A unbiased for (A,,) then

(

ai)

(

ai)

= D i.e. .

The classical points are eigenvectors in a suitable sense:

Lemma 2. If ai : I A is classical for (A,,) and : I A arbitrary then

D (() ai) = ( ai) ai i.e. .

The monoid multiplication on points carries over to scalars:

Lemma 3. Let ai : I

A be classical and , : I

A arbitrary then

D (ai ( )) = (ai ) (a ) i.e. .

Remark 1. The reader may find the scalar factors in the above equations mys-terious, not to say vexing. But recall that in FdHilb the equation

= D isrequired to satisfy the comonoid laws; this scalar factor reappears here.6

Definition 3. Two classical structures (A, X , X) and (A, Z , Z) in a -SMC

are called complementary if they obey the following rules: whenever zi : I A is classical for (X , X) it is unbiased for (Z , Z); whenever xj : I A is classical for (Z , Z) it is unbiased for (X , X); X is classical for (Z , Z) and

Z is classical for (X , X).

We abbreviate complementary classical structure as CCS.

Notation. The reason that we refer to the classical points of (A, X , X) by zi,and vice versa, is because zi is unbiased to (Z , Z) and hence can participate in

the generalised spider theorem for the classical structure (Z , Z).

For any non-degenerate quantum observable we can find a pair of complementaryclassical structures in FdHilbwp merely by picking || for one observable fromamong the eigenstates of the other observable.6 An alternative would be to replace ( 1H) = 1H with ( 1H) = 1

D1H.


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5.2 Derivation of the Scaled Bialgebra Law from Abstract Bases

Definition 4. A set of points {ai}i is a basis for an object A if for all f, g :A B, if f ai = g ai for all ai, then f = g. A basis is classical, or unbiased,

with respect to some classical structure (A,,) if its elements are respectivelyclassical, or unbiased, with respect to this structure. An unbiased basis is calledclosed if for all ai, aj there exists ak such that ai aj = ak and a0 = . Wesay that a -SMC has monoidal bases when, for each basis {ai}i for A, and eachbasis {bj}j for B, the set {ai bj}ij is a basis for A B.An immediate consequence of this definition is that whenever b is an elementof a closed unbiased basis {bi}i, then (b) is a permutation on the set {bi}i.Further, by Lemma 2 every classical point is an eigenvector of (b).

Lemma 4. Let {ai}i be a classical basis for A suppose p : A A acts as apermutation on this set; then

(p p) = p i.e. .

Lemma 5. Let (X , X), (Z , Z) be CCSs, let x be in a closed classical basis of(Z , Z) and let z be unbiased for (Z , Z), then

D

(X(x)

Z(z)) = (x

z)

(Z(z)

X(x)) i.e. .

The Pauli matrices provide an example of these commutation relations.

Lemma 6. Let(Z, Z) and(X , X) be CCSs and letUZ denote all the unbiasedpoints and CZ a basis of classical points for (Z , Z). Suppose x CZ and letX = X(x). If CZ is closed under X then:

X is a permutation on CZ ; X is an automorphism on UZ such that

X ( Z ) = (X)Z (X ); , X Z = Z and (X )1= X 1.

Corollary 1. (CZ , X) is an abelian group with a group action on UZ definedby (x, z) X(x) z.Lemma 7. Consider a -SMC with monoidal bases and let be the monoidalsymmetry. Let(A, X , X) and(A, Z , Z) be CCSs with classical bases {zj}j and{xi}i; {xi}i is closed if and only if

D (X X)(1A 1A)(Z Z) = D ZX i.e. .Corollary 2. In the above situation {xi}i is closed if and only if {zi} is.

Theorem 3. Let (X , X) and (Z, Z) be CCSs with closed bases including thepoints z and x respectively. Then, graphically,


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. (3)

We call the morphisms obeying eq.(3) a scaled bialgebra.

Proposition 3. If (X , X) and (Z , Z) form a scaled bialgebra then

i.e. it is a scaled Hopf algebra with dim(A)

1A as its antipode.

5.3 Complementary Classical Observables in FdHilbwp

Classical structures in FdHilb are bases [6] so complementary pairs of baseswhich satisfy the closedness condition of Definition 4 induce scaled bialgebrasin the sense of Theorem 3. These scaled bialgebra laws carry over to the CCSsin FdHilbwp consisting of the states spanned by the basis vectors. Moreover, inFdHilbwp, since all scalars are positive reals, all scalars in eqs.(3) coincide, socancellation simplifies eqs.(3) to

. (4)

Conversely, any pair of complementary observables yields a family of CCSs inFdHilbwp mediated by a group of permutations on the respective sets of classicalstates, and one can always construct a corresponding underlying family of CCSsin FdHilb. What we so far failed to prove is that in general we can alwaysconstruct a corresponding underlying family of closed CCSs. However: (i) CCSs

in FdHilb on C2

and C3

are closed; (ii) CCSs can be chosen to be closed forall (to us) known constructions of mutually unbiased bases (e.g. [10]); (iii) weconstructed closed CCSs on Cn in FdHilbwp for all n. Hence for all practicalsituations involving complementary observables eq.(4) hold. We conjecture thatclosed CCSs can be derived from any pair of mutually unbiased observables.7

6 Applications and Examples in Quantum Informatics

Inevitably, the examples from this field are constructed from the ubiquitous qubiti.e. C2. Take the green classical structure (Z , Z) as in eq.(1) for {| 0, | 1}.The unbiased points for (Z , Z) are of the form |Z = |0+ ei |1, and |ZZ7 The study of mutually unbiased bases is an active area of research; characterisation

of the maximal number of mutually unbiased bases is one of the important openproblems in quantum informatics.


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|Z = | + Z. Further, Z() =

1 00 ei

, in particular, Z() = Z. Notice

that Z = |0+ |1 and |Z = |0|1 form a basis, which is closed and unbiasedwith respect to (Z , Z) and define a complementary red classical structure

(X , X). The unbiased points for (X , X) have the form |X = 2(cos2 |0 +

sin 2

|1) and X() =

cos 2

sin 2

sin 2 cos2

, in particular, Z() = X. We have Z

|X = |X, and, upto a global phase, X |Z = |Z. In the language ofLemma 6 we have: if |CZ | = 2, then (CZ , X) is the symmetric group S2, itsunique non-identity element X is self-adjoint, and for UZ we have X = i.e. X assigns the inverses for the group UZ .

For some of the examples below it will also be convenient to explicitly have theunitary operation which changes the green dots into red dots, that is, concretely,

the unitary operation which establishes the corresponding change of basis. In thecase of (Z , Z) and (X , X) given above, the two structures are connected viathe familiar Hadamard map H. As well as being unitary, H = H so this map isparticularly well behaved.We will introduce H into the graphical language with

the following equations , and .

Below we disregard scalar factors which only distract from the essential point.

6.1 Quantum Gates, Circuits, and AlgorithmsAbove we introduced 1-qubit unitaries Z() and X() corresponding to ro-tations in the X-Y and the Y-Z planes respectively; these suffice to representall 1-qubit unitaries, and their basic equational properties follow from the var-ious lemmas introduced in the preceding sections. We demonstrate how to de-fine the X and Z gates, and prove two elementary equations involving them.The addition of these gates will provide a computationally universal set ofgates.

Example 1 (X gate). Setting one verifies by concrete cal-culation that X = . We can also give an abstract proof . Let |i be a classicalpoint for the green classical structure; by evaluating it with an input to its con-

trol qubit (the green end) we have , which for |i = |0Xis the identity, and in the binary case, for |i = |1X, is the unique operationX. By applying it three times, alternating the target and control input, we ob-

tain, , i.e. . while this is a well-known

property ofX, our proof uses only the bialgebra structure hence it will hold inmuch greater generality than just for qubits.


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Example 2 (Z gate). One can derive the Z from that of X by augmentingthe target qubit of the X with H gates i.e. . We have that

Z Z = 1 since .

Example 3 (An algorithm : the quantum Fourier transform). The quantumFourier transform is one of the most important quantum algorithms, lying atthe centre of Shors famous factoring algorithm. The equations we have enablethis algorithm to be simulated in the diagrammatic language. Unlike the pre-ceding examples, here we require the interaction between the two phase groups.

In our language the Z gate is and the circuit involving it

realises the quantum Fourier transform for 2 qubits. The algorithm can be sim-ulated graphically, as shown below:

This example makes use of classical values coded as quantum states to control theinterference of phases: this is the archetypal behaviour of quantum algorithms.

6.2 Multi-partite Entanglement

In our graphical language, a quantum state is nothing more than a circuit withno inputs; output edges correspond to the individual qubits making up the state.The interior of the diagram, i.e. its graph structure, describes how these qubitsare related. Hence this notation is ideal for representing large entangled states.

Example 4. The cluster states used in measurement-based quantum comput-ing, can be prepared in several ways; the graphical calculus provides shortproofs of their equivalence. For example, the original scheme describes a Zinteraction between qubits initially prepared in the state |+; in our nota-

tion this is |0Z, or . So 1D cluster arises as where

the boxes delineate the individual |+ preparations and Z operations. Al-ternatively, the cluster state can be prepared by fusion of states of the form|0+ + |1. Our Z is in fact this fusion operation, so a 1D cluster arises as


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. Using the spider theorem, these are equiva-

lent . Ongoing work seeks

to classify multipartite entangled states in terms of their graphical representa-tives, and to formalises general matrix product states.

6.3 Properties of Quantum Computational Models

Our formalism axiomatises two key features of quantum mechanics: the underly-ing monoidal structure and the interaction of complementary observables. Fur-thermore it is a semantic, which is to say extensional, framework which makesit ideal for unifying various approaches to quantum computation. E.g. we candemonstrate equivalence between different quantum computational models.

Example 5 (Verifying one-way quantum computations). We show how to verifysome example programs for the one-way model, taken from [7], by translation toequivalent quantum circuits. Post-selected qubit measurements8 can be repre-sented by copoints such a . The spider theorem allows the post-selected one-way

program implementing a

X opera-

tion upon its inputs to be rewritten to a X gate in no more than two steps. Now recall that any single qubit unitary map U has an Euler decomposi-tion as such that U = ZXZ. In our notation this is Z =

Z() andX =

X(). Again a sequence of simple rewrites shows that the one-way pro-

gram to compute such a uni-

tary indeed computes the desired map.

References

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2. Carboni, A., Walters, R.F.C.: Cartesian bicategories I. Journal of Pure and AppliedAlgebra 49, 1132 (1987)

3. Coecke, B.: De-linearizing linearity: projective quantum axiomatics from strongcompact closure. ENTCS 170, 4972 (2007)

8 Classical structures were initially introduced in [5] to represent classical control struc-ture so this example can easily be extended with the required unitary corrections.


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4. Coecke, B., Paquette, E.O.: POVMs and Naimarks theorem without sums (2006)(to appear in ENTCS), arXiv:quant-ph/0608072

5. Coecke, B., Pavlovic, D.: Quantum measurements without sums. In: Chen, G.,Kauffman, L., Lamonaco, S. (eds.) Mathematics of Quantum Computing and Tech-nology, pp. 567604. Taylor and Francis, Abington (2007)

6. Coecke, B., Pavlovic, D., Vicary, J.: Dagger Frobenius algebras in FdHilb are bases.Oxford University Computing Laboratory Research Report RR-08-03 (2008)

7. Danos, V., Kashefi, E., Panangaden, P.: The measurement calculus. Journal of theACM 54(2) (2007), arXiv:quant-ph/0412135

8. Joyal, A., Street, R.: The Geometry of tensor calculus I. Advances in Mathemat-ics 88, 55112 (1991)

9. Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. Journal ofPure and Applied Algebra 19, 193213 (1980)

10. Klappenecker, A., Rotteler, M.: Constructions of mutually unbiased bases. LNCS,

vol. 2948, pp. 137144. Springer, Heidelberg (2004)11. Kock, J.: Frobenius Algebras and 2D Topological Quantum Field Theories. In:Composing PROPs. Theory and Applications of Categories, vol. 13, pp. 147163.Cambridge University Press, Cambridge (2003)

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13. Street, R.: Quantum Groups: A Path to Current Algebra, Cambridge UP (2007)
http://www.mathstat.dal.ca/~selinger/papers.htmldaggerhttp://www.mathstat.dal.ca/~selinger/papers.htmldaggerhttp://www.mathstat.dal.ca/~selinger/papers.htmldaggerhttp://www.mathstat.dal.ca/~selinger/papers.htmldagger

Bob Coecke and Ross Duncan- Interacting Quantum Observables

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