A 2-category for mixed states protocols

Author Thomas CuvillierSupervisor Jamie Vicary

17/08/2012

The author would like to advertise the reader that this short dissertation is a sum-mary of a much larger one, which contains far more details, examples and definitions.Please e-mail me at thomas-cuvillier@ens-cachan.fr if you wish to see it. As the intern-ship took place in the United-Kingdom, it is written in English.

1 Introduction

1.1 General context

Quantum computation is traditionally studied at a very low level: in terms of gates,circuits and qbits, which are akin to bit programming with string manipulations at thebirth of computing science. Filling the need for a higher language, symmetric daggermonoidal categories have emerged as a convenient categorical formalization of quantummechanics. The upshot of this approach is that properties can be displayed in a suitablefashion, relying on graphical language for a category.

This field was first investigated by Abramksy and Coecke [AC04], and has notablyevolved since then. However, one of the biggest issues was to distinguish quantumsystems from their classical counterparts, and then to axiomatize their interfaces ; themeasurement.

A breakthrough for this problem was then brought by Vicary in [Vic12]; he useda semantics based on a 2-category to describe quantum protocols. There, the objectsare spaces of classical data, whereas the 1-morphisms are spaces of quantum systems.Finally, 2-morphisms captures the quantum evolutions. Moreover, there is still a graph-ical fashion to look at it, and the graphical calculus is as powerful as the one used for1-categories. This category arises in a meaningful fashion using the Bimod constructionon Hilbert spaces, leading to a 2-category called 2Hilb. However, one may wonder, cansuch a construction be carried to other categories ?

Nowadays, two different mathematical formalisms are used to describe quantum me-chanics. Although both are based on Hilbert spaces, one considers that quantum systemscan live in different states at the same time. Such a state is called a mixed one. Subse-quently the founder article publication of categorical semantics for quantum protocols,Selinger argued in [Sel07], that the exhibited category was not the right one to carryquantum mechanics as it couldn’t axiomatize mixed states, and then proposed a con-struction, called CPM (for completely positive maps), to adapt the just-created categoryto fit this requirement.

Despite that, investigating this new category in [Coe08] and [BCP09], Coecke re-veals that in this setting a new definition of measurement can be given. Indeed, in

1

such a category, the measurement is not a unitary operation anymore. Besides, the oldcharacterization of classical, which used to be cloneable elements, turns out to be un-adapted. Using a so-called environment structure, it gives rise to a new axiomatisationto distinguish classical from quantum, and to define the interface between both.

1.2 Studied problem

The first problem I was addressed was to give some axiomatizations of a so-called com-plementary controlled measurement in the 2Hilb category. This problem is now solvedand the answer is given in (A.1).

After that, I was looking into a connection between the CPM construction, where,as shown in [BCP09], some similarities between the graphical calculus and the one com-ing from the 2-category can be found. But, by seeing the Coecke article [BCP09] onenvironments and remembering a construction done by Selinger in [Sel07], I got the ideathat the Bimod construction could be carried to form a 2-category of mixed states. Sothis problem is entirely new, and has not been yet investigated as the paper of Vicarydescribing 2-quantum protocols just got published (the end of July).

1.3 My contribution

First, I had the idea that this construction could be carried from 2Hilb to CPM, byusing some tricks. Moreover, it turns out that because of the new axiomatization ofmeasurement and classical data, some categorical works had to be performed before theconstruction itself, and some definitions had to be given as well. The main differencefrom this 2-category and 2Hilb comes from the fact that a measurement is not unitaryanymore and so cannot be forgotten. The consequences of such a property are explainedin (7.3).

1.4 Arguments supporting its validity

My construction carried properties from the 1-category and the environment structureinto a 2-category. Most of the technical work was already done: the additive completionof CPM has been done by Selinger, the Bimod construction has been deeply investigatedin [Ost03]. However, I have no computational proofs of the works I have done. Thesolution given is a particular solution for a particular problem: a 2-category for mixedstates protocols. Since this field was created, one of the main goal is to capture the mostuseful category and the most suitable properties for doing quantum mechanics in it. Icannot state that this 2-category is the right answer to this general problem.

1.5 What is next

A future (giant) step could be to do the Bimod construction in the general case of adagger-compact category, and then to explore his general properties, and conclude abouthis ability to be a 2-category for quantum mechanics. However, some more little stepsare presented at the end of the dissertation (8).

2

2 An introduction to categories for quantum protocols

In this section, we will introduce the reader to the state of the art of describing quantumprotocols with categorical tools, and how the required properties for the category arisenaturally from FdHilb.

2.1 Symmetric monoidal compact category

2.1.1 General definitions

In the following, we will always work in FdHilb.Such a category turned out to have interesting property, which will be listed here.

1. FdHilb is monoidal : A category C is said to be monoidal if there exists a functor⊗ : C × C → C, a unit object I, two unit natural isomorphisms ρ : A ⊗ I ' Aand λ : I ⊗A ' A, and a natural associativity isomorphism αa,b,c : A⊗ (B ⊗C) '(A⊗B)⊗ C which satisfy certain coherence diagrams, which we omit.

2. We have a symmetry: A symmetric monoidal category is one equipped with anatural symmetry isomorphism

σA,B : A⊗B ' B ⊗A (1)

3. A dual property : A category is compact closed if for every object A, there existsa object A∗ (called the dual), and two morphisms

ηA : I → A∗ ⊗A εA : A⊗A∗ → I (2)

such that the following diagram commutes

A

idA��

1A⊗ηA// A⊗A ∗ ⊗A

εA⊗1Avvmmmmmmmmmmmmmm

A

(3)

in FdHilb we haveηA : C→ A∗ ⊗A :: 1 7→

∑ni=1 ei ⊗ ei

εA : A⊗A∗ → C :: ei ⊗ ej 7→ ej(ei)(4)

where (ei) is a basis of A, Hilbert space of dimension n.

2.1.2 Morphisms and dagger compact properties

In every compact closed category, given a morphism f : A → B, one can constructf∗ : B∗ → A∗ the following way :

B∗

f∗

��

ρA // B ∗⊗ I1B∗⊗ηA // B∗ ⊗A∗ ⊗A

1B∗⊗1A⊗f // B∗ ⊗A∗ ⊗B1B∗⊗σA∗,B

��A∗ I ⊗A∗

λ−1A

oo B ⊗B∗ ⊗A∗εB⊗1A∗

oo B∗ ⊗B ⊗A∗σB∗,B⊗1A∗oo

(5)

3

We will now try to capture the notion of adjoint, by adding the notion of dagger.

Definition.

A symmetric monoidal dagger category C means a symmetric monoidal category with acontravariant functor

()† : Cop → C (6)

such that ()† is the identity on objects, involutive and preserves the monoidal structure.

However, to complete the structure, we would like the dagger to be related to thecompact structure.

Definition. A dagger compact closed category is a compact closed category with a daggersuch that the following diagram commutes.

I

ε†A��

ηA // A∗ ⊗A

σA,A∗yyrrrrrrrrrr

A⊗A∗

(7)

Combining both the compact and the dagger structure, we can construct a thirdinvolutive functor ()∗ : C → C defined by

(A)∗ = A∗ f∗ = (f∗)† = (f †)∗ (8)

Moreover, in the following, we will consider that A∗ = A.

2.2 Graphical calculus for compact closed SMC

In this paper, a lot of results will rest on graphical calculus. In this section we present thegraphical calculus for SMC, and we show as well how it extends to carry the propertiesof the compact structure in it too.

Morphisms are drawn upward, the identity on an object A is just represented bya simple vertical line, and a morphism : f : A1 ⊗ A2 ⊗ ... ⊗ An → B1 ⊗ ... ⊗ Bm isrepresented as the following

A1 An· · ·

B1 Bm· · ·

f (9)

with each wire in input corresponds to an Ai, whereas the output ones correspond toan object Bj . The parallel composition of wires expresses the tensor product of objects,or of morphisms, whereas the vertical juxtaposition of morphisms expresses the usual ◦composition, as depicted in the following diagram.

f ⊗ g = f g g ◦ f =f

g(10)

4

The white part can be seen as being the unit element C. Each equation expresses inthe graphical language can be turn upside down, describing the action of the dagger.

A

B

f

g

C

D

h

†

=

A

B

f

g

C

D

h (11)

where

f = f † f = f∗ and f = f∗ (12)

We are now able to represent the structure morphisms of the compact category (εAand ηA) in our graphical calculus.

A A

ηA A A

εA(13)

The consistence and justification of the graphical calculus comes from the followingtheorem, proved independently by Selinger [Sel11] and Joyal and Street [JS91]

Theorem 1. An equation follows from the axioms of dagger compact categories if andonly if it can be derived in the graphical language via isotopy.

By isotopy we express the fact that we only care about the number of inputs andoutputs.

2.3 Classical structure and spider theorem

There are two points of view leading us to the same definition :

1. A classical data can be represented by a quantum system which has just been mea-sured. So, this way we construct a relation between classical data and orthogonalbases, which are seen as the eigenvalues of an observable operator.

2. The second point of view is that the classical data can, in opposition to a quantumsystem, be easily copied or erased.

Starting from the second point of view, we can described classical data as an objectA endowed with a commutative Frobenius algebra, which is defined by the data of twomaps on this object, the copying and deleting maps :

(14)

5

such that if one copies or deletes the classical data, then the data remains the same.

= = = = (15)

We have moreover maps coming from the dagger, which act coherently with the others.

= = (16)

The Frobenius algebras are a characterization of orthogonal bases in FdHilb. Theequation (19) displays this property; the Frobenius structure maps rest on a orthogonalbasis.

We finish this section by pointing out that a Frobenius structure induces a compactone, with

εA : A⊗A→ I := ∇ ◦ >∇A : I → A⊗A := ⊥ ◦∆

(17)

Given a classical structure, one can, by composing and tensoring their structuralmorphisms, creates a family of morphisms. The spider theorem states that no matterhow one composes or tensors them, only the number of inputs and outputs matters.

Theorem 2. If f, g : A⊗n → A⊗m are morphisms generated from the classical structure(A,∇,⊥), the symmetric monoidal structure maps, the adjoints of all of these, and ifthe graphical representations of f and g are connected, then

f = g = Ξmn (18)

Remark. In FdHilb, given a orthonormal basis (|i〉)i associated with a classical struc-ture, we then have

Ξmn :: |i . . . i〉︸ ︷︷ ︸n

7→ |i . . . i〉︸ ︷︷ ︸m

(19)

3 Positivity and new categories

This short section aims to establish two new categories starting from a dagger compactone C: the CPM and the WP ones.

Before going into details, we will present how one can define positivity and com-pletely positivity in a dagger compact category, allowing us to, then, define CPM andemphasizing the inclusion functor from WP to CPM. Then, we will briefly explain whatmixed states are, in opposition to pure ones, and what is, physically, the category comingfrom the CPM construction.

6

3.1 Definitions

The following definitions take place in any dagger compact category.

Definition. A morphism f : A → A is positive if there exists B, g : A → B such thatf = g† ◦ g

We would like to find out the maps under which the positive matrices are stable, bothby composition and tensoring, so we can construct an associated monoidal category. Suchmaps are said to be completely positive.

Definition. A morphism f : A∗ ⊗A→ B∗ ⊗B is completely positive if there exists anobject C and a morphism g : A→ C ⊗B making the following holds .

AA

f

BB

=

A A

g g

B B

(20)

The object C is called the ancilla of f , while the morphism g is called its Krausmorphism. They are not unique.

3.2 WP and CPM construction

Given a dagger compact closed symmetric monoidal category C, one can make the bothfollowing constructions. The first one is an attempt to erase the global phases in FdHilb,whereas the second one tries to capture the notion of mixed state.

The original problem is that, while doing quantum physics in FdHilb, two vectorswhich are equal modulo a global phase represent in fact the same physical state. There-fore, we would like to create a new category with no more redundancy. Starting from acategory C, we create the category WP(C).

Definition. [Coe07b] The WP construction : WP(C) is defined to have

1. as objects the same as those of C

2. as maps A→ B the maps f∗ ⊗ f : A∗ ⊗A→ B∗ ⊗B,

Theorem 3. The functor defined by F : C→ WP (C) defined by F (A) = A on objectsand F (f) = f∗ ⊗ f on morphisms respects the monoidal structure. Moreover, if C isdagger closed compact, then so is WP (C). However, the functor does not carry thebiproduct.

Remark. Taking FdHilb as C, the functor F : C→WP (C) is almost faithfull. Indeed,we have, for two morphisms f, g : A→ B

F (f) = F (g)⇔ ∃φ ∈ R, f = eiφg

7

The WP category is the same as the original one modulo the global phases.

Remark. A morphism f : A∗ ⊗A→ B∗ ⊗B which is such that

∃g : A→ B.f = g∗ ⊗ g (21)

is a completely positive morphism.

Starting from the previous remark, one can also think of a larger category than WP,which would have completely positive maps as morphisms.

Definition. [Sel07] The CPM construction : Starting from a symmetric compact closedcategory C, one can create the category CPM(C) with

1. as objects : the same as those of C

2. as morphism f : A→ B, the completely positive maps A∗ ⊗A→ B∗ ⊗B.

Theorem 4. CPM(C) is again a dagger compact closed category.

In particular, all the structural morphisms, such as the ones coming from the compactstructure or the ones coming from the classical structure, lift into CPM.

3.3 Mixed states and CPM construction

In quantum mechanics, a physical system may turn out to be in several states as thesame time, it is then represented by a statistical superposition of states. A systemwhich lives in only one state, would be called a pure one. Otherwise, to represent thephysical system, we need a so-called density matrix. The density matrices of an Hilbertspace A are the semi-positive hermitian ones A → A, so completely positive matricesρ : I → A⊗A. Indeed, given such a matrix ρ, one can also write it as :

ρ =∑i

pi|ψi〉〈ψi| (22)

where ψi are the eigenvectors with pi the eigenvalues associated, representing the prob-ability that the physical system lives in the state ψi.

Definition. Given a category CPM(C), a pure state of an object A is a morphismΦ : I∗ ⊗ I → A∗ ⊗A which is such that

∃ψ,Φ = ψ∗ ⊗ ψ (23)

Generalizing, in CPM(C) the pure morphisms are the ones which belong to the imageof WP(C) under the inclusion functor WP (C) ↪→ CPM(C)

Definition. With C = FdHilb, the category CPM(FdHilb) is called Dens.It is the category with

8

1. Cn∗n, for n ∈ N as objects

2. the completely positive maps F : Cn∗n → Cm∗m as morphisms. The identity is theidentity matrix, and the composition is the usual composition of matrices.

In this category, a morphism f : Cn∗n → Cm∗m is pure if there exists a linear mapL : Cn → Cm such that f := p→ LpL†.

4 Environment and classical channel

While working in FdHilb, we restrict ourselves to unitary operators: we can forgetabout a measurement. By working with mixed states, it is not the case anymore; we arenow working with completely positive matrices, and some might be not invertible. Thatis the case of the measurement.

We will explain in this section how one can translate some properties in our newcategorical language.

4.1 Environment

Definition. For a dagger compact category C and a dagger compact subcategory Cpure

of C such that |C| = |Cpure|, an environment structure consists of a co-state >A : A→ Ifor each object A of the category C, which satisfies the following properties.

1.

A⊗B=

A B

A = A

2. for all A, B objects of C and all f, g : A→ B ∈ Cpure(A,B) we have

A

f

f

A

=

A

g

g

A

⇐⇒

A

f =

A

g (24)

Remark. In Dens one can construct an environment structure using the subcategoryWP (FdHilb) as the pure one. The environment structure is then the trace.

>Cn∗n : Cn∗n → C : p 7→ tr(p)

A measurement has to have as output, a classical element.So before introducing it,let us give a characterization, in our setting, of what is a classical data.

Definition. We say that a element e : I → A is classical for a given classical structureΞ over A, if its matrix representation, using the basis coming from A, is diagonal.

9

4.2 Environmental measurement

Definition. An environmental measurement is defined by the following morphism m :

m

AA

A

=δ

AA

A

(25)

Then the result is diagonal in the basis |ii > coming from the classical structure. Wethen have the following properties:

m

m

AAA

A

=m

AA

A

A

m

A

A

6=

S

A

(26)

Theses pictures correspond to the definition of a non-unital co-module (26). Theclassical data are indeed a ring, and co-act by m on the density matrices, which are amonoid. However, as we can see in the following equation (27), we have what we mightcall a “pseudo-co-unit”, which is a co-unit just over the co-image of m.

m

m

AA

A

=m

AA

B

m

H

=

H

(27)

The co-unit property reveals an interesting one; the classical data form a (unital)co-module under the application of m. That is, the right notion to capture the resultof a measurement is the co-module. An other interesting fact, is that the environmentstructure coming from an object A is universal among the classical structures over A, asit comes from the compactness of the category. A such property is exposed in [Coe08].

Proposition. The state ⊥A = >†A is diagonal in any basis

So the measurement is not done by the application of > but by the application of δAcoming from the classical structure.

5 Towards a 2-cat

In this section, we will present successively some constructions which will give rise to a2-category. This part is highly inspired from the chapter 6 of [Vic12], the constructionsbeing the same, but taking place in Dens instead of FdHilb. Though, a lot of diagrams

10

are taken directly from this paper. The first part of this section is dedicated to theconstruction of a co-module for an environmental measurement. We will first definewhat’s an environment interacting with the quantum system, which will, after analysis,give us the key ingredients of a new category that will capture the new notions we want.During the second part, we will exhibit some constructions based on this category, whichwill turn out to be the objects and morphisms of a new 2-category, defined in the nextsection.

5.1 Interacting environment

Definition. Given a classical structure Ξ in a category C, a co-module H is an objectof the category endowed with a morphism f : H → H ⊗ Ξ called interaction map whichsatisfies the following properties :

f

f

HΞΞ

H

=f

H

δA

Ξ

H

Ξ

f

H

H

=

H

H

(28)

Definition. An environmental interaction map is a map m satisfying the followingaxioms (26) (27) .

More concretely, the environmental interaction map would then be :

mΞ := h 7→∑i

|Pii(h)〉|i〉

:= h 7→∑i

|Pi ◦ h ◦ Pi〉|i〉(29)

where the Pii are a family of orthogonal projector of Dens which are pure. If we onlyrely on the classical elements, which means the ones that are diagonal (with respect tothe environmental interaction map), then the environmental interaction map acts as asimple interaction map (28), and so the classical elements endowed with such a mapform a co-module. But to create the object of the classical elements with respect to anenvironmental interaction map, we have to use kernel and quotients; we therefore needa new category endowed with such objects, that we will look at in the next section.

5.1.1 Bi-co-module

Generalizing, a physical system can interact with two different environments in a con-sistent way; the order we perform the measurement does not matter. Such a physicalsystem is called a bi-co-module.

11

Concretely, there are f1 : H → E1 and f2 : H → E2 which satisfy the previousequations (of the co-module), and such that moreover the following equation holds.

f1

f2

HE2E1

H

=

f2

f1

HE1 E2

H

(30)

Two interaction maps form a co-module if their families of projectors are mutuallyorthogonal :

PiiQjjPkkQll = δikδjl

5.2 The additive completion of CPM

Definition. Starting from an Ab-category C, we can construct the additive completionof C, called C⊕, which has

1. as objects, tuples of objects 〈A1, A2, ...., An〉 of C

2. as morphisms, matrices of morphisms of C.

The composition of morphisms is given by the composition of matrices in the usual way,using the Ab-structure for the +, the usual composition of morphisms for the ×.

Proposition. If C is monoidal closed, then so is C⊕. Furthermore, if C is a daggercompact closed category, such that the dagger functor is linear on morphisms, that is(g + f)† = g† + f † and 0† = 0, then C⊕ is a biproduct dagger compact closed category.

Let us point out that the the inclusion functor F :C→CPM(C)⊕ does not carrythe bi-product. The bi-products of Dens⊕ are not the same as the ones coming fromFdHilb.

For example, a state of 〈C,C〉 is called a classical bit, and is of the form (a, b), witha, b ∈ R+. On the other hand, a state of C ⊕ C is of the form of a completely positive

matrix

(a bc d

).

Theorem 5. Dens⊕ is a category which have both kernels and quotients.

The interested reader may find more details in [Lan].From now on we denote [n] = 〈C,C, . . . ,C〉︸ ︷︷ ︸

n

the space of the classical n-bits.

A way to see what can represent Φ = 〈A1, ..., An〉 in Dens is to look at it as aunique completely positive matrix with the blocks Ai on the diagonal, and 0 everywhereelse. The space 〈Ca1 ,Ca2 . . .Can〉, is the subspace of Ca1+a2+...+an spanned by the stateswhich have been measured over an environment which has made them collapse to the ndifferent subspaces defined by ai.

12

5.3 Formalism

We adopt some new names for our category Dens⊕.The objects are now signatures σ = (n,m, s...., t), so finite sequences, or polynoms,

of natural numbers. They represent the tuples of spaces 〈Cn,Cm, ....,Ct〉, which aredenoted Vσ, but, by abuse of notation, we merge the both. The sum of two sequences isnoted with u, and the product ⊗ as it naturally extend the original one.

(a1, a2, .., an) u (b1, b2, ..., bm) = (a1, a2, .., am, b1, b2, ..., bm)(a1, a2, ...an)⊗ (b1, b2, .., bm) = (a1 ∗ b1, a1 ∗ b2, ...a1 ∗ bm, a2 ∗ b1, ...., a2 ∗ bm, ...an ∗ b1, .., an ∗ bm)

(31)Given a state Φ = 〈A1, ...., An〉 in a signature σ, we can define the notion of trace,

which extends the usual onetr(Φ) =

∑i

tr(Ai) (32)

This allows us to extend the notion of environment structure for signatures. A environ-ment structure for a signature is the adjoint of the trace.

5.4 New sum and measurement

We can give an evident classical structure on [n], as well as an environment structureone. Using such a structure, an environmental co-module for this environment is thedata of a signature σ and n maps, from this signature to [n], whose restrictions to σ areprojectors. The environmental interaction map is then of the form

f : Vσ → Vσ ⊗ [n] : |Ψ〉 7→∑i

|P †i ΨPi〉|i〉 (33)

So a co-module for such an environmental interaction map is the data of a signature,or, equivalently, a tuple of Hilbert spaces, n orthogonal projectors on this signature,which span it entirely, with the interaction map acting:

f : uiVσi → (uiVσi)⊗ [n] : uΨi 7→ uΨi ⊗ |i〉 (34)

So the classical elements of a environmental co-module are isomorphic to a tuple ofsignatures, which is then again a signature, which represents the orthogonal decomposi-tion and collapse of the original signature into n orthogonal subspaces. Such an objectcan be constructed as the quotient σ/Ker(f). We represent it by a vector (1, n) ofsignatures, each component then corresponds to the image of the ith projector, that isthe elements stable under its action.

The result can be generalized in the case of an environmental bi-co-module, whoseclassical elements are then a matrix of signatures.

13

5.5 Simultaneous interaction

Given two co-modules σ and τ , interacting with the same environment [n], we wouldlike to define the subspace σ ⊗[n] τ of σ ⊗ τ of states that are compatible with [n].

fH

HE H ′

ψ

= fH′

H ′E

ψ

H

(35)

Moreover, given two bi-co-modules H : m→ n and H ′ : n→ p, one would like to beable to make the product of them : m→ p, that is H ⊗nH ′ endowed with bi-co-moduleinteraction maps. Writing both bi-co-modules as matrices m ∗ n and n ∗ p of signatures,the bi-co-module H ⊗n H ′ turns then out to be the matrix-product of the both, usingthe product and the sum of signatures previously defined.

For example, a simple case: given a bi-co-module [1] → [n] of n Hilbert spaces(H1, ...,Hn), and one [n]→ [1] of also n Hilbert spaces (H ′1, ...H

′n), then the product of

the both is be a tuple of Hilbert spaces Vσ = 〈H1 ⊗H ′1, H2 ⊗H ′2, ...,Hn ⊗H ′n〉.

5.6 Co-module homomorphism

One might be interested , given an environment and an interaction map, in the physicalprocesses that let the classical data invariant. Graphically, denoting f our interactionmap, and Φ our process.

f

[n]

Ψ

Vσ

Vσ

=f

[n]

Vσ

Ψ

Vσ

(36)

Such a map Φ is called a co-module homomorphism. One can see that having this prop-erty (36) is equivalent to say that Pi(Ψ(h)) = Ψ(Pi(h)), for each Pi. So each subspace σiis stable under Ψ. Therefore, the data of n completely positive maps σi → σi perfectlydescribes a physical process acting on a co-module and which let invariant the classicaldata.

Generalizing, a bi-co-module homomorphism is perfectly described by a matrix ofn*m of completely positive maps σij → σij . The action of such a process to a matrix ofn*m signatures is component wise.

14

6 The 2-category 2Dens

6.1 Definition

Definition. The 2-cat 2Dens is defined by the following.

1. 0 object : classical data : [n].

2. 1-morphism Cn → Cm : quantum system : bi-co-module for the two environments.

3. 2-morphism : physical process : bi-co-module homomorphism.

Remark. Our 2-category highly remains on the category Dens⊕. But in this category,two different types of objects belong. The classical data, which are under the form [n],and the physical systems, which are given by the objects that belong in Dens (withoutthe additive completion), so simple objects, in the sense that they cannot be decomposedinto a sum of other ones. Interestingly, the ones of the form ui=1..nHi, where Hi isdifferent of C, represent systems which are correlated with a n-bit of classical data.

Remark. As emphasized in the introduction, this 2-cat looks really similar at the 2Hilbone. In fact, the main difference comes from the bi-product structure. In 2Hilb, a sumof Hilbert spaces is still an Hilbert space, whereas in our setting it is just a formal sumof Hilbert spaces, resulting in a tuple of them.

6.2 Graphical calculus

The areas represent the objects, lines between areas the 1-morphisms, and boxes the2-morphisms. Parallel vertical lines correspond to composition of 1-morphism, whilethe composition of 2-morphism is depicted by stacking them vertically. A 2-cell alwaysrepresents a 2-morphism, which has its input below and its output above.

f

A B

C

H

J K

Figure 1: This 2-cell represents the 2 morphism from H : A → B to K ◦ J : A → B,with J : A→ C and k : C → B

f

g

†7→f†

g†

(37)

15

By applying the dagger, each 2-cell can be turned upside-down, as shown in (37), in asimilar fashion to the morphisms in the graphical calculus for dagger compact categories.

6.3 Basic constructions

A witness for an object [n] is a pair of 1-morphisms [1]→ [n] and [n]→ [1], representingthe natural encodement of the classical data n into a quantum-system. In our category,it has to come equipped with some 2-cells satisfying the properties exhibited below inthis definition-proposition.

Proposition. Eeach object n has a unique witness up to unitary isomorphism, for whichthe 1cell part consists of the n-element matrices,

=

CC...C

=(C C · · · C

)(38)

and endowed with the 2-cells displayed below,

which satisfy the following topological properties :

= =

= =

= =

(39)

Here we make use of the symmetric monoidal 2-category structure to define the swapfunction.

6.4 Measurement

A measurement is represented by a 2-cell like this one, drawing out the creation of clas-sical data.

16

C C

Cn(40)

We define a measurement in a 2-categorical way.

Definition. A measurement is a 2-morphism ( 1σ−→ 1 )

m−→ ( (1Pσ−−→ n) ◦ (n

>σ−−→ 1) )such that :

1. A measurement is left unitary. That is, given classical data, to encode it into aphysical system and then measure it will result in the same classical data.

= (41)

2. The right output is a witness.

3. The following map has to satisfy the axioms (27) (26) of a complete environmentalinteraction map.

(42)

Definition. A measurement such that the left and right output are witnesses is called atotal measurement.

It is the case when the measurement is performed in a quantum space of dimensionn, and the environment associated is [n].

7 Some constructions and comparison with 2Hilb

In order to understand this section, the reader has to know that all the propertiesdescribed until now in 2Dens hold as well in 2Hilb, except the measurement, thatturns out to be defined as a 2-cell like (40) but which is unitary. We will see how thisproperty turns out to have a physical and experimental meaning.

7.1 Complementary structures in FdHilb

Given a Hilbert space H endowed with a classical structure, associated to an orthonormalbasis (ei), a normalized element h is said to be unbiased if,

|〈h, ei〉| =1√n

(43)

where n is the dimension of H.

17

Definition. [CD11] Two classical structures Ξ and Ξ′ over the same object H are saidto be complementary if whenever a state is classical for one, it is unbiased for the other.

Let us remind that here classical means cloneable.

Remark. The reason we define complementary structures is highlighted by the followingexperiment: imagine one measures the state of a qbit in the basis (|0〉, |1〉), then in thebasis (|+〉, |−〉). Whatever the result he got from his first measure, the second one seemsrandom for him.

Definition. [Vic12] A pair of total measurements indexed by the same classical infor-mation are physically complementary if they satisfy the following equation

=1√n

φ(44)

Where the green node depicts the measurement/encodement coming from one of theclassical structure, the red from the complementary other one.

7.2 Complementary structures in 2Dens

A good characterization of complementarity in WP(FdHilb) comes from [CD11], and isdepicted below. Such a property does not hold in FdHilb because of some global phasesproblems. However, as there is no global phase in Dens, this definition still holds inthis category.

= (45)

To each classical structure and environment structure, a total measurement can beassociated. Using this, and the following proposition coming from [CP10] :

Proposition.

' (46)

where the white square denotes a morphism which maps each basis to a complemen-tary one, we can show the following proposition.

18

Proposition. In 2Dens, we have :

Complementarity ⇐⇒ ' ⇐⇒ ' (47)

7.3 Quantum erasement and comparison between the two categories

This section is mainly inspired from [Vic12].The study of complementarity structures provides a good framework to distinguish

clearly the difference between 2Hilb and 2Dens. Indeed, one can remark that in Fd-Hilb in order to characterize the complementarity of two measurements, we have tooutput the results of the both measurements, whereas, in 2Dens, one classical output isenough. The origin of this difference appears clearly in the following protocol, proposedby Jamie Vicary in [Vic12].

We are given two complementary structures (Ξ and Ξ′), that is, two complementarymeasurements and encodes morphisms. Some classical data are given in input, we encodeit using Ξ, and measure the resulting system with Ξ′. In 2Dens, as depicted in (46), wethen have lost all the classical informations given in input. Nevertheless let us performsome more new operations. Using the 2-cell associated, we copy the classical data. Fromthis classical data, using Ξ again, we encode a physical system in the associated state.We then measure it using Ξ′ and output the measurement. We then have two classicaloutputs at the end.Using the graphical framework provides by 2Hilb and 2Dens we have :

' 1√n

φ

(48)

That is, as expected, the right classical data is uncorrelated with the input one.However, let us see what happen if we forget the left classical data, using rules coming

19

from topology and of the 2-category structure.

' ' ' Unitary measurement−−−−−−−−−−−−−−→' '

(49)which means : the classical data flow stays invariant under the action of this protocol.So, if we look at the left classical data, the right output is uncorrelated with the input.If we forget it, then right output, which is now the only output, is exactly what wasgiven in input.

However, if we are working in 2Dens, then we have

' (50)

Which means, there is no way we can forget about the measurement ; the action itdid is not reversible.

8 Future work

Although this 2-category extends in a interestingly way the initial category 2Hilb, itdoes not capture all the physics that could be done with mixed states. Indeed, whileworking with mixed states, some measurements (the POVM ones) can be done in sucha way that we don’t have the co-module property; while performing twice the samemeasurement, one cannot be sure to get the same result anymore. A possibly futurework would be then to look for the axioms that rule such a measurement, and thentrying to see if a 2-category can arise from them. An other possibility would be tointerest myself to higher dimensions. Indeed, some works tend to prove that the we canextend the CPM construction to infinite dimensional spaces [CH11] [Coe07a] and stillhaving nice properties, especially a graphical language.

Right now I’m working (as shown in the long report), on a construction of a 2Denscategory with unitary measurement, highlighting the difference of the construction be-tween 2Dens and 2Hilb as a difference coming from the co-unit, allowing us to translatethis work in a more general purpose.

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References

[AC04] Samson Abramsky and Bob Coecke. “A categorical semantics of quantumprotocols.” In: Proceedings of the 19th Annual IEEE Symposium on Logic inComputer Science (2004). Ed. by IEEE Computer Science Press, pp. 415–425.

[Bae97] John C Baez. “Higher-dimensional algebra II : 2-Hilbert spaces.” In: Advancesin Mathematics (1997), 127:125–189.

[BCP09] Eric Oliver Paquette Bob Coecke and Dusko Pavlovic. “Classical and quan-tum structuralism”. In: Semantical Techniques in Quantum Computation(2009). Ed. by Ian Mackie Simon Gay, pp. 29–69.

[BCV08] Dusko Pavlovic Bob Coecke and Jamie Vicary. “Commutative dagger Frobe-nius algebras in FdHilb are orthogonal bases.” In: Technical Report (2008).

[CD11] Bob Coecke and Ross Duncan. “Interacting quantum observables: categoricalalgebra and diagrammatics”. In: New Journal of Physic 13 (2011).

[CH11] Bob Coecke and Chris Heunen. “Pictures of complete positivity in arbitrarydimension”. In: Proceedings of QPL VIII, ENTCS (2011).

[Coe07a] Bob Coecke. “Complete positivity without positivity and without compact-ness.” In: (2007). Ed. by Oxford University Computing Laboratory TechnicalReport PRG-RR-07-05.

[Coe07b] Bob Coecke. “De-linearizing Linearity: Projective Quantum Axiomatics FromStrong Compact Closure”. In: Electronic Notes in Theoretical Computer Sci-ence 170 (2007). Ed. by ENTCS, pp. 49–72.

[Coe08] Bob Coecke. “Axiomatics description of mixed states from Selinger’s CPMconstruction”. In: Electronic Notes in Theoretical Computer Science (2008).Ed. by ENTCS, pp. 3–13.

[CP06] Bob Coecke and Dusko Pavlovic. “The Mathematics of Quantum Computa-tion and Technology”. In: (2006). Ed. by Taylor and Francis.

[CP10] Bob Coecke and Simon Perdrix. “Environment and classical channels in cat-egorical quantum mechanics”. In: Lecture Notes in Computer Science (2010).Ed. by Proceedings of the 19th EACSL Annual Conference on ComputerScience Logic number 6247.

[HBS96] Chirtopher A.Fuchs Richard Jozsa Howard Barnum Carlton M.Caves andBenjamin Schumacher. “Noncommuting states cannot be broadcast”. In: Phys-ical reviews letters 76 (1996), 28182821.

[HV08] Chris Heunen and Jamie Vicary. Lectures on categorical quantum mechanics.2008.

[JS91] Andre Joyal and Ross Street. “The geometry of tensor calculus I”. In: Ad-vances in Mathematics (1991), 88:55–112.

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[Lan] S.Mac Lane. Categories of the working mathematician. Ed. by Springer-Verlag.Chap. 8.

[Ost03] Victor Ostrik. “Module categories, weak Hopf algebras and modular invari-ants”. In: Transformation Groups (2003), pp. 177–206.

[Sel04] Peter Selinger. “Towards a quantum programming language”. In: Mathemat-ical Structures in Computer Science (2004).

[Sel07] Peter Selinger. “Dagger compact closed categories and completely positivemaps”. In: Proceedings of the 3rd International Workshop on Quantum Pro-gramming Languages (QPL 2005) (2007).

[Sel11] Peter Selinger. In: New Structures for Physics (2011). Ed. by Springer, pp. 289–355.

[Vic12] Jamie Vicary. “Higher quantum theory”. In: (2012). Ed. by arxiv.

A Controlled complementary measurement

A.1 Controlled operation

A controlled operation is a morphism which acts on a physical system, whose actiondepends on some classical data, which stay unvariant under its action.

Definition. Given a co-module H, and so an interaction map f : H → E⊗H, where Eis an environment, a controlled operation is a morphism φ : E ⊗H → E ⊗H such that

φ

E E H

HE

=φ

E E H

HE

(51)

We then have the following property:

Proposition. The protected maps f : E ⊗H → E ⊗H defined as above have the fol-lowing form:

f : |i〉 ⊗ |σ〉 7→ |i〉 ⊗ fi(|σ〉) (52)

where |σ〉 is a state of H, and fi : H → H is an indexed family of completely positivemaps on H.

A controlled operation is depicted in a 2-category graphical language like that.

22

On the right side, the line represents a quantum system, whereas the left wire is awitness for the classical data. As a witness is perfectly encoded by the classical data itrepresents, the controlled operation let it invariant.

We now define a controlled unitary operation, which is a controlled operation f, suchthat

f†

f

' (53)

Definition. A permutation for a classical structure is a morphism π : A → A, suchthat

π

=

π ππ

=

π π

(54)

It is moreover fixed-point free if for all ψ classical

ψ

π 6=

ψ

(55)

We now have all the definitions we need to define a controlled complementary mea-surement, which is a controlled measurement performed on some quantum system, de-pending on some classical data. Moreover, for two different classical bits given in input,the measurement is then performed in two complementary bases.

Definition. A controlled complementary measurement is a measurement such that wehave, for every π fixed-point free permutation :

m

π

m m

m m

= (56)

23

An other characterization of the complementary controlled measurement is the fol-lowing:

m m

m m

=1

n+

n− 1

n(57)

24