PROLOGUE

‘‘BEAM US UP SCOTTY!’’‘‘BEAM US UP SCOTTY!’’‘‘BEAM US UP SCOTTY!’’

•••

‘‘HOW DO I DO THAT?’’‘‘HOW DO I DO THAT?’’

•••

‘‘HERE’S THE CODE’’‘‘HERE’S THE CODE’’‘‘HERE’S THE CODE’’

1

The physicist’s description

Alice has an ‘unknown’ qubit |φ〉 and wants to send itto Bob. They have the ability to communicate classicalbits, and they share an entangled pair in the EPR-state, that is 1√

2(|00〉 + |11〉), which Alice produced

by first applying a Hadamard-gate 1√2

„1 11 −1

«to

the first qubit of a qubit pair in the ground state |00〉,and by then applying a CNOT-gate, that is

|00〉 #→ |00〉 |01〉 #→ |01〉 |10〉 #→ |11〉 |11〉 #→ |10〉,

then she sends the first qubit of the pair to Bob.

To teleport her qubit, Alice first performs a bipar-tite measurement on the unknown qubit and her halfof the entangled pair in the Bell-base, that isn

|0x〉+ (−1)z|1(1− x)〉˛̨̨

x, z ∈ {0, 1}o

,

where we denote the four possible outcomes of themeasurement by xz. Then she sends the 2-bit out-come xz to Bob using the classical channel.

Then, if x = 1, Bob performs the unitary oper-

ation σx =

„0 11 0

«on its half of the shared en-

tangled pair, and he also performs a unitary operation

σz =

„1 00 −1

«on it if z = 1. Now Bob’s half of

the initially entangled pair is in state |φ〉.

2

Ours

ρQ ; 1⊗ !1" ;α ; 〈#βi$〉i=4i=1 ⊗ 1 ; DIST ; ⊕i=4

i=1 β−1i

3

I.e.

Q

produce EPR-pair

Q⊗ (Q∗⊗Q)

(1⊗ !1") ◦ ρ

!

spatial relocation

(Q⊗Q∗)⊗Q

α

!

Bell-base measurement

(4 · I)⊗Q

〈#βi$〉i=4i=1 ⊗ 1

!

classical communication

4 · Q

DIST

!

unitary correction

4 · Q

⊕i=4i=1β−1

i

!

4

The physicist’s proof

In the case that the measurement outcome of the Bell-base measurement is xz, for

Pxz := 〈0x + (−1)z1(1− x)|−〉|0x + (−1)z1(1− x)〉we have to apply Pxz ⊗ id to the input state

|φ〉 ⊗ 1√2(|00〉+ |11〉) .

Setting |φ〉 := φ0|0〉+ φ1|1〉 we rewrite the input as

1√2(φ0|000〉+ φ0|011〉+ φ1|100〉+ φ1|111〉)

=1√2(φ0

Xx=0,1

|0xx〉+ φ1X

x=0,1

|1(1− x)(1− x)〉)

and application of Pxz ⊗ id then yields

1√2|0x + (−1)z1(1− x)〉 ⊗ (φ0|x〉+ (−1)zφ1|1− x〉).

There are four cases concerning the unitary correctionsUxz which have to be applied. For x = z = 0 the thirdqubit is φ0|0〉 + φ1|1〉 = |φ〉. If x = 0 and z = 1 it is

φ0|0〉 − φ1|1〉 which after applying σz =

„1 00 −1

«becomes |φ〉. If x = 1 it is φ0|1〉 + (−1)zφ1|0〉 which

after applying σx =

„0 11 0

«brings us back to the

previous two cases. Hence the result follows.

5

Ours

=

=β−1

i β−1i

βi

βi

1

6

Quantum formalism := vonNeumann [1932]

“I would like to make a confession which mayseem immoral: I do not believe absolutely inHilbert space no more.”

von Neumann [1935]

???von Neumann QM

! high-level languagelow-level language

Defects of von Neumann’s quantum formal-ism for Quantum Informatics:

• Types do not reflect kinds,

• Classical data-flow isn’t addressed,

• Quantum data-flow i.e. the featurewhich enables teleportation, is hidden.

7

OUTLINE

Qubits and quantum information flow.

B. Coecke. “The logic of entanglement”. OUCL-

PRG-RR-03-12 http://web.comlab.ox.ac.uk/oucl/

publications/tr/rr-03-12.html

From von Neumann quantum mechan-ics to high-level quantum mechanics.

S. Abramsky and B. Coecke. “A categorical seman-

tics of quantum protocols”. 19th IEEE Conference on

Logic in Computer Science (LiCS’04) www.arXiv.org/

quant-ph/0402130

On transdisciplinary opportunities andmethodological contributions of com-puter science to the other sciences.

8

A bit admits

• two values 0, 1,

• it is freely readable,

• it admits arbitrary transformations.

A qubit admits

• a sphere of values spanned by |0〉, |1〉,

|1〉

|0〉

|ψ〉|+〉

|−〉

• a measurement of it

– has two outcomes |−〉, |+〉,– changes the value |ψ〉,

• it admits unitary transformations i.e. op-posites and angles are preserved.

9

|ψ〉

|+〉

|−〉

θ−θ+

P+ : |ψ〉 &→ |+〉 P− : |ψ〉 &→ |−〉have chance prob(θ&) for & ∈ {+,−}. On thesphere Q we obtain partial constant maps

P& : Q → Q :: |ψ〉 &→ | & 〉.We know the value after a measurement, butnot what it was before it.

• Bad: measurements destroy data.

• Good: measurements act on data.

Quantum computing and quantum protocols:acrobatics between “the good” and “the bad”.

10

Q.M. in a nutshell

Systems are described by vectors (up to ascalar) of an inner-product space over C.

Compoundness is described by −⊗−.

Operations is described by unitaries.

Measurements “are described” by self-adjointoperators H : H→ H i.e.

H = a1 · P1 + . . . + a1 · Pn .

The measurement process constitutes:

• One Pi in (P1, . . . ,Pn) happens.

• The observer receives the token ai.

• The probability of this is |Pi(ψ)|2.

11

Quantum teleportation [Bennett et al. 1993]

|Ψ〉

{Px}x

Ux

|φ〉

|φ〉

Continuous data transmission through 2 bits?

So what causes the magic?

A pair of qubits isn’t described by a pair

|ψ1,ψ2〉 ∈ Q1 ×Q2

but by a function

|f : Q1→ Q2〉 ∈ Q1 ⊗Q2

12

Indeed, the tensor product H1⊗H2 of Hilbertspaces H1,H2 is the Hilbert space spanned by

|11〉 · · · |1n〉...

. . ....

|k1〉 · · · |kn〉hence,

∑ij

mij |ij〉 +←→ m11 · · · m1n

.... . .

...mk1 · · · mkn

+←→ | i 〉 &→

∑j

mij | j 〉

e.g. for k = n,

|11〉+ . . . + |nn〉 +←→ | i 〉 &→ | i 〉

Pairs |ψ1,ψ2〉 are a special case in H1⊗ H2

— reducing to |ij〉 in a well-chosen base.

13

The identity

|id : Q → Q 〉 ∈ Q⊗Q

is the Einstein-Podolsky-Rosen state.

A measurement of Q⊗Q has four outcomes

|f1〉, |f2〉, |f3〉, |f4〉cf.

|00〉, |01〉, |10〉, |11〉and corresponding

Pf : Q⊗Q → Q⊗Q :: |g〉 &→ | f 〉e.g.

Pid : Q⊗Q → Q⊗Q :: |g〉 &→ |id〉produces the EPR state.

Do Q⊗Q-functions compose (in some way)?

They do!

14

|φin〉

|φout〉?

f1

f2

f3

f4

φout =(f2 ; f1 ; f †3 ; f †

2 ; f4 ; f3)(φin)

15

f1

f2

f3

f4

f2 ; f1 ; f †3 ; f †

2 ; f4 ; f3

16

Permitted

Forbidden

17

id

id

id ; id = id

⇒ Teleportation

18

1 ≤ i ≤ 4

id

βi

β−1i

βi ; id; β−1i = id

19

f

βi

γi

βi ; f ; γi = f

⇒ Logic-gateteleportation

20

id

id id

id ; id; id = id

⇒ Entanglementswapping

21

f2

f1

φin

φout = (f2 ◦ f1)(φin)

((Pf1⊗ 1) ◦ (1⊗Pf2)

)(φin ⊗ . . .)

= . . .⊗(f2◦f1)(φin)

Hilbert space ! setlinear function ! relationtensor product ! product

PR := R×R

={((x1, y1), (x2, y2))

∣∣ x1Ry1, x2Ry2

}(sin, · , · )((PR1⊗1)◦(1⊗PR2)

)( · , · , sout)

⇒ sin(R2◦R1)sout

22

High-level quantum mechanics

A category C has objects A,B,C, . . . andfor each two objects A,B a set of morphismsC(A,B). It also has idA ∈ C(A,A) andcomposition g ◦ f when types match.

Why categories? Since they allow to deal ex-plicitly with processes e.g.

Logic ProgrammingPropositions I/O-Types

Proofs Programs

Variables (e.g. qubits Q) ∼ objects

Processes (e.g. measurements) ∼ morphisms

Compoundness (e.g. Q⊗Q) ∼ morphism sets

23

We also want an associative connective ⊗which encodes compound systems as objects.

A symmetric monoidal category is a cat-egory in which we can “plug things together”,things being both objects and morphisms:

A⊗B f1 ⊗ f2 : A1 ⊗A2 → B1 ⊗B2

There exist natural operations e.g. swapping,

σA,B : A⊗B → B ⊗A,

which satisfy obvious behavioral rules e.g.

A1 ⊗A2f1 ⊗ f2" B1 ⊗B2

A2 ⊗A1

σA1,A2

! f2 ⊗ f1" B2 ⊗B1

σB1,B2

!

24

There is some locality for components

A1 ⊗A2f1 ⊗ id" B1 ⊗A2

A1 ⊗B2

id⊗ f2

!

f1 ⊗ id" B1 ⊗B2

id⊗ f2

!

But we don’t assume a pair-like structure, i.e.

whole /= ∑components

This turns out to comprise the absence of

A∆−→ A⊗A A⊗B

π−→ A

25

No-cloning & No-deleting

Wooters-Zurek 1982; Pati-Braunstein 2000

Crucial for secure quantum cryptography.

Is this connected to resource sensitive lan-guages and recent models for concurrency andinteraction in distributed computation?

Yes

Categories will enable us to capture the quan-tum information flow (= geometry of the line)syntactically, and compare it to the resource-sensitive and concurrent CS semantics.

No-cloning & no-deleting refute existence of

A∆−→ A⊗A A⊗B

π−→ A

A 0 A ∧A A ∧B 0 A

What more do we need for quantum theory?

26

Dissection of Pf

id

id

f

f

Pf = f ⊗ id ; #id$ ; !id" ; id⊗ f

(cf. unraveling a program into commands)

27

The algebra of entanglement

1. a monoidal involution on objects,

A &→ A∗

2. for each involutive pair a morphism

!id" : I → A⊗A∗

3. a monoidal duality on morphisms,

f : A → B &→ f† : B → A

such that

id

id

=

28

=

=

=

29

Scalars:

• Unit 1 as idI.

• Integers as #idA$ ◦ !idA"

• Reals as s ◦ s†

• Probabilities [0, 1] as |〈φ | ψ〉|.contained in C(I, I) for I the ⊗-unit, and eachscalar induces “natural” scalar multiplication.

Unitarity:U† = U−1

Inner-product:

〈φ | ψ〉 := φ† ◦ ψ

for φ : I →A and ψ : I → B. We have:

〈U ◦ ψ | U ◦ φ〉 = 〈ψ | φ〉〈f† ◦ ψ | φ〉 = 〈ψ | f ◦ φ〉

30

Exampleobjects: sets

morphisms: relations

tensor: cartesian product

Additional data:

X∗ := X R† := Rc

η := {(∗, (x, x)) | x ∈ X} ⊆ {∗}× (X ×X)

Matrices in any involutive abelian semiringprovide a model generalizing relations, thatis, matrices in the Boolean semiring.

Other examples are inner-product spaces, freecompact closed categories generated by self-dual categories, n-cobordisms, . . .

31

What do we miss to have all ingredi-ents of von Neumann’s formalism?

An operation ⊕ on C which now stands for

“plugging histories together”

where history = picture = branch = world,and which has pair-like nature (= biproduct).

Disjunctive/probabilistic content:

Q⊕Q

Producing a “sphere” from two “germs”:

Q 4 I⊕ I

32

A qubit measurement becomes

〈P+,P−〉 : Q → Q⊕Q

whereP& = π†

& ◦ π& : Q → Q

with〈π†

+,π†−〉 : Q → I⊕ I

unitary.

Classical information flow is distributivity,

(I⊕ I)⊗QDIST" (I⊗Q)⊕ (I⊗Q) 4 Q⊕Q

(Q1 ⊕Q2)⊗QDIST" (Q1 ⊗Q)⊕ (Q2 ⊗Q)

33

Theorem.

Q ============== Q

produce EPR-pair

Q⊗ (Q∗⊗Q)

(id⊗ !id") ◦ ρ

!

spatial relocation

(Q⊗Q∗)⊗Q

+!

Bell-base measurement

(4 · I)⊗Q

〈#βi$〉i=4i=1 ⊗ id

!

classical communication

4 · Q

DIST

!

unitary correction

4 · Q

〈 id 〉i=4i=1

!=========== 4 · Q

⊕i=4i=1β−1

i

!

i.e.

ρQ ; id⊗!id" ; α ; 〈#βi$〉i=4i=1⊗id ; DIST ;⊕i=4

i=1β−1i = id

34

=

=β−1

i β−1i

βi

βi

1

35

Beyond von Neumann’s 1932 Q.M.

We increased resolution. We now see thequantum information flow. We extendedscope. We added classical data and informa-tion flow. Types reflect kinds, and thereare many more of them.

The formalism is high-level. We have acompact purely formal language and intuitivegeometrical proofs, but with a categorical log-ical and syntactic counterpart, namely linearlogic with dual self-dual connectives.

We emphasized essence. We abstractover non-crucial things and allowed new de-grees of axiomatic freedom, enabling rela-tional reasoning: sets, relations and thecartesian product satisfy the axiomatics.

A bonus. The probability rule can be de-rived in our formalism while in von Neumann’sformalism it is a postulate.

36

Direct applications

Intuitive analysis and design of computa-tional schemes and protocols, and general op-erational reasoning about quantum systems.

A step stone towards automated analysis,design and reasoning about quantum systems

E.g. measurement based quantum com-puting is both from soft- and hardware per-spective very promising, and relies completelyon the structure of entanglement.

In particular, qualifying and quantifying mul-tipartite entanglement and obtaining abetter understanding of its properties is thecurrent holy grail of quantum information.

37

A transdisciplinary perspective

What did we do so far? We definitely did

• high-level quantum informatics,

but, also

• physics in informatic perspective.

Why did it take 60 years and 6 persons for

• teleportation to be invented?

Why did it took 70 years to have

• a complete quantum formalism and logic

• which recognizes the compositional na-ture of entanglement,

and as such

• trivializes teleportation?

38

Since many of the tools were either

• not available (yet), or,

• not being considered (yet).

Now they do exist and were developed by

• computer scientists.

So why didn’t physicist invent them?

• Many physicists couldn’t care less.

• They looked in the wrong direction.

• They used inappropriate methodologies.

• They followed old-fashion paradigms.

39

Computer science has something to offer toother sciences — other than “the computer”!

• Logical and structural reasoning.

• The operational methodology.

• Fresh paradigms.

E.g.

• Interaction and concurrency.

• Open systems.

• Qualitative reasoning about information.

• Continuous vs. discrete.

• Hybrid systems.

40

EPILOGUE

id

βi

β−1i

41