Bob Coecke- Kindergarten Quantum Mechanics

8/3/2019 Bob Coecke- Kindergarten Quantum Mechanics

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Kindergarten Quantum Mechanics

bOB cOECKE @ QTRF-3

Oxford University

se10.comlab.ox.ac.uk:8080/BobCoecke/Home en.html

(or Google Bob Coecke)


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THE CHALLENGE


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Why did discovering quantum teleportation take 60 year?

Claim: bad formalism since too low level cf.GOOD QM

von Neumann QM

HIGH-LEVEL language

low-level language

Wouldnt it be nice to have a such a good formalism,in which discovering teleportation would be trivial?

Claim: it exists! And Ill present it to you.

Isnt it absurdly abstract coming from you guys?

Claim: It could be taught in kindergarten!


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1. Analyse quantum compoundness. A notion ofquantum information-flow emerges.

Physical Traces. Abramsky & Coecke (2003) CTCS02; cs/0207057

The Logic of Entanglement. Coecke (2003) PRG-RR; quant-ph/0402014

Quantum Information-flow, Concretely, and Axiomatically. quant-ph/0506132

2. Axiomatize quantum compoundness. ... full quantum mechanics emerges! A Categorical Semantics of Quantum Protocols. Abramsky & Coecke

(2004) IEEE-LiCS04; quant-ph/0402130

Abstract Physical Traces. Abramsky & Coecke (2005) TAC05.

... & quantum logic ... & open systems/CPMs! De-linearizing Linearity I: Projective Quantum Axiomatics from SCC.

Coecke (2005) QPL05; quant-ph/0506134.

-CCCs and Completely Positive Maps. Selinger (2005) QPL05.


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NO DOGMAS nor TABOOS!

Do you want ...

states to be ontological or empirical?

vectorial, projective, POVM-/CPM-/open system-style?

hidden variables, quantum potential, contextuality,(non-)locality, Bayesianism, ... ?

The bulk of the developments ignores these choices, but,

they can be implemented formally since we both have

great axiomatic freedom

great expressiveness


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CATEGORY THEORY!

Audience: Seriously, you dont expect us to learn that?

Bob: No! Of course not!

Bob: We are gonna go far back in time, ...Bob: to the time you were all still at kindergarten, ...

Bob: Were gonna draw pictures!

The sheer magic of the kind of category theory we needhere is that it formally justifies its own formal absence.


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THE SOLE AXIOM

=


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Since

= = =

the axiom is equivalent to

=


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When setting

=: f =:

fff

we obtain

=f

g

=f

g

f

g

=

f

g


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COMPOSITIONALITY bis

=g

f h

h g fo o


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that is, approximately, as

Pf : A B

A

Bas

f

f

f

f

=


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14th-TELEPORTATION

=

since id id = id


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HILBERT SPACE QM

f : H1 H2 is a linear map

: C H cf. (1) H

s : C C cf. s(1) C

H := conjugate Hilbert space of H

f := linear adjoint of f

= | = | for :=

= |


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A key role is played by

H1

H2

H1

H2

i.e. bipartite states H1 H2 are representable bylinear functions f : H1 H2 and vice versa. Indeed

=

ij

mij |ij m11 m1n... . . . ...

mk1 mkn

f =

ij mij |ji|

e.g.

|00 + |11

id = |00| + |11|


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Adjointness implies

f | =

f

= | f

Unitarity means U1= U i.e.

U

= =

U

U

U

hence

U | U =

=

=

U

U

= |


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Hence the star operations

=:ff* and

=:ff

*

provide a decomposition of the adjoint:

f = (f) = (f)

In particular, for the Hilbert space model we have

() := transposition

() := complex conjugation


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From

=:ff

*

follows

ff =

f f

*

*

=

and hence

=g

f

f

g


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ALGEBRA BEHIND THE SCENE

Symmetric monoidal bifunctor : CC C and

-involution dual A A;

contravariant -involution adjoint fAB fBA ;

Units A : I A

A with A = A,A A;

A

I A A 1A (A A) A

A

1A

6

- A I

1A A- A (A A)

6


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ALGEBRA BEHIND THE SCENE

Symmetric monoidal bifunctor : CC C and

-involution dual A A;

contravariant -involution adjoint fAB fBA ;

Units A : I A

A with A = A,A A;

=


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NATURAL SCALAR MULTIPLES

Scalars satisfy

s t = I

- I Is t

- I I

- Is t =

and we define scalar multiplication as

s f := A

- A If s

- B I

- Bs t =

for which we can then prove

(s f) (t g) = (s t) (f g)

(s f) (t g) = (s t) (f g)

i.e. diamonds can move around freely in time and space


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NATURAL SCALAR MULTIPLES

f fs

s

f

s

= =

and similarly

B

A

B

A

=

i.e.

= A

- I A

- B I

- Bs t =


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NO-CLONING NO-DELETING

Cf. Dieks-Wooters-Zurek 1982 & Pati-Braunstein 2000

Obviously we do not want to be a categorical(co-)product since that would imply existence of

A

A A A Bp

A

i.e. there are no logical rules

A A A A B A


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Proof


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Proof.

1 s := (f) f and t := (g) f

f

f

s=: g

f

t=:

2 f f = g g

gf= f g

Proof


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Proof.

3 s f = t g with s/t := (f /g) f

g

f

=

f

gff

Proof


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Proof.

4 s s = t t with s/t := (f /g) f

g

f

=

f

f

f

f

g

f


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OPEN SYSTEMS AND CPMs


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OPEN SYSTEMS AND CPMs

f

B

A

f

B

A

C

C

C*

projective process with hidden ancila

= open process on open system


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ABSTRACT QM


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ABSTRACT QM

System of type A := Object AComposite of A and B := Tensor A B

Process of type A B := Morphism f : A B

State of A := Element : I A

Evolution of A := Unitary U : A A

Measurement on A := Projectors {Pi : A A}i

Data := {i}i Dynamics := P

Probability := P = Tr(P ) : I I


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DIGEST


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DIGEST

... first full formal descrition of protocols

... types reflect kinds

... classical data-flow is included

... quantum info-flow is explicit

... kindergarten description/correctness proofs

... space for formal/conceptual choices

... the thing people call QM-relationalism?

APPLICATIONS


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APPLICATIONS

why computer scientists care about this stuff

Quantum programing language design

Quantum program logics for verification

Quantum protocol specification

Quantum protocol verfication

Appropriate semantics for new quantum computational

paradigms e.g. one-way (Briegel), teleportation based(Gottesman-Chuang), measurement based in general,topological quantum computing (Kitaev et al.) etc.


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Bob Coecke- Kindergarten Quantum Mechanics