# Sémantique des Jeux Quantiques Quantum Game Semantics Marc de Visme

## Texte intégral

(1)

### Sémantique des Jeux Quantiques

Quantum Game Semantics

Marc de Visme

LIP, ENS Lyon

(2)

## I. Semantics of Programs

“Program”

Ingredients: Bread, Jam Tools: Toaster

1 With a bread knife, cut two slices of bread.

2 Put both slices in the toaster and turn on the toaster.

3 When toasted, take out the slices.

4 Put the jam of your choice on them.

What is the semantics of this “program”?

Are both “programs” equivalent?

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## I. Semantics of Programs

“Program”

Ingredients: Bread, Jam Tools: Toaster

1 With a bread knife, cut two slices of bread.

2 Put both slices in the toaster and turn on the toaster.

3 When toasted, take out the slices.

4 Put the jam of your choice on them.

What is the semantics of this “program”?

Are both “programs” equivalent?

(4)

## I. Semantics of Programs

“Program”

Ingredients: Bread, Jam Tools: Toaster

1 With a bread knife, cut two slices of bread.

2 Put both slices in the toaster and turn on the toaster.

3 When toasted, take out the slices.

4 Put the jam of your choice on them.

What is the semantics of this “program”?

Are both “programs” equivalent?

(5)

## I. Semantics of Programs

Another “Program”

Ingredients: Bread, Jam Tools: Toaster

1 With a bread knife, cut one slice of bread.

2 Put the slice in the toaster.

3 With a bread knife, cut one slice of bread.

4 Put the slice in the toaster.

5 Turn on the toaster.

6 When toasted, take out the slices.

7 Put the jam of your choice on them.

“Program”

Ingredients: Bread, Jam Tools: Toaster

1 With a bread knife, cut two slices of bread.

2 Put both slices in the toaster and turn on the toaster.

3 When toasted, take out the slices.

4 Put the jam of your choice on them.

What is the semantics of this “program”?

Are both “programs” equivalent?

(6)

### First Approach: Operational Semantics

Start 1 2 3 4

On the Table

The Toaster OFF OFF ON OFF OFF

(7)

//// //

(8)

_

ON

_ ""*

_

_

_

ON

_ ##+

_

_

(9)

### Different Semantics

Banana Semantics?

Program //

))

Relational Semantics

Operational Semantics

KS

Game Semantics

Execution

Focus

Operational Step by step.

Game Interactions.

Relational Inputs-Outputs.

Banana? Every program produces bananas?

→ Different semantics but same behaviour.

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### Different Semantics

Banana Semantics? Program //

))

oo Relational Semantics

Operational Semantics

KS

Game Semantics

Execution

Focus

Operational Step by step.

Game Interactions.

Relational Inputs-Outputs.

Banana? Every program produces bananas?

→ Different semantics but same behaviour.

(11)

### Different Semantics

Banana Semantics? Program //

))

oo Relational Semantics55

uu OO

Operational Semantics oo //

KS

Game Semantics

Execution

Focus

Operational Step by step.

Game Interactions.

Relational Inputs-Outputs.

Banana? Every program produces bananas?

→ Different semantics but same behaviour.

(12)

### Which Programs?

"I think I can safely say that nobody understands quantum mechanics."

—Richard Feynman Quantum Programming

Quantum data Quantum operations

→ Non locality (entanglement)

→ No cloning

Higher Order

Functions: n7→n+ 2

Functions over functions: f 7→f(f(2)) etc.

Quantum λ-calculus

Terms Description

λx.M M N

Functions Application

()

M;N

Skip Divergence

Sequence tt

ff

ifP thenM elseN

True False Conditional MN

letxy=M inN

Pairing Pair Destructor M::N

split

Lists List Destructor letrecf x =M inN Recursive Definition

new,meas, andU Quantum Primitives

=⇒ Quantum data and classical control flow

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### Which Programs?

"I think I can safely say that nobody understands quantum mechanics."

—Richard Feynman Quantum Programming

Quantum data Quantum operations

→ Non locality (entanglement)

→ No cloning

Higher Order

Functions: n7→n+ 2

Functions over functions: f 7→f(f(2)) etc.

Quantum λ-calculus

Terms Description

λx.M M N

Functions Application

()

M;N

Skip Divergence

Sequence tt

ff

ifP thenM elseN

True False Conditional MN

letxy =M inN

Pairing Pair Destructor M::N

split

Lists List Destructor letrecf x =M inN Recursive Definition

new,meas, andU Quantum Primitives

=⇒ Quantum data and classical control flow

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### Which Programs?

"I think I can safely say that nobody understands quantum mechanics."

—Richard Feynman Quantum Programming

Quantum data Quantum operations

→ Non locality (entanglement)

→ No cloning

Higher Order

Functions: n7→n+ 2

Functions over functions: f 7→f(f(2)) etc.

Quantum λ-calculus

Terms Description

λx.M M N

Functions Application ()

M;N

Skip Divergence

Sequence tt

ff

ifP thenM elseN

True False Conditional MN

letxy =M inN

Pairing Pair Destructor M::N

split

Lists List Destructor letrecf x =MinN Recursive Definition

new,meas, andU Quantum Primitives

=⇒ Quantum data and classical control flow

(15)

### Which Programs?

"I think I can safely say that nobody understands quantum mechanics."

—Richard Feynman Quantum Programming

Quantum data Quantum operations

→ Non locality (entanglement)

→ No cloning

Higher Order

Functions: n7→n+ 2

Functions over functions: f 7→f(f(2)) etc.

Quantum λ-calculus

Terms Description

λx.M M N

Functions Application ()

M;N

Skip Divergence

Sequence tt

ff

ifP thenM elseN

True False Conditional MN

letxy =M inN

Pairing Pair Destructor M::N

split

Lists List Destructor letrecf x =MinN Recursive Definition

new,meas, andU Quantum Primitives

=⇒ Quantum data and classical control flow

(16)

### Which Programs?

"I think I can safely say that nobody understands quantum mechanics."

—Richard Feynman Quantum Programming

Quantum data Quantum operations

→ Non locality (entanglement)

→ No cloning

Higher Order

Functions: n7→n+ 2

Functions over functions: f 7→f(f(2)) etc.

Quantum λ-calculus

Terms Description

λx.M M N

Functions Application ()

M;N

Skip Divergence

Sequence tt

ff

ifP thenM elseN

True False Conditional MN

letxy =M inN

Pairing Pair Destructor M::N

split

Lists List Destructor letrecf x =MinN Recursive Definition

new,meas, andU Quantum Primitives

=⇒ Quantum data and classical control flow

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### Contributions

Quantum λ-calculus

++

//Quantum Relational Semantics Pagani, Selinger, Valiron, 2014OO

Collapse

CBV Operational Semantics Selinger, Valiron, 2006 oo

FA //

ss

FA 33

Quantum Game Semantics

Clairambault, de Visme, Winskel, 2019

Full Abstraction (FA)

A semantics~−is fully abstract if

M =obsN ⇐⇒ ~M=~N

Pros of our game model

Bounded quantum annotation Dynamic model

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### Plan

1 Semantics of Programs

2 Classical Control Flow

3 Quantum Data

4 Contribution: Quantum Game Semantics

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## II. Classical Control Flow

(In English from now on) Rideau, Winskel, 2011

Concurrent game semantics based on event structures.

(20)

### Event Structures

Events (set)

Causality (partial order) Conflict (relation)

→ ConfigurationsC:

down-closed & conflict-free

Start of the program

Calling “print” on

“Hello World!”

”Hello World!”

is printed

Calling “+” on

“30” and “12”

Receiving 42 Receiving 10

(21)

### Event Structures

Events (set)

Causality (partial order)

Conflict (relation)

→ ConfigurationsC:

down-closed & conflict-free

Start of the program

_

Calling “print” on

“Hello World!”

,,2 ”Hello World!”

is printed

ss{/ Calling “+” on

“30” and “12”

_ \$\$,

Receiving 42 Receiving 10

(22)

### Event Structures

Events (set)

Causality (partial order)

Conflict (relation)

→ ConfigurationsC:

down-closed & conflict-free

Start of the program

_

(

Calling “print” on

“Hello World!”

,,2 ”Hello World!”

is printed

Calling “+” on

“30” and “12”

_ \$\$,

Receiving 42 Receiving 10

(23)

### Event Structures

Events (set)

Causality (partial order) Conflict (relation)

→ ConfigurationsC:

down-closed & conflict-free

Start of the program

_

(

Calling “print” on

“Hello World!”

,,2 ”Hello World!”

is printed

Calling “+” on

“30” and “12”

_ \$\$,

Receiving 42 Receiving 10

(24)

### Event Structures

Events (set)

Causality (partial order) Conflict (relation)

→ ConfigurationsC:

down-closed & conflict-free

Start of the program

_

(

Calling “print” on

“Hello World!”

,,2 ”Hello World!”

is printed

Calling “+” on

“30” and “12”

_ \$\$,

Receiving 42 Receiving 10

(25)

### Games as Event Structures with Polarities

Events (set)

Causality (partial order) Conflict (relation)

→ ConfigurationsC:

down-closed & conflict-free

Polarities

Player Program; Positive polarity.

Opponent Environment; Negativepolarity.

Game for bit

ff tt

Game for 1(bit λ

xx;

?

% \$\$, ff tt

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### Games as Event Structures with Polarities

Events (set)

Causality (partial order) Conflict (relation)

→ ConfigurationsC:

down-closed & conflict-free

Polarities

Player Program; Positive polarity.

Opponent Environment; Negativepolarity.

Game for bit

ff tt

Game for 1(bit λ

xx;

?

% \$\$, ff tt

(27)

### Games and Strategies

Type A 7→ Game~A

= An event structure

Term Γ`t :A

7→ Strategy ~t:~Γ→~A

= Strategy ~ton~Γk~A

= An event structure T

and a labelling map τ :T~Γk~A which ensures that T abides by the rules of the game.

Example: f :1( bit`f() :bit

λ

?

ff tt

ff tt

(

1 ( bit

) k

bit

λ tt}1

ff tt

?

!!*ff ''.tt

(28)

### Games and Strategies

Type A 7→ Game~A

= An event structure

Term Γ`t :A

7→ Strategy ~t:~Γ→~A

= Strategy ~ton~Γk~A

= An event structure T

and a labelling map τ :T~Γk~A which ensures that T abides by the rules of the game.

Example: f :1( bit`f() :bit

λ

?

ff tt

ff tt

(1 ( bit) k bit

λ tt}1

ff tt

?

!!*ff ''.tt

(29)

### Games and Strategies

Type A 7→ Game~A

= An event structure

Term Γ`t :A

7→ Strategy ~t:~Γ→~A

= Strategy ~ton~Γk~A

= An event structure T

and a labelling map τ :T~Γk~A which ensures that T abides by the rules of the game.

Example: f :1( bit`f() :bit

λ tt}1

?

!!*ff ''.

))/

tt

))/ ff tt

(1 ( bit) k bit

λ tt}1

ff tt

?

!!*ff ''.tt

(30)

### Games and Strategies

Type A 7→ Game~A

= An event structure

Term Γ`t :A

7→ Strategy ~t:~Γ→~A

= Strategy ~ton~Γk~A

= An event structure T and a labelling map τ :T~Γk~A which ensures thatT abides by the rules of the game.

Example: f :1( bit`f() :bit

λ tt}1

?

!!*ff ''.

))/

tt

))/

τ

ff tt

(1 ( bit) k bit

λ tt}1

ff tt

?

!!*ff ''.tt

(31)

### Composition of Strategies

b :bit`λ().b :1( bit

ff

''.

tt

''.

x

λ xx;

λ xx:

?

%

?

&

σ

ff tt

bit k 1 ( bit

f :1(bit`f() :bit

λ yy>

y

?

%ff \$\$,

''.

tt

&&-

τ

ff tt

(1 ( bit) k bit

σ:~bit→~1(bit

τ :~1(bit→~bit τσ:~bit→~bit

b :bit`(λ().b) () :bit

ff

&&- tt

&&-

z

τσ

ff tt

bit k bit

Configurations

∀z ∈ C(τ σ),∃!x ∈ C(σ),∃!y∈ C(τ),z =yx

(32)

### Composition of Strategies

b :bit`λ().b :1( bit

ff

''.

tt

''.

x

λ xx;

λ xx:

?

%

?

&

σ

ff tt

bit k 1 ( bit

f :1(bit`f() :bit

λ yy>

y

?

%ff \$\$,

''.

tt

&&-

τ

ff tt

(1 ( bit) k bit

σ:~bit→~1(bit

τ :~1(bit→~bit τσ:~bit→~bit

b :bit`(λ().b) () :bit

ff

&&- tt

&&-

z

τσ

ff tt

bit k bit

Configurations

∀z ∈ C(τ σ),∃!x ∈ C(σ),∃!y∈ C(τ),z =yx

(33)

### Composition of Strategies

b :bit`λ().b :1( bit

ff

''.

tt

''.

x

λ yy=

λ yy<

?

%

?

%

σ

ff tt

bit k 1 ( bit

f :1(bit`f() :bit

λ zz?

y

?

% \$\$,ff

&&- tt

&&-

τ

ff tt

(1 ( bit) k bit

σ:~bit→~1(bit

τ :~1(bit→~bit τσ:~bit→~bit

b :bit`(λ().b) () :bit

ff

&&- tt

&&- z

τσ

ff tt

bit k bit

Configurations

∀z ∈ C(τσ),∃!x ∈ C(σ),∃!y∈ C(τ),z =yx

(34)

## III. Quantum Data

Quantum Data and Operations

Mixed states and completely positive maps.

(35)

### Quantum Computation

A quantum bit Multiple quantum bits: qubit⊗n New: bitqubit

Measure: qubitbit Unitary: qubit⊗nqubit⊗n Superposition and entanglement

(36)

### Representation of Quantum States: (Sub)Density Matrices

A quantum-less quantum bit

### q 0

P(0) P(1)

p,q∈R≥0 with p+q ≤1.

a∈C with |a|2pq

1/2 0 0 1/2

!

is a fair coin.

1/2 0

0 0

! half false half divergence

Two quantum-less quantum bits 0

p q 0 0 0

0 0 0 0 0 0

0 0 r

s

P(00) P(01)P(10) P(11)

1/2 0 0 1/2

0 0 0 0

0 0 0 0

1/2 0 0 1/2

is a pair of entangled quantumbits

(37)

A quantumbit

P(0) P(1)

### a a

Superposition

p,q∈R≥0 with p+q ≤1.

a∈C with |a|2pq 1/2 i/2

−i/2 1/2

!

is a quantumcoin.

1/2 0

0 0

! half false half divergence

Two quantumbits

p q

r s

P(00) P(01)P(10) P(11)

a a

d d

Superposition

b x y c b y x c

& Entanglement

1/2 0 0 1/2

0 0 0 0

0 0 0 0

1/2 0 0 1/2

is a pair of entangled quantumbits

(38)

A quantumbit

P(0) P(1)

### a a

Superposition

p,q∈R≥0 with p+q ≤1.

a∈C with |a|2pq 1/2 i/2

−i/2 1/2

!

is a quantumcoin.

1/2 0

0 0

! half false half divergence

Two quantumbits

p q

r s

P(00) P(01)P(10) P(11)

a a

d d

Superposition

b x y c b y x c

& Entanglement

1/2 0 0 1/2

0 0 0 0

0 0 0 0

1/2 0 0 1/2

is a pair of entangled quantumbits

(39)

### Quantum States and Operations

Object

Quantum datatypes qubit

qubit⊗n

qubit⊗nqubit⊗m

Representation

Positive hermitian matrices Pos(C2)⊆Mat2(C) Pos(C2

n)⊆Mat2n(C)

f ∈ L(Mat2n(C),Mat2m(C)) Completely positive maps f ∈CPM(C2

n,C2

m)

(40)

### Quantum States and Operations

Object

Quantum datatypes qubit

qubit⊗n

qubit⊗nqubit⊗m

Representation

Positive hermitian matrices Pos(C2)⊆Mat2(C) Pos(C2

n)⊆Mat2n(C) f ∈ L(Mat2n(C),Mat2m(C)) Completely positive maps f ∈CPM(C2

n,C2

m)

(41)

### Examples

new0,new1∈CPM(C,C2)

meas0,meas1∈CPM(C2,C)

new0 :z 7→ z 0 0 0

!

meas0 : a b c d

! 7→a

new1 :z 7→ 0 0 0 z

!

meas1 : a b c d

! 7→d

U unitary of size 2n bUc ∈CPM(C2

n,C2

n)

bUc:M 7→UMU

(42)

### Examples

new0,new1∈CPM(C,C2) meas0,meas1∈CPM(C2,C) new0 :z 7→ z 0

0 0

!

meas0 : a b c d

! 7→a new1 :z 7→ 0 0

0 z

!

meas1 : a b c d

! 7→d

U unitary of size 2n bUc ∈CPM(C2

n,C2

n)

bUc:M 7→UMU

(43)

### Examples

new0,new1∈CPM(C,C2) meas0,meas1∈CPM(C2,C) new0 :z 7→ z 0

0 0

!

meas0 : a b c d

! 7→a new1 :z 7→ 0 0

0 z

!

meas1 : a b c d

! 7→d

U unitary of size 2n bUc ∈CPM(C2

n,C2

n)

bUc:M 7→UMU

(44)

### Examples

new0,new1∈CPM(C,C2) meas0,meas1∈CPM(C2,C) new0 :z 7→ z 0

0 0

!

meas0 : a b c d

! 7→a new1 :z 7→ 0 0

0 z

!

meas1 : a b c d

! 7→d

U unitary of size 2n bUc ∈CPM(C2

n,C2

n)

bUc:M 7→UMU

(45)

## IV. Quantum Game Semantics

(46)

### Overview of the Quantum Game Model

Term of the quantum λ-calculus

q :qubit,q0:qubit`t :qubit⊗(bit(qubit) t :=qλbbit.if b then q0 elseH q0

Associated strategy (classical strategy) (qb,qb)

&&-

(qb, λ)

uu~3 _

ff

) tt

)

qb qb

Associated strategy (CPM annotation)

(qb,qb) +3

(λ,qb)

qb

tt +3

ks ff

qb

(47)

### Overview of the Quantum Game Model

Term of the quantum λ-calculus

q :qubit,q0:qubit`t :qubit⊗(bit(qubit) t :=qλbbit.if b then q0 elseH q0 Associated strategy (classical strategy)

(qb,qb)

&&-

(qb, λ)

uu~3 _

ff

) tt

)

qb qb

Associated strategy (CPM annotation)

(qb,qb) +3

(λ,qb)

qb

tt +3

ks ff

qb

(48)

### Overview of the Quantum Game Model

Term of the quantum λ-calculus

q :qubit,q0:qubit`t :qubit⊗(bit(qubit) t :=qλbbit.if b then q0 elseH q0 Associated strategy (classical strategy)

(qb,qb)

&&-

(qb, λ)

uu~3 _

ff

) tt

)

qb qb

Associated strategy (CPM annotation)

(qb,qb) +3

(λ,qb)

qb

tt +3

ks ff

qb

(49)

### Quantum Games and Strategies

Type A7→ Game ~A

= An event structure

together with

for every eventa an Hilbert space HA(a)

Term Γ`t :A7→ Strategy ~Γ→~A

= An event structure T together with a labelling mapτ :T →(~Γk~A)

and a quantum valuationQτ on configuration Qτ :C(T)→CPM

Qτ(x)∈CPM(HΓ(x),HA(x))

Qτ({qb, λ, ?})∈CPM(C2,C⊗C)

q:qubit ` λ().Coin()⊗(H q) :1((bit⊗qubit)

qb

))/λ rrz,

?

))/

\$\$,

τ

(ff,qb) (tt,qb)

qubit k 1 ( (bitqubit)

qb

C2

λ

C

ss{/

?

C

((/

##+

(ff,qb)

C2

(tt,qb)

C2

(50)

### Quantum Games and Strategies

Type A7→ Game ~A

= An event structure together with

for every eventa an Hilbert space HA(a)

Term Γ`t :A7→ Strategy ~Γ→~A

= An event structure T together with a labelling mapτ :T →(~Γk~A)

and a quantum valuationQτ on configuration Qτ :C(T)→CPM

Qτ(x)∈CPM(HΓ(x),HA(x))

Qτ({qb, λ, ?})∈CPM(C2,C⊗C)

q:qubit ` λ().Coin()⊗(H q) :1((bit⊗qubit)

qb

))/λ rrz,

?

))/

\$\$,

τ

(ff,qb) (tt,qb)

qubit k 1 ( (bitqubit)

qbC2 λ

C

ss{/

?C

((/

##+

(ff,qb)

C2 (tt,qb)

C2

(51)

### Quantum Games and Strategies

Type A7→ Game ~A

= An event structure together with

for every eventa an Hilbert space HA(a)

Term Γ`t :A7→ Strategy ~Γ→~A

= An event structure T together with a labelling mapτ :T →(~Γk~A) and a quantum valuationQτ on configuration Qτ :C(T)→CPM

Qτ(x)∈CPM(HΓ(x),HA(x))

Qτ({qb, λ, ?})∈CPM(C2,C⊗C)

q:qubit ` λ().Coin()⊗(H q) :1((bit⊗qubit)

qb

))/λ rrz,

?

))/

\$\$,

τ

(ff,qb) (tt,qb)

qubit k 1 ( (bitqubit)

qbC2 λ

C

ss{/

?C

((/

##+

(ff,qb)

C2 (tt,qb)

C2

(52)

### Quantum Games and Strategies

Type A7→ Game ~A

= An event structure together with

for every eventa an Hilbert space HA(a)

Term Γ`t :A7→ Strategy ~Γ→~A

= An event structure T together with a labelling mapτ :T →(~Γk~A) and a quantum valuationQτ on configuration Qτ :C(T)→CPM

Qτ(x)∈CPM(HΓ(x),HA(x))

Qτ({qb, λ, ?})∈CPM(C2,C⊗C)

q:qubit ` λ().Coin()⊗(H q) :1((bit⊗qubit) qb

))/λ ssz-

?

))/

##+

τ

(ff,qb) (tt,qb)

qubit k 1 ( (bitqubit)

qbC2 λC

tt|0

?C

((/

##+

(ff,qb)

C2 (tt,qb)

C2

(53)

### The Quantum Valuation

For x∈ C(T),

Q(x)∈CPM(HΓ(x),HA(x)).

Q({qb, λ, ?}) CPM(C2,CC)

Q({qb, λ, ?,(ff,qb)}) CPM(C2,CCC2) Q({qb, λ, ?,(tt,qb)}) CPM(C2,CCC2)

q:qubit ` λ().Coin()⊗(H q) :1((bit⊗qubit) qb

))/λ ssz-

?

))/

##+

τ

(ff,qb) (tt,qb)

qubit k 1 ( (bitqubit)

qbC2 λC

tt|0

?C

((/

##+

(ff,qb)

C2 (tt,qb)

C2

(54)

### The Quantum Valuation

For x∈ C(T),

Q(x)∈CPM(HΓ(x),HA(x)).

Q({qb, λ, ?}) CPM(C2,CC) Q({qb, λ, ?,(ff,qb)}) CPM(C2,CCC2)

Q({qb, λ, ?,(tt,qb)}) CPM(C2,CCC2)

q:qubit ` λ().Coin()⊗(H q) :1((bit⊗qubit) qb

))/λ ssz-

?

))/

##+

τ

(ff,qb) (tt,qb)

qubit k 1 ( (bitqubit)

qbC2 λC

tt|0

?C

((/

""*

(ff,qb)

C2 (tt,qb)

C2

(55)

### The Quantum Valuation

For x∈ C(T),

Q(x)∈CPM(HΓ(x),HA(x)).

Q({qb, λ, ?}) CPM(C2,CC) Q({qb, λ, ?,(ff,qb)}) CPM(C2,CCC2) Q({qb, λ, ?,(tt,qb)}) CPM(C2,CCC2)

q:qubit ` λ().Coin()⊗(H q) :1((bit⊗qubit) qb

))/λ ssz-

?

))/

##+

τ

(ff,qb) (tt,qb)

qubit k 1 ( (bitqubit)

qbC2 λC

tt|0

?C

((/

""*

(ff,qb)

C2 (tt,qb)

C2

(56)

### The Quantum Valuation

For x∈ C(T),

Q(x)∈CPM(HΓ(x),HA(x)).

Q({qb, λ, ?})

=trC2

CPM(C2,C) Q({qb, λ, ?,(ff,qb)})

= 12bHc

CPM(C2,C2) Q({qb, λ, ?,(tt,qb)})

= 12bHc

CPM(C2,C2)

q:qubit ` λ().Coin()⊗(H q) :1((bit⊗qubit) qb

))/λ ssz-

?

))/

##+

τ

(ff,qb) (tt,qb)

qubit k 1 ( (bitqubit)

qbC2 λC

tt|0

?C

((/

##+

(ff,qb)

C2 (tt,qb)

C2

(57)

### The Quantum Valuation

For x∈ C(T),

Q(x)∈CPM(HΓ(x),HA(x)).

Q({qb, λ, ?}) =trC2 CPM(C2,C) Q({qb, λ, ?,(ff,qb)})

= 12bHc

CPM(C2,C2) Q({qb, λ, ?,(tt,qb)})

= 12bHc

CPM(C2,C2)

q:qubit ` λ().Coin()⊗(H q) :1((bit⊗qubit) qb

))/λ ssz-

?

))/

##+

τ

(ff,qb) (tt,qb)

qubit k 1 ( (bitqubit)

qbC2 λC

tt|0

?C

((/

##+

(ff,qb)

C2 (tt,qb)

C2

(58)

### The Quantum Valuation

For x∈ C(T),

Q(x)∈CPM(HΓ(x),HA(x)).

Q({qb, λ, ?}) =trC2 CPM(C2,C) Q({qb, λ, ?,(ff,qb)}) = 12bHc CPM(C2,C2) Q({qb, λ, ?,(tt,qb)}) = 12bHc CPM(C2,C2)

q:qubit ` λ().Coin()⊗(H q) :1((bit⊗qubit) qb

))/λ ssz-

?

))/

##+

τ

(ff,qb) (tt,qb)

qubit k 1 ( (bitqubit)

qbC2 λC

tt|0

?C

((/

""*

(ff,qb)

C2 (tt,qb)

C2

(59)

### Composition of Strategies

b :bit`λ().b :1( bit

ff

''.

tt

''.

x

λ zz@

λ zz?

? }\$

?

\$

σ

ff tt

bit k 1 ( bit

f :1(bit`f() :bit

λ zzA

y

?

}\$ ##+

ff

&&- tt

&&-

τ

ff tt

(1 ( bit) k bit

σ :~bit→~1(bit τ :~1(bit→~bit τσ:~bit→~bit b :bit`(λ().b) () :bit

ff

&&- tt

&&- yx

τσ

ff tt

bit k bit

Valuations

Qτσ(yx) :=Qτ(y)◦ Qσ(x)

Updating...

## Références

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