Sémantique des Jeux Quantiques Quantum Game Semantics Marc de Visme

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(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)

Second Approach: Relational Semantics

//// //

(8)

Third Approach: Game Semantics

?

_

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?

Linking the Semantics

→ Different semantics but same behaviour.

(10)

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?

Linking the Semantics

→ 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?

Linking the Semantics

→ 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

(13)

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

(14)

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

(17)

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

(18)

Plan

1 Semantics of Programs

2 Classical Control Flow

3 Quantum Data

4 Contribution: Quantum Game Semantics

(19)

Quantum Game Semantics

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

(26)

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 ~ton~Γk~A

= An event structure T

and a labelling map τ :T~Γk~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 ~ton~Γk~A

= An event structure T

and a labelling map τ :T~Γk~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 ~ton~Γk~A

= An event structure T

and a labelling map τ :T~Γk~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 ~ton~Γk~A

= An event structure T and a labelling map τ :T~Γk~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)

Quantum Game Semantics

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

0 p

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)

Representation of Quantum States: (Sub)Density Matrices

A quantumbit

p q

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)

Representation of Quantum States: (Sub)Density Matrices

A quantumbit

p q

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)

Quantum Game Semantics

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 →(~Γk~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 →(~Γk~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 →(~Γk~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 →(~Γk~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)

Figure

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Références

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