Numbers • A number is a process with no inputs or outputs, written as: or just: λ λ Interpret as: what happens when a state meets an effect effect π possibility state ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 11 / 49
Numbers • A number is a process with no inputs or outputs, written as: or just: λ λ Interpret as: what happens when a state meets an effect effect π number state ψ This is called the (generalised) Born rule Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 11 / 49
Numbers • A number is a process with no inputs or outputs, written as: or just: λ λ Interpret as: what happens when a state meets an effect effect π number state ψ This is called the (generalised) Born rule • From properties of diagrams, we get: µ µ · := 1 := λ λ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 11 / 49
Numbers • A number is a process with no inputs or outputs, written as: or just: λ λ Interpret as: what happens when a state meets an effect effect π number state ψ This is called the (generalised) Born rule • From properties of diagrams, we get: µ µ · := 1 := λ λ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 11 / 49
Process theories in general Q: What kinds of behaviour can we study using just diagrams, and nothing else? Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 12 / 49
Process theories in general Q: What kinds of behaviour can we study using just diagrams, and nothing else? A: (Non-)separability Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 12 / 49
Separability for states • Separable: = ψ 1 ψ 2 ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 13 / 49
Separability for states • Separable: = ψ 1 ψ 2 ψ • vs. ‘completely non-separable’: Definition A state ψ is called cup-state if there exists an effect φ , called a cap-effect , such that: φ φ = = ψ ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 13 / 49
Cup-states • By introducing some clever notation: φ := := ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 14 / 49
Cup-states • By introducing some clever notation: φ := := ψ • Then these equations: φ φ = = ψ ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 14 / 49
Cup-states • By introducing some clever notation: φ := := ψ • Then these equations: φ φ = = ψ ψ • ...look like this: = = Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 14 / 49
Yank the wire! = = Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 15 / 49
Yank the wire! = = Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 15 / 49
A no-go theorem for separability Theorem If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics. Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 16 / 49
A no-go theorem for separability Theorem If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics. Proof. Suppose a cup-state separates: = ψ 1 ψ 2 Then for any f : Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 16 / 49
A no-go theorem for separability Theorem If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics. Proof. Suppose a cup-state separates: = ψ 1 ψ 2 Then for any f : f =
A no-go theorem for separability Theorem If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics. Proof. Suppose a cup-state separates: = ψ 1 ψ 2 Then for any f : f f = =
A no-go theorem for separability Theorem If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics. Proof. Suppose a cup-state separates: = ψ 1 ψ 2 Then for any f : f f f = = = ψ 1 ψ 2
A no-go theorem for separability Theorem If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics. Proof. Suppose a cup-state separates: = ψ 1 ψ 2 Then for any f : f f f f ψ 2 = = = =: ψ 1 ψ 2 ψ 1
A no-go theorem for separability Theorem If a process theory (i) has cup-states for every type and (ii) every state separates, then it has trivial dynamics. Proof. Suppose a cup-state separates: = ψ 1 ψ 2 Then for any f : f φ f f f ψ 2 = = = =: ψ 1 ψ 2 π ψ 1 Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 16 / 49
Transpose B A = ∼ =: f T ← → f f A B Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 17 / 49
Transpose B A = ∼ =: f T ← → f f A B = f f i.e. ( f T ) T = f Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 17 / 49
Tranpose = rotation A bit of a deformation: f f � Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 18 / 49
Tranpose = rotation A bit of a deformation: f f � allows some clever notation: := f f Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 18 / 49
Tranpose = rotation A bit of a deformation: f f � allows some clever notation: := f f = = = = Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 18 / 49
Transpose = rotation Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 19 / 49
Adjoint = reflection Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 20 / 49
Adjoint = reflection † �→ ψ ψ state ψ testing for ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 20 / 49
Adjoint = reflection † �→ ψ ψ state ψ testing for ψ Extends from states/effects to all processes: B A † �→ f f A B Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 20 / 49
4 kinds of box conjugate B B f f A A adjoint adjoint transpose A A f f B B conjugate Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 21 / 49
Doubling Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 22 / 49
Doubling If the ‘numbers’ of our process theory are complex numbers (e.g. as in linear maps ), then we have a problem: Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 22 / 49
Doubling If the ‘numbers’ of our process theory are complex numbers (e.g. as in linear maps ), then we have a problem: effect φ complex number � = probability! ψ state Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 22 / 49
Doubling Solution: multiply by the conjugate: φ φ φ � ψ ψ ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 23 / 49
Doubling Solution: multiply by the conjugate: φ φ φ � ψ ψ ψ (i.e. use the ‘plain old’ Born rule: � φ | ψ �� φ | ψ � = |� φ | ψ �| 2 ) Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 23 / 49
Doubling New problem: We lost this: effect π probability ψ state Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 24 / 49
Doubling New problem: We lost this: effect π probability ψ state ...which was the basis of our interpretation for states, effects, and numbers. Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 24 / 49
Doubling Solution: Make a new process theory with doubling ‘baked in’: Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 25 / 49
Doubling Solution: Make a new process theory with doubling ‘baked in’: φ φ � φ := := � ψ ψ ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 25 / 49
Doubling Solution: Make a new process theory with doubling ‘baked in’: φ φ � φ := := � ψ ψ ψ Then: effect φ φ � := φ probability � := ψ state ψ ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 25 / 49
Doubling The new process theory has doubled systems � H := H ⊗ H : := Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 26 / 49
Doubling The new process theory has doubled systems � H := H ⊗ H : := and processes: f = double := � f f f Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 26 / 49
Doubling preserves diagrams � l g l � g = = ⇒ = � � � f h k f h k Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 27 / 49
...but kills global phases (i.e. λ = e i α ) = 1 λ λ = ⇒ λ = double = = � f f f f f λ f λ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 28 / 49
Discarding Doubling also lets us do something we couldn’t do before: Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 29 / 49
Discarding Doubling also lets us do something we couldn’t do before: throw stuff away! � ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 29 / 49
Discarding Doubling also lets us do something we couldn’t do before: throw stuff away! � ψ How? Like this: := Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 29 / 49
Discarding For normalised ψ , the two copies annihilate: ψ = = = � ψ ψ ψ ψ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 30 / 49
Quantum maps Definition The process theory of quantum maps has as types (doubled) Hilbert spaces � H and as processes: . . . � � � f . . . Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 31 / 49
Causality A quantum map is called causal if: = Φ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 32 / 49
Causality A quantum map is called causal if: = Φ If we discard the output of a process, it doesn’t matter which process happened. Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 32 / 49
Causality A quantum map is called causal if: = Φ If we discard the output of a process, it doesn’t matter which process happened. causal ⇐ ⇒ deterministically physically realisable Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 32 / 49
Consequence: no signalling Aleks Claire Bob Φ Ψ ρ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 33 / 49
Consequence: no signalling Aleks Claire Bob Φ Ψ ρ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 33 / 49
Consequence: no signalling Aleks Claire Bob Φ Ψ ρ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 33 / 49
Consequence: no signalling Aleks Claire Bob Ψ ρ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 33 / 49
Consequence: no signalling Aleks Claire Bob Ψ ρ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 33 / 49
Consequences of doubling + causality • Impossibility of deterministic teleporation: Aleks Bob Aleks Bob = ρ • Purification/Stinespring dilation Φ = � f • Quantum no-broadcasting theorem ρ = ⇒ = = = Φ ∆ ∆ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 34 / 49
Classical and quantum interaction Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 35 / 49
Classical and quantum interaction := quantum Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 35 / 49
Classical and quantum interaction classical := � = := quantum Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 35 / 49
Classical and quantum interaction encode := measure := Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 36 / 49
Quantum teleportation: take 2 Aleks Bob ? ? ρ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 37 / 49
Quantum teleportation: take 2 Aleks Bob ? � U ρ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 37 / 49
Quantum teleportation: take 2 Aleks Bob � U � U ρ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 37 / 49
Quantum teleportation: take 2 Aleks Bob � U � U ρ Aleks Kissinger Foundations 2016, LSE Quantum Picturalism 37 / 49
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