Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Characterising non-separability • ...which is why non-separable states are way more interesting! • But, how do we know we’ve found one? • i.e. that there do not exist states ψ 1 , ψ 2 such that: = ψ 1 ψ 2 ψ • Problem: Showing that something doesn’t exist can be hard. Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 22 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Characterising non-separability Solution: Replace a negative property with a postive one: 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 28th June 2016 Process Theories and Graphical Language 23 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Cup-states • By introducing some clever notation: φ := := ψ • Then these equations: φ φ = = ψ ψ • ...look like this: = = Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 24 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Yank the wire! = = Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 25 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality 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 is trivial. 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 28th June 2016 Process Theories and Graphical Language 26 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Transpose B A ∼ = =: f T ← → f f A B = f f i.e. ( f T ) T = f Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 27 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Tranpose = rotation A bit of a deformation: f f � allows some clever notation: := f f = = = = Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 28 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Tranpose = rotation Specialised to states: ψ := ψ Aleks Bob Aleks Bob = ψ ψ as soon as Aleks obtains ψ Bob’s system will be in state ψ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 29 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality State/effect correspondence ∼ states of system A effects for correlated system B = ψ ψ transpose But what about... ∼ states of system A effects for system A = ψ ψ adjoint Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 30 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Adjoints † �→ ψ ψ state ψ testing for ψ Extends from states/effects to all processes: B A † �→ f f A B ψ f = = φ = ⇒ φ f ψ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 31 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Normalised states and isometries • Adjoints increase expressiveness, for instance can say when ψ is normalised : ψ = ψ • U is an isometry : A U = B A U A • ...and unitary, self-adjoint, positive, etc. Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 32 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Conjugates If we: � � �→ �→ ...we get horizontal reflection.The conjugate : �→ := f f f Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 33 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality 4 kinds of box conjugate B B f f A A adjoint adjoint transpose A A f f B B conjugate Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 34 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum teleportation: take 1 Can we fill in ‘?’ to get this? Aleks Bob Aleks Bob ψ ? ? = ψ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 35 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum teleportation: take 1 Here’s a simple solution: Aleks Bob Aleks Bob ψ = ψ Problem: ‘cap’ can’t be performed deterministically Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 36 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum teleportation: take 1 Aleks Bob Aleks Bob U i error = ψ U i ψ Bob’s problem now! Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 37 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum teleportation: take 1 Solution: Bob fixes the error. U i fix = ψ U i error ψ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 38 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum teleportation: take 1 Aleks Bob Aleks Bob Aleks Bob U i U i U i U i ψ ψ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 39 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Hilbert space The starting point for quantum theory is the process theory of linear maps , which has: 1 systems: Hilbert spaces 2 processes: complex linear maps ...in particular, numbers are complex numbers . Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 41 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Hilbert space Looking at the ‘Born rule’ for linear maps , we have a problem: effect φ complex number � = probability! state ψ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 42 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Doubling Solution: multiply by the conjugate: φ φ φ � ψ ψ ψ Then, for normalised ψ, φ : φ φ 0 ≤ ≤ 1 ψ ψ (i.e. the ‘usual’ Born rule: � φ | ψ �� φ | ψ � = |� φ | ψ �| 2 ) Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 43 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Doubling New problem: We lost this: test π probability state ψ ...which was the basis of our interpretation for states, effects, and numbers. Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 44 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Doubling Solution: Make a new process theory with doubling ‘baked in’: φ φ � φ := := � ψ ψ ψ Then: test φ φ � := φ probability � := ψ state ψ ψ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 45 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Doubling The new process theory has doubled systems � H := H ⊗ H : := and processes: f = double := � f f f Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 46 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Doubling preserves diagrams � g l l g � = = ⇒ = � � � f h k f h k Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 47 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality ...but kills global phases (i.e. λ = e i α ) = λ λ = ⇒ λ = double = = � f f f f f λ λ f Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 48 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Discarding Doubling also lets us do something we couldn’t do before: throw stuff away! � ψ How? Like this: := Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 49 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Discarding For normalised ψ , the two copies annihilate: ψ = = = � ψ ψ ψ ψ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 50 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum maps Definition The process theory of quantum maps has as types (doubled) Hilbert spaces � H and as processes: . . . � � � f . . . Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 51 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Purification Theorem All quantum maps are of the form: � := = f f f f f for some linear map f . Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 52 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Purification Proof. Pretty much by construction: g � � g �→ � � f h � � f h then note that: ... := H 1 ⊗ . . . ⊗ � � � � � H n H 1 H 2 H n Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 53 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality 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 28th June 2016 Process Theories and Graphical Language 54 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Consequence: no cap effect ☹ Consequence: there is a unique causal effect, discarding: = e Hence ‘deterministic quantum teleportation’ must fail: Aleks Bob Aleks Bob Aleks ? ρ = ρ ρ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 55 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Consequence: no signalling ☺ Aleks Claire Bob Aleks Claire Bob Aleks Φ Ψ Φ Ψ Φ ρ ρ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 56 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Stinespring’s theorem ☺ Lemma Pure quantum maps � U are causal if and only if they are isometries. Proof. Unfold the causality equation: = U U and bend the wire: U = = = U U U Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 57 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Stinespring’s theorem ☺ Theorem (Stinespring) For any causal quantum map Φ , there exists an isometry � f such that: = Φ � f Proof. Purify Φ, then apply the lemma to � f . Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 58 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Double vs. single wires classical := � = := quantum Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 60 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Classical values := ‘providing classical value i ’ i i := ‘testing for classical value i ’ � j 1 if i = j = 0 if i � = j i ( ⇒ ONB) Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 61 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Classical states General state of a classical system: := � p i ← probability distributions p i i Hence: ← point distributions i Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 62 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Copy and delete Unlike quantum states, classical values can be copied : = i i i and deleted : = j Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 63 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Copy and delete These satisfy some equations you would expect: = = = = Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 64 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Other classical maps := � := � i i i i i i := � := � i i i i i i := � := � i i i i i i Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 65 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality ...satisfying lots of equations = = = = = = = ... When does it end??? Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 66 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Spiders All of these are special cases of spiders : n n ... ... i i i � := ... i ... i i i m m Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 67 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Spiders The only equation you need to remember is this one: ... ... ... ... = ... ... ... When spiders meet, they fuse together. Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 68 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Spider reasoning = For example: = = Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 69 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Spider reasoning ⇒ string diagram reasoning = = = Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 70 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality How do we recognise spiders? Suppose we have something that ‘behaves like’ a spider: Do we know it is one? Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 71 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Spiders = ‘diagrammatic ONBs’ Yes! n � � ... ← → i ... i m m , n Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 72 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Classical and quantum interaction Classical values can be encoded as quantum states, via doubling: :: �→ := i i i i This is our first classical-quantum map, encode . It’s a copy-spider in disguise: := = i i i i Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 73 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Measuring quantum states The adjoint of encode is measure : probability distribution quantum state ρ This represents measuring w.r.t. � � i i ...where probabilities come from the Born rule: i i P ( i | ρ ) := = ρ ρ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 74 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Classical-quantum maps Definition The process theory of cq-maps has as processes diagrams of quantum maps and encode/decode: . . . � � Φ . . . Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 75 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum processes Causality generalises to cq-maps: = Φ quantum processes := causal cq-maps Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 76 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Special case: classical processes Classical processes are quantum processes with no quantum inputs/outputs: := f Φ These correspond exactly to stochastic maps. Positivity comes from doubling, and normalisation from causality: = = f p Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 77 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Special case: quantum measurements A measurement is any quantum process from a quantum system to a classical one: ∼ = ← → POVMs Φ Special case: ONB-measurement := � unitary U Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 78 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Special case: controlled-operations A quantum process with a classical input is a controlled operation : Φ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 79 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Special case: controlled-operations A controlled isometry furthermore satisfies: � U = � U Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 80 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Special case: controlled-operations Suppose we can use a single � U to build a controlled isometry : � U ...and an ONB measurement: measurement unitary � U Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 81 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum teleportation: take 2 ...then teleportation is a snap! Aleks Bob Aleks Bob ? ? ? � U ρ ρ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 82 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum spiders Doubling a classical spider gives a quantum spider : ... � � ... ... := double = ... ... ... Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 83 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum spiders Since doubling preserves diagrams, these fuse when they meet: ... ... ... ... = ... ... ... Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 84 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Quantum meets classical Q: What happens if a quantum spider meets a classical spider, via measure or encode? ... ... ... ... A: Bastard spiders! ... ... ... ... ... ... = =: ... ... ... ... ... ... Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 85 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Bastard spider fusion ... ... ... ... = ... ... ... Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 86 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Phase states 0 α 1 Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 87 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Phase spiders ... ... := α ... α ... ... ... ... α ... = α + β ... β ... ... Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 88 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Example: phase gates phase gate := α - α β = = α + β α α Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 89 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Complementary bases 0 = = π 1 0 0 1 Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 90 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Complementary bases n � � ... ← → i ... i m m , n n � � ... ← → i ... i m m , n Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 91 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Complementarity = Interpretation: (encode in ) THEN (measure in ) = (no data flow) Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 92 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Consequence: Stern-Gerlach N S S N N blocked! S 0 0 2nd Z -measurement = X -measurement 1st Z -measurement Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 93 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Strong complementarity ... ... ... ... = = Interpretation: Mathematically: Fourier transform. Operationally: ??? Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 94 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Consequences • strong complementarity = ⇒ complementarity • ONB of forms a subgroup of phase states, e.g. � � � � = = ⊆ , 0 1 π α 0 α ∈ [0 , 2 π ) • GHZ/Mermin non-locality Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 95 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality The setup Y Y 1 1 Z Z P P Y Y Y Y Y 3 2 3 2 Z Z Z Z Z 1 ZZZ 2 ZYY 3 YZY 4 YYZ Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 97 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality A locally realistic model 1st system 2nd system 3rd system � �� � � �� � � �� � z A y A z B y B z C y C Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 98 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality A locally realistic model ZZZ ZYY YZY YYZ ZZZ z A z B z C z A y B y C y A z B y C y A y B z C z B i i i i i i i i i i i i i Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 99 / 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality A quantum model measurements α β γ GHZ state Z -measurement := Y -measurement := π 0 2 100 / Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality A quantum model ZZZ ZYY YZY YYZ π π π π π π 0 0 0 0 0 0 2 2 2 2 2 2 101 / Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Deriving the contradiction We prove the correlations from the quantum model are inconsistent with any locally realistic one, by computing: parity := ... 102 / Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Deriving the contradiction z A y A z B y B z C y C z A y i i i i i i i 103 / Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 106
Process theories and diagrams Quantum processes Radboud University Nijmegen Classical and quantum interaction Application: Non-locality Deriving the contradiction π π π π π π 0 0 0 0 0 0 0 0 2 2 2 2 2 2 104 / Aleks Kissinger 28th June 2016 Process Theories and Graphical Language 106
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