From semantics of computation to physics, and back Samson Abramsky - - PowerPoint PPT Presentation

From semantics of computation to physics, and back

Samson Abramsky

Department of Computer Science, University of Oxford

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 1 / 29

Adriaan van Wijngaarden

From his Wikipedia article: His education was in mechanical engineering, for which he received a degree from Delft University of Technology in 1939. He then studied for a doctorate in hydrodynamics, but then abandoned the area. He joined the Nationaal Luchtvaartlaboratorium in 1945 and went with a group to England the following year to learn about new technologies that had been developed there during World War II. Van Wijngaarden was intrigued by the new idea of automatic computing, and on 1 January 1947 he became the head of the Computing Department of the brand-new Mathematisch Centrum (MC) in Amsterdam.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 2 / 29

Some Themes

Even though he was trained as an engineer, van Wijngaarden would emphasize and promote the mathematical aspects of computing, first in numerical analysis, then in programming languages and finally in design principles of programming languages.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 3 / 29

Some Themes

Even though he was trained as an engineer, van Wijngaarden would emphasize and promote the mathematical aspects of computing, first in numerical analysis, then in programming languages and finally in design principles of programming languages. Current work in quantum information and computation draws on mathematical tools developed in the study of mathematical foundations of computation, and delivers exciting new challenges and possibilities for computer science.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 3 / 29

Some Themes

Even though he was trained as an engineer, van Wijngaarden would emphasize and promote the mathematical aspects of computing, first in numerical analysis, then in programming languages and finally in design principles of programming languages. Current work in quantum information and computation draws on mathematical tools developed in the study of mathematical foundations of computation, and delivers exciting new challenges and possibilities for computer science. New possibilities for quantum advantage sit at the very boundaries of paradox, reminiscent of the way that paradoxical combinators and fixpoints are turned into a basis for recursion, as a working tool for the programmer.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 3 / 29

From the Y Combinator to Y Combinator

Wikipedia disambiguation:

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 4 / 29

From the Y Combinator to Y Combinator

Wikipedia disambiguation: Y Combinator may refer to: Y combinator, one of the fixed-point combinators in untyped lambda calculus Y Combinator (company), a startup-company accelerator

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 4 / 29

From the Y Combinator to Y Combinator

Wikipedia disambiguation: Y Combinator may refer to: Y combinator, one of the fixed-point combinators in untyped lambda calculus Y Combinator (company), a startup-company accelerator Fast Company has called YC ”the world’s most powerful start-up incubator”. Fortune has called Y Combinator ”a spawning ground for emerging tech giants”.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 4 / 29

From the Y Combinator to Y Combinator

Wikipedia disambiguation: Y Combinator may refer to: Y combinator, one of the fixed-point combinators in untyped lambda calculus Y Combinator (company), a startup-company accelerator Fast Company has called YC ”the world’s most powerful start-up incubator”. Fortune has called Y Combinator ”a spawning ground for emerging tech giants”. Y Combinator was started by Paul Graham and colleagues, LISP gurus — LISP is based on the lambda calculus.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 4 / 29

From the Y Combinator to Y Combinator

Wikipedia disambiguation: Y Combinator may refer to: Y combinator, one of the fixed-point combinators in untyped lambda calculus Y Combinator (company), a startup-company accelerator Fast Company has called YC ”the world’s most powerful start-up incubator”. Fortune has called Y Combinator ”a spawning ground for emerging tech giants”. Y Combinator was started by Paul Graham and colleagues, LISP gurus — LISP is based on the lambda calculus. The Y combinator was introduced by Haskell Curry, in his analysis of Russell’s

• paradox. Self-application (reflexivity) plays a key rˆ
• le.

Y ≡ λF. (λx. F(xx))(λx. F(xx))

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 4 / 29

From the Y Combinator to Y Combinator

Wikipedia disambiguation: Y Combinator may refer to: Y combinator, one of the fixed-point combinators in untyped lambda calculus Y Combinator (company), a startup-company accelerator Fast Company has called YC ”the world’s most powerful start-up incubator”. Fortune has called Y Combinator ”a spawning ground for emerging tech giants”. Y Combinator was started by Paul Graham and colleagues, LISP gurus — LISP is based on the lambda calculus. The Y combinator was introduced by Haskell Curry, in his analysis of Russell’s

• paradox. Self-application (reflexivity) plays a key rˆ
• le.

Y ≡ λF. (λx. F(xx))(λx. F(xx)) For any term M, YM = M(YM).

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 4 / 29

From the Y Combinator to Y Combinator

Wikipedia disambiguation: Y Combinator may refer to: Y combinator, one of the fixed-point combinators in untyped lambda calculus Y Combinator (company), a startup-company accelerator Fast Company has called YC ”the world’s most powerful start-up incubator”. Fortune has called Y Combinator ”a spawning ground for emerging tech giants”. Y Combinator was started by Paul Graham and colleagues, LISP gurus — LISP is based on the lambda calculus. The Y combinator was introduced by Haskell Curry, in his analysis of Russell’s

• paradox. Self-application (reflexivity) plays a key rˆ
• le.

Y ≡ λF. (λx. F(xx))(λx. F(xx)) For any term M, YM = M(YM). The Haskell programming language is named after Curry.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 4 / 29

From paradox to technology

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 5 / 29

From paradox to technology

“Strange loops” (Hofstadter)

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 5 / 29

From paradox to technology

“Strange loops” (Hofstadter) “It’s not a bug, it’s a feature”.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 5 / 29

From paradox to technology

“Strange loops” (Hofstadter) “It’s not a bug, it’s a feature”. Resolution of the paradox — add new values (partial values).

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 5 / 29

From paradox to technology

“Strange loops” (Hofstadter) “It’s not a bug, it’s a feature”. Resolution of the paradox — add new values (partial values). Scott’s reflexive domain models of the untyped lambda calculus.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 5 / 29

Quantum Paradoxes and Quantum Technologies

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 6 / 29

Quantum Paradoxes and Quantum Technologies

We are witnessing the beginnings of quantum technologies for information processing:

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 6 / 29

Quantum Paradoxes and Quantum Technologies

We are witnessing the beginnings of quantum technologies for information processing: randomness certification and amplification

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 6 / 29

Quantum Paradoxes and Quantum Technologies

We are witnessing the beginnings of quantum technologies for information processing: randomness certification and amplification quantum key distribution and other security protocols (and post-quantum crypto)

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 6 / 29

Quantum Paradoxes and Quantum Technologies

We are witnessing the beginnings of quantum technologies for information processing: randomness certification and amplification quantum key distribution and other security protocols (and post-quantum crypto) simulation of quantum chemistry, machine learning, optimization may soon be in reach

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 6 / 29

Quantum Paradoxes and Quantum Technologies

We are witnessing the beginnings of quantum technologies for information processing: randomness certification and amplification quantum key distribution and other security protocols (and post-quantum crypto) simulation of quantum chemistry, machine learning, optimization may soon be in reach These remarkable developments are directly connected with ideas from quantum foundations, closely associated with paradoxes or quasi-paradoxes: Bell’s theorem, Kochen-Specker paradox, Hardy’s paradox, teleportation, pseudo-telepathy, non-locality, contextuality, . . .

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 6 / 29

Quantum Paradoxes and Quantum Technologies

We are witnessing the beginnings of quantum technologies for information processing: randomness certification and amplification quantum key distribution and other security protocols (and post-quantum crypto) simulation of quantum chemistry, machine learning, optimization may soon be in reach These remarkable developments are directly connected with ideas from quantum foundations, closely associated with paradoxes or quasi-paradoxes: Bell’s theorem, Kochen-Specker paradox, Hardy’s paradox, teleportation, pseudo-telepathy, non-locality, contextuality, . . . The borders of paradox are a fruitful place to be!

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 6 / 29

Alice-Bob games

Verifier Alice Bob

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 7 / 29

The XOR Game

Alice and Bob play a cooperative game against Verifier (or Nature!):

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 8 / 29

The XOR Game

Alice and Bob play a cooperative game against Verifier (or Nature!): Verifier chooses an input x ∈ {0, 1} for Alice, and similarly an input y for

• Bob. We assume the uniform distribution for Nature’s choices.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 8 / 29

The XOR Game

Alice and Bob play a cooperative game against Verifier (or Nature!): Verifier chooses an input x ∈ {0, 1} for Alice, and similarly an input y for

• Bob. We assume the uniform distribution for Nature’s choices.

Alice and Bob each have to choose an output, a ∈ {0, 1} for Alice, b ∈ {0, 1} for Bob, depending on their input. They are not allowed to communicate during the game.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 8 / 29

The XOR Game

Alice and Bob play a cooperative game against Verifier (or Nature!): Verifier chooses an input x ∈ {0, 1} for Alice, and similarly an input y for

• Bob. We assume the uniform distribution for Nature’s choices.

Alice and Bob each have to choose an output, a ∈ {0, 1} for Alice, b ∈ {0, 1} for Bob, depending on their input. They are not allowed to communicate during the game. The winning condition: a ⊕ b = x ∧ y.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 8 / 29

The XOR Game

Alice and Bob play a cooperative game against Verifier (or Nature!): Verifier chooses an input x ∈ {0, 1} for Alice, and similarly an input y for

• Bob. We assume the uniform distribution for Nature’s choices.

Alice and Bob each have to choose an output, a ∈ {0, 1} for Alice, b ∈ {0, 1} for Bob, depending on their input. They are not allowed to communicate during the game. The winning condition: a ⊕ b = x ∧ y. A table of conditional probabilities p(a, b|x, y) defines a probabilistic strategy for this game. The success probability for this strategy is: 1/4[p(a = b|x = 0, y = 0) + p(a = b|x = 0, y = 1) + p(a = b|x = 1, y = 0) +p(a = b|x = 1, y = 1)]

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 8 / 29

A Strategy for the Alice-Bob game

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 9 / 29

A Strategy for the Alice-Bob game

Example: The Bell Model A B (0, 0) (1, 0) (0, 1) (1, 1) 1/2 1/2 1 3/8 1/8 1/8 3/8 1 3/8 1/8 1/8 3/8 1 1 1/8 3/8 3/8 1/8

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 9 / 29

A Strategy for the Alice-Bob game

Example: The Bell Model A B (0, 0) (1, 0) (0, 1) (1, 1) 1/2 1/2 1 3/8 1/8 1/8 3/8 1 3/8 1/8 1/8 3/8 1 1 1/8 3/8 3/8 1/8 The entry in row 2 column 3 says: If the Verifier sends Alice a1 and Bob b2, then with probability 1/8, Alice outputs a 0 and Bob outputs a 1.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 9 / 29

A Strategy for the Alice-Bob game

Example: The Bell Model A B (0, 0) (1, 0) (0, 1) (1, 1) 1/2 1/2 1 3/8 1/8 1/8 3/8 1 3/8 1/8 1/8 3/8 1 1 1/8 3/8 3/8 1/8 The entry in row 2 column 3 says: If the Verifier sends Alice a1 and Bob b2, then with probability 1/8, Alice outputs a 0 and Bob outputs a 1. This gives a winning probability of 3.25

4

≈ 0.81.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 9 / 29

A Strategy for the Alice-Bob game

Example: The Bell Model The entry in row 2 column 3 says: If the Verifier sends Alice a1 and Bob b2, then with probability 1/8, Alice outputs a 0 and Bob outputs a 1. This gives a winning probability of 3.25

4

≈ 0.81. The optimal classical probability is 0.75!

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 9 / 29

A Strategy for the Alice-Bob game

Example: The Bell Model The entry in row 2 column 3 says: If the Verifier sends Alice a1 and Bob b2, then with probability 1/8, Alice outputs a 0 and Bob outputs a 1. This gives a winning probability of 3.25

4

≈ 0.81. The optimal classical probability is 0.75! The proof of this uses (and is essentially the same as) the use of Bell inequalities.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 9 / 29

A Strategy for the Alice-Bob game

Example: The Bell Model The entry in row 2 column 3 says: If the Verifier sends Alice a1 and Bob b2, then with probability 1/8, Alice outputs a 0 and Bob outputs a 1. This gives a winning probability of 3.25

4

≈ 0.81. The optimal classical probability is 0.75! The proof of this uses (and is essentially the same as) the use of Bell inequalities. The Bell table exceeds this bound. Since it is quantum realizable using an entangled pair of qubits, it shows that quantum resources yield a quantum advantage in an information-processing task.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 9 / 29

A Simple Observation

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 10 / 29

A Simple Observation

Suppose we have propositional formulas φ1, . . . , φN

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 10 / 29

A Simple Observation

Suppose we have propositional formulas φ1, . . . , φN Suppose further we can assign a probability pi = Prob(φi) to each φi.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 10 / 29

A Simple Observation

Suppose we have propositional formulas φ1, . . . , φN Suppose further we can assign a probability pi = Prob(φi) to each φi. (Story: perform experiment to test the variables in φi; pi is the relative frequency

• f the trials satisfying φi.)

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 10 / 29

A Simple Observation

Suppose we have propositional formulas φ1, . . . , φN Suppose further we can assign a probability pi = Prob(φi) to each φi. (Story: perform experiment to test the variables in φi; pi is the relative frequency

• f the trials satisfying φi.)

Suppose that these formulas are not simultaneously satisfiable. Then (e.g.)

N−1

• i=1

φi ⇒ ¬φN,

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 10 / 29

A Simple Observation

Suppose we have propositional formulas φ1, . . . , φN Suppose further we can assign a probability pi = Prob(φi) to each φi. (Story: perform experiment to test the variables in φi; pi is the relative frequency

• f the trials satisfying φi.)

Suppose that these formulas are not simultaneously satisfiable. Then (e.g.)

N−1

• i=1

φi ⇒ ¬φN,

• r equivalently

φN ⇒

N−1

• i=1

¬φi.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 10 / 29

A Simple Observation

Suppose we have propositional formulas φ1, . . . , φN Suppose further we can assign a probability pi = Prob(φi) to each φi. (Story: perform experiment to test the variables in φi; pi is the relative frequency

• f the trials satisfying φi.)

Suppose that these formulas are not simultaneously satisfiable. Then (e.g.)

N−1

• i=1

φi ⇒ ¬φN,

• r equivalently

φN ⇒

N−1

• i=1

¬φi. Using elementary probability theory, we can calculate: pN ≤ Prob(

N−1

• i=1

¬φi) ≤

N−1

• i=1

Prob(¬φi) =

N−1

• i=1

(1 − pi) = (N − 1) −

N−1

• i=1

pi.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 10 / 29

A Simple Observation

Suppose we have propositional formulas φ1, . . . , φN Suppose further we can assign a probability pi = Prob(φi) to each φi. (Story: perform experiment to test the variables in φi; pi is the relative frequency

• f the trials satisfying φi.)

Suppose that these formulas are not simultaneously satisfiable. Then (e.g.)

N−1

• i=1

φi ⇒ ¬φN,

• r equivalently

φN ⇒

N−1

• i=1

¬φi. Using elementary probability theory, we can calculate: pN ≤ Prob(

N−1

• i=1

¬φi) ≤

N−1

• i=1

Prob(¬φi) =

N−1

• i=1

(1 − pi) = (N − 1) −

N−1

• i=1

pi. Hence we obtain the inequality

N

• i=1

pi ≤ N − 1.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 10 / 29

Logical analysis of the Bell table

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 11 / 29

Logical analysis of the Bell table

(0, 0) (1, 0) (0, 1) (1, 1) (a1, b1) 1/2 1/2 (a1, b2) 3/8 1/8 1/8 3/8 (a2, b1) 3/8 1/8 1/8 3/8 (a2, b2) 1/8 3/8 3/8 1/8

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 11 / 29

Logical analysis of the Bell table

(0, 0) (1, 0) (0, 1) (1, 1) (a1, b1) 1/2 1/2 (a1, b2) 3/8 1/8 1/8 3/8 (a2, b1) 3/8 1/8 1/8 3/8 (a2, b2) 1/8 3/8 3/8 1/8 If we read 0 as true and 1 as false, the highlighted entries in each row of the table are represented by the following propositions: ϕ1 = (a1 ∧ b1) ∨ (¬a1 ∧ ¬b1) = a1 ↔ b1 ϕ2 = (a1 ∧ b2) ∨ (¬a1 ∧ ¬b2) = a1 ↔ b2 ϕ3 = (a2 ∧ b1) ∨ (¬a2 ∧ ¬b1) = a2 ↔ b1 ϕ4 = (¬a2 ∧ b2) ∨ (a2 ∧ ¬b2) = a2 ⊕ b2.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 11 / 29

Logical analysis of the Bell table

(0, 0) (1, 0) (0, 1) (1, 1) (a1, b1) 1/2 1/2 (a1, b2) 3/8 1/8 1/8 3/8 (a2, b1) 3/8 1/8 1/8 3/8 (a2, b2) 1/8 3/8 3/8 1/8 If we read 0 as true and 1 as false, the highlighted entries in each row of the table are represented by the following propositions: ϕ1 = (a1 ∧ b1) ∨ (¬a1 ∧ ¬b1) = a1 ↔ b1 ϕ2 = (a1 ∧ b2) ∨ (¬a1 ∧ ¬b2) = a1 ↔ b2 ϕ3 = (a2 ∧ b1) ∨ (¬a2 ∧ ¬b1) = a2 ↔ b1 ϕ4 = (¬a2 ∧ b2) ∨ (a2 ∧ ¬b2) = a2 ⊕ b2. These propositions are easily seen to be contradictory.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 11 / 29

Logical analysis of the Bell table

(0, 0) (1, 0) (0, 1) (1, 1) (a1, b1) 1/2 1/2 (a1, b2) 3/8 1/8 1/8 3/8 (a2, b1) 3/8 1/8 1/8 3/8 (a2, b2) 1/8 3/8 3/8 1/8 If we read 0 as true and 1 as false, the highlighted entries in each row of the table are represented by the following propositions: ϕ1 = (a1 ∧ b1) ∨ (¬a1 ∧ ¬b1) = a1 ↔ b1 ϕ2 = (a1 ∧ b2) ∨ (¬a1 ∧ ¬b2) = a1 ↔ b2 ϕ3 = (a2 ∧ b1) ∨ (¬a2 ∧ ¬b1) = a2 ↔ b1 ϕ4 = (¬a2 ∧ b2) ∨ (a2 ∧ ¬b2) = a2 ⊕ b2. These propositions are easily seen to be contradictory. The violation of the logical Bell inequality is 1/4.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 11 / 29

Science Fiction? – The News from Delft

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 12 / 29

Science Fiction? – The News from Delft

First Loophole-free Bell test, 2015

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 12 / 29

Science Fiction? – The News from Delft

First Loophole-free Bell test, 2015

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 12 / 29

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 13 / 29

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 14 / 29

Timeline

1932 von Neumann’s Mathematical Foundations of Quantum Mechanics 1935 EPR Paradox, the Einstein-Bohr debate 1964 Bell’s Theorem 1982 First experimental test of EPR and Bell inequalities (Aspect, Grangier, Roger, Dalibard) 1984 Bennett-Brassard quantum key distribution protocol 1985 Deutch Quantum Computing paper 1993 Quantum teleportation (Bennett, Brassard, Cr´ epeau, Jozsa, Peres, Wooters) 1994 Shor’s algorithm 2015 First loophole-free Bell tests (Delft, NIST, Vienna)

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 15 / 29

Qubits: Spin Measurements

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 16 / 29

Qubits: Spin Measurements

States of the system can be described by complex unit vectors in C2. These can be visualized as points on the unit 2-sphere: |+ |− |+ |− |Ψ

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 16 / 29

Qubits: Spin Measurements

States of the system can be described by complex unit vectors in C2. These can be visualized as points on the unit 2-sphere: |+ |− |+ |− |Ψ Spin can be measured in any direction; so there are a continuum of possible

• measurements. There are two possible outcomes for each such measurement;

spin in the specified direction, or in the opposite direction. These two directions are represented by a pair of orthogonal vectors. They are represented on the sphere as a pair of antipodal points.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 16 / 29

Qubits: Spin Measurements

States of the system can be described by complex unit vectors in C2. These can be visualized as points on the unit 2-sphere: |+ |− |+ |− |Ψ Spin can be measured in any direction; so there are a continuum of possible

• measurements. There are two possible outcomes for each such measurement;

spin in the specified direction, or in the opposite direction. These two directions are represented by a pair of orthogonal vectors. They are represented on the sphere as a pair of antipodal points. Note the appearance of quantization here: there are not a continuum of possible

• utcomes for each measurement, but only two!

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 16 / 29

Quantum Entanglement

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 17 / 29

Quantum Entanglement

Bell state: |↑↑ + |↓↓

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 17 / 29

Quantum Entanglement

Bell state: |↑↑ + |↓↓ Compound systems are represented by tensor product: H1 ⊗ H2. Typical element:

• i

λi · φi ⊗ ψi Superposition encodes correlation.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 17 / 29

Quantum Entanglement

Bell state: |↑↑ + |↓↓ Compound systems are represented by tensor product: H1 ⊗ H2. Typical element:

• i

λi · φi ⊗ ψi Superposition encodes correlation. Einstein’s ‘spooky action at a distance’. Even if the particles are spatially separated, measuring one has an effect on the state of the other.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 17 / 29

Quantum Entanglement

Bell state: |↑↑ + |↓↓ Compound systems are represented by tensor product: H1 ⊗ H2. Typical element:

• i

λi · φi ⊗ ψi Superposition encodes correlation. Einstein’s ‘spooky action at a distance’. Even if the particles are spatially separated, measuring one has an effect on the state of the other. Entangled pairs of qubits provide quantum resources which can be used to gain quantum advantage in information processing tasks.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 17 / 29

The Mermin Magic Square

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 18 / 29

The Mermin Magic Square

A B C D E F G H I

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 18 / 29

The Mermin Magic Square

A B C D E F G H I The values we can observe for these variables are 0 or 1.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 18 / 29

The Mermin Magic Square

A B C D E F G H I The values we can observe for these variables are 0 or 1. We require that each row and the first two columns have even parity, and the final column has odd parity.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 18 / 29

The Mermin Magic Square

A B C D E F G H I The values we can observe for these variables are 0 or 1. We require that each row and the first two columns have even parity, and the final column has odd parity. This translates into 6 linear equations over Z2: A ⊕ B ⊕ C = 0 A ⊕ D ⊕ G = 0 D ⊕ E ⊕ F = 0 B ⊕ E ⊕ H = 0 G ⊕ H ⊕ I = 0 C ⊕ F ⊕ I = 1

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 18 / 29

The Mermin Magic Square

A B C D E F G H I The values we can observe for these variables are 0 or 1. We require that each row and the first two columns have even parity, and the final column has odd parity. This translates into 6 linear equations over Z2: A ⊕ B ⊕ C = 0 A ⊕ D ⊕ G = 0 D ⊕ E ⊕ F = 0 B ⊕ E ⊕ H = 0 G ⊕ H ⊕ I = 0 C ⊕ F ⊕ I = 1 Of course, the equations are not satisfiable in Z2!

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 18 / 29

Alice-Bob games for binary constraint systems

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 19 / 29

Alice-Bob games for binary constraint systems

Alice and Bob can share prior information, but cannot communicate once the game starts.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 19 / 29

Alice-Bob games for binary constraint systems

Alice and Bob can share prior information, but cannot communicate once the game starts. Verifier sends an equation to Alice, and a variable to Bob.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 19 / 29

Alice-Bob games for binary constraint systems

Alice and Bob can share prior information, but cannot communicate once the game starts. Verifier sends an equation to Alice, and a variable to Bob. They win if Alice returns a satisfying assignment for the equation, and Bob returns a value for the variable consistent with Alice’s assignment.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 19 / 29

Alice-Bob games for binary constraint systems

Alice and Bob can share prior information, but cannot communicate once the game starts. Verifier sends an equation to Alice, and a variable to Bob. They win if Alice returns a satisfying assignment for the equation, and Bob returns a value for the variable consistent with Alice’s assignment. A perfect strategy is one which wins with probability 1.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 19 / 29

Alice-Bob games for binary constraint systems

Alice and Bob can share prior information, but cannot communicate once the game starts. Verifier sends an equation to Alice, and a variable to Bob. They win if Alice returns a satisfying assignment for the equation, and Bob returns a value for the variable consistent with Alice’s assignment. A perfect strategy is one which wins with probability 1. Classically, A-B have a perfect strategy if and only if there is a satisfying assignment for the equations.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 19 / 29

Alice-Bob games for binary constraint systems

Alice and Bob can share prior information, but cannot communicate once the game starts. Verifier sends an equation to Alice, and a variable to Bob. They win if Alice returns a satisfying assignment for the equation, and Bob returns a value for the variable consistent with Alice’s assignment. A perfect strategy is one which wins with probability 1. Classically, A-B have a perfect strategy if and only if there is a satisfying assignment for the equations. Mermin’s construction shows that there is a quantum perfect strategy for the magic square.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 19 / 29

Recent results

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 20 / 29

Recent results

These games for general binary constraint systems studied by Cleve, Mittal, Liu and Slofstra.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 20 / 29

Recent results

These games for general binary constraint systems studied by Cleve, Mittal, Liu and Slofstra. They show that have a quantum perfect strategy is equivalent to a purely group-theoretic condition on a solution group which can be associated to each system of binary equations.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 20 / 29

Recent results

These games for general binary constraint systems studied by Cleve, Mittal, Liu and Slofstra. They show that have a quantum perfect strategy is equivalent to a purely group-theoretic condition on a solution group which can be associated to each system of binary equations. Major recent result by Slofstra:

Theorem

Every finitely presented group can be embedded in a solution group.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 20 / 29

Recent results

These games for general binary constraint systems studied by Cleve, Mittal, Liu and Slofstra. They show that have a quantum perfect strategy is equivalent to a purely group-theoretic condition on a solution group which can be associated to each system of binary equations. Major recent result by Slofstra:

Theorem

Every finitely presented group can be embedded in a solution group. Corollaries: There are finite systems of boolean equations which have quantum perfect strategies in infinite-dimensional Hilbert space, but not in any finite dimension. The question: Given a binary constraint system, does a quantum perfect strategy exist? is undecidable.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 20 / 29

Alice-Bob games for Graph Homomorphisms1

1Studied by Mancinska and Robertson, following Cameron, Montanaro, Newman, Severini and

Winter on the quantum chromatic number.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 21 / 29

Alice-Bob games for Graph Homomorphisms1

Given graphs G and H, does there exist a homomorphism G → H?

1Studied by Mancinska and Robertson, following Cameron, Montanaro, Newman, Severini and

Winter on the quantum chromatic number.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 21 / 29

Alice-Bob games for Graph Homomorphisms1

Given graphs G and H, does there exist a homomorphism G → H? Verifier sends a vertex of G to Alice, and a vertex to Bob. They output vertices of H.

1Studied by Mancinska and Robertson, following Cameron, Montanaro, Newman, Severini and

Winter on the quantum chromatic number.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 21 / 29

Alice-Bob games for Graph Homomorphisms1

Given graphs G and H, does there exist a homomorphism G → H? Verifier sends a vertex of G to Alice, and a vertex to Bob. They output vertices of H. They win if . . . ?

1Studied by Mancinska and Robertson, following Cameron, Montanaro, Newman, Severini and

Winter on the quantum chromatic number.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 21 / 29

Alice-Bob games for Graph Homomorphisms1

Given graphs G and H, does there exist a homomorphism G → H? Verifier sends a vertex of G to Alice, and a vertex to Bob. They output vertices of H. They win if . . . ? So we get a notion of “quantum graph homomorphism”. What does it mean?

1Studied by Mancinska and Robertson, following Cameron, Montanaro, Newman, Severini and

Winter on the quantum chromatic number.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 21 / 29

Alice-Bob games for Graph Homomorphisms1

Given graphs G and H, does there exist a homomorphism G → H? Verifier sends a vertex of G to Alice, and a vertex to Bob. They output vertices of H. They win if . . . ? So we get a notion of “quantum graph homomorphism”. What does it mean? What is the general underlying notion? How far can we generalize? Does it lead to a notion of “quantum mathematics”?

1Studied by Mancinska and Robertson, following Cameron, Montanaro, Newman, Severini and

Winter on the quantum chromatic number.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 21 / 29

Alice-Bob games for Graph Homomorphisms1

Given graphs G and H, does there exist a homomorphism G → H? Verifier sends a vertex of G to Alice, and a vertex to Bob. They output vertices of H. They win if . . . ? So we get a notion of “quantum graph homomorphism”. What does it mean? What is the general underlying notion? How far can we generalize? Does it lead to a notion of “quantum mathematics”? Are there connections to description in various kinds of logic? E.g. a kind of “quantum finite model theory”?

1Studied by Mancinska and Robertson, following Cameron, Montanaro, Newman, Severini and

Winter on the quantum chromatic number.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 21 / 29

Contextuality Analogy: Local Consistency

a a′ b b′

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 22 / 29

Contextuality Analogy: Local Consistency

a a′ b b′

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 22 / 29

Contextuality Analogy: Global Inconsistency

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 23 / 29

Strong Contextuality2

A B (0, 0) (1, 0) (0, 1) (1, 1) a1 b1 1 1 a1 b2 1 1 a2 b1 1 1 a2 b2 1 1 The PR Box: winning conditions for the XOR game!

2SA and A. Brandenburger, The Sheaf-theoretic structure of non-locality and contextuality

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 24 / 29

Bundle Pictures3

Strong Contextuality E.g. the PR box: 00 01 10 11 ab

• ×

×

• ab′
• ×

×

• a′b
• ×

×

• a′b′

×

• ×
• a
• b
• a′
• b′
• 1
• 1
• 1
• 3SA, R. Barbosa, K. Kishida, R. Lal, S. Mansfield, Contextuality, Cohomology and Paradox.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 25 / 29

Contextuality, Logic and Paradoxes

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 26 / 29

Contextuality, Logic and Paradoxes

Liar cycles. A Liar cycle of length N is a sequence of statements S1 : S2 is true, S2 : S3 is true, . . . SN−1 : SN is true, SN : S1 is false. For N = 1, this is the classic Liar sentence S : S is false.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 26 / 29

Contextuality, Logic and Paradoxes

Liar cycles. A Liar cycle of length N is a sequence of statements S1 : S2 is true, S2 : S3 is true, . . . SN−1 : SN is true, SN : S1 is false. For N = 1, this is the classic Liar sentence S : S is false. We can model the situation by boolean equations: x1 = x2, . . . , xn−1 = xn, xn = ¬x1

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 26 / 29

Contextuality, Logic and Paradoxes

Liar cycles. A Liar cycle of length N is a sequence of statements S1 : S2 is true, S2 : S3 is true, . . . SN−1 : SN is true, SN : S1 is false. For N = 1, this is the classic Liar sentence S : S is false. We can model the situation by boolean equations: x1 = x2, . . . , xn−1 = xn, xn = ¬x1 The “paradoxical” nature of the original statements is captured by the inconsistency of these equations.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 26 / 29

Contextuality in the Liar; Liar cycles in the PR Box

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 27 / 29

Contextuality in the Liar; Liar cycles in the PR Box

We can regard each of these equations as fibered over the set of variables which

• ccur in it:

{x1, x2} : x1 = x2 {x2, x3} : x2 = x3 . . . {xn−1, xn} : xn−1 = xn {xn, x1} : xn = ¬x1

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 27 / 29

Contextuality in the Liar; Liar cycles in the PR Box

We can regard each of these equations as fibered over the set of variables which

• ccur in it:

{x1, x2} : x1 = x2 {x2, x3} : x2 = x3 . . . {xn−1, xn} : xn−1 = xn {xn, x1} : xn = ¬x1 Any subset of up to n − 1 of these equations is consistent; while the whole set is inconsistent.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 27 / 29

Contextuality in the Liar; Liar cycles in the PR Box

We can regard each of these equations as fibered over the set of variables which

• ccur in it:

{x1, x2} : x1 = x2 {x2, x3} : x2 = x3 . . . {xn−1, xn} : xn−1 = xn {xn, x1} : xn = ¬x1 Any subset of up to n − 1 of these equations is consistent; while the whole set is inconsistent. Up to rearrangement, the Liar cycle of length 4 corresponds exactly to the PR box.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 27 / 29

Contextuality in the Liar; Liar cycles in the PR Box

We can regard each of these equations as fibered over the set of variables which

• ccur in it:

{x1, x2} : x1 = x2 {x2, x3} : x2 = x3 . . . {xn−1, xn} : xn−1 = xn {xn, x1} : xn = ¬x1 Any subset of up to n − 1 of these equations is consistent; while the whole set is inconsistent. Up to rearrangement, the Liar cycle of length 4 corresponds exactly to the PR box. The usual reasoning to derive a contradiction from the Liar cycle corresponds precisely to the attempt to find a univocal path in the bundle diagram.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 27 / 29

Paths to contradiction

• a1
• b1
• a2
• b2
• 1
• 1
• 1
• Samson Abramsky

(Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 28 / 29

Paths to contradiction

• a1
• b1
• a2
• b2
• 1
• 1
• 1
• Suppose that we try to set a2 to 1. Following the path on the right leads to the

following local propagation of values: a2 = 1 b1 = 1 a1 = 1 b2 = 1 a2 = 0 a2 = 0 b1 = 0 a1 = 0 b2 = 0 a2 = 1

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 28 / 29

Paths to contradiction

• a1
• b1
• a2
• b2
• 1
• 1
• 1
• Suppose that we try to set a2 to 1. Following the path on the right leads to the

following local propagation of values: a2 = 1 b1 = 1 a1 = 1 b2 = 1 a2 = 0 a2 = 0 b1 = 0 a1 = 0 b2 = 0 a2 = 1 We have discussed a specific case here, but the analysis can be generalised to a large class of examples.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 28 / 29

Envoi

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 29 / 29

Envoi

Contextuality in physics raises deep questions about the nature of reality. But it is also a new kind of resource, which yields new possibilities in information processing tasks. The challenge is to find methods to harness this resource, and understand its structure.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 29 / 29

Envoi

Contextuality in physics raises deep questions about the nature of reality. But it is also a new kind of resource, which yields new possibilities in information processing tasks. The challenge is to find methods to harness this resource, and understand its structure. By using these notions, we may come to understand them better. This may be the only way!

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 29 / 29

Envoi

Contextuality in physics raises deep questions about the nature of reality. But it is also a new kind of resource, which yields new possibilities in information processing tasks. The challenge is to find methods to harness this resource, and understand its structure. By using these notions, we may come to understand them better. This may be the only way! Under the rubric of ”local consistency, global inconsistency” contextuality is a pervasive notion, arising e.g. in constraint satisfaction, databases, distributed computation and elsewhere in classical computation.

Samson Abramsky (Department of Computer Science, University of Oxford) From semantics of computation to physics, and back 29 / 29