Quantum Sets. Andre Kornell. University of California, Davis. SYCO September 21, 2018

Transcription

1 Quantum Sets Andre Kornell University of California, Davis SYCO September 21, 2018 Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

2 two categories Fun: sets and functions qfun: quantum sets and quantum functions Inc Pts Inc is full and faithful Inc Fun qfun Fun = Id Pts Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

3 quantum sets definition A quantum set X is a set of nonzero finite-dimensional Hilbert spaces. Inc(S) = S = { } C {s} s S Pts(X ) = {X X dim(x ) = 1} qfun has terminal object 1 = {C} = { }. Pts(X ) = qfun(1, X ) Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

4 + and Let X and Y be quantum sets. definition Cartesian product X Y = {X Y X X, Y Y} X Y is not the product of X and Y definition disjoint union X + Y = (X {1}) (Y {2}) X + Y is the coproduct of X and Y Inc(S + T ) = Inc(S) + Inc(T ) Pts(S + T ) = Pts(S) + Pts(T ) Inc(S T ) = Inc(S) Inc(T ) Pts(S T ) = Pts(S) Pts(T ) Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

5 quantum functions expository definition A quantum function F from a quantum set X to a quantum set Y assigns to each element X of X a unitary operator X = (H 1 Y 1 ) (H 2 Y 2 ) (H n Y n ), up to unitary equivalence of the coefficients H 1,..., H n. example: qubit measurement X = {C 2 } X S F S = { 1 2, 1 2 } C 2 = (C C { 1 2 } ) (C C { 1 2 } ) Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

6 composing quantum functions = X = i,j F G X Y Z X = H i Y i Y i = i j K j i Z j H i K j i Z j = j ( i H i K j i ) Z j Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

7 qfun is like a topos theorem (K) The symmetric monoidal category (qfun, ) 1 has finite colimits, 2 has finite limits, 3 has a terminal monoidal unit, 4 is closed monoidal, and 5 classifies subobjects by classical quantum functions into 1 + 1: Z 1! T X For a symmetric monoidal category (C, ) satisfying (1) (5): (C, ) is a topos is a category-theoretic product Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

8 compatible quantum functions we say that F 1 and F 2 are compatible just in case Z F 1 F 2 F Y 1 Y 1 1 Y 1 Y 2 1 Y 2 Y 2 id!! id P 1 P 2 definition A quantum function out of X is classical iff it is compatible with every quantum function out of X. A quantum set is classical iff I X is classical. Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

9 classical quantum sets and classical quantum functions proposition (K) A quantum set X is classical iff there is a set S such that X = S. proposition (K) A quantum function F : X Y is classical iff there is a function f : X Pts(Y) making the following diagram commute: X F Y Q J X Pts(Y) f X = Q X C {X } C {Y } = J C Y Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

10 von Neumann algebras proposition (K) There is a full and faithful contravariant functor l q : qfun vnalg. l q (X ) = X X { L(X ) l q (X ) = a l q (X ) sup X X } a(x ) < theorem (K) Let A be a von Neumann algebra. The following are equivalent: 1 A = l q (X ) for some quantum set X 2 every von Neumann subalgebra of A is atomic 3 if a = a, then there is an orthogonal family of projections (p α α R) a = α R α p α Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

11 internal ring of quantum complex numbers Write X Y for category theoretic product of X and Y. Write C = R R. There are quantum functions C C + C C C C C C C C such that the set qfun(x, C) has the structure of a -algebra over C. proposition (K) We have a natural isomorphism qfun(x, C) = l q (X ). Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

12 interlude: quantum relations definition (essentially, Weaver) A quantum relation R from a quantum set X to a quantum set Y assigns to each element X of X and each element Y of Y a subspace R(X, Y ) L(X, Y ) Quantum relations correspond to quantum functions X Y The category qrel of quantum sets and quantum relations is a dagger compact category enriched over ortholattices. Definition A quantum function from X to Y is a quantum relation such that R R I X R R I Y Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

13 the graph coloring game parameters: a finite simple graph G and a finite set S players: Alice and Bob, cooperating blindly against a Referee round 1: Referee plays a pair (g A, g B ) G G (Alice sees only g A, and Bob sees only g B ) round 2: Alice plays a color s A and Bob plays a color s B (Alice sees only s A and Bob sees only s B ) scoring: Alice and Bob lose iff (g A = g B and s A s B ) or (g A g B and s A = s B ) Alice and Bob have a winning strategy G can be properly colored by S true if Alice and Bob share classical randomness false if Alice and Bob share quantum randomness Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

14 from the graph coloring game to quantum functions G. Brassard, R. Cleve, and A. Tapp, Cost of Exactly Simulating Quantum Entaglement with Classical Communication, Phys. Rev. Lett. 83, no 9 (1999). V. Galliard and S. Wolf, Pseudo-telepathy, entanglement, and graph colorings, Proc. ISIT 2002 (2002). V. Galliard, A. Tapp, and S. Wolf, The impossibility of pseudotelepathy without quantum entaglement, Proc. ISIT 2003 (2004). P. J. Cameron, A. Montanaro, M. W. Newman, S. Severini, and A. Winter, On the quantum chromatic number of a graph, Electron. J. Combin. 14, no. 1 (2007). L. Mančinska and D. E. Roberson, Quantum homomorphisms, J. Combin. Theory, Series B 118 (2016). S. Abramsky, R. S. Barbosa, N. De Silva, and O. Zapata, The quantum Monad on Relational Structures, Proc. MFCS 2017 (2017). B. Musto, D. J. Reutter, and D. Verdon, A compositional approach to quantum functions, to appear in J. Math. Phys. (2017/2018). Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15

15 quantum families of graph colorings F G Z S F (E G I Z ) ( I S ) F proposition Alice and Bob have a winning strategy using quantum entanglement iff there is a quantum family of graph colorings of G by S. Andre Kornell (UC Davis) Quantum Sets SYCO September 21, / 15