The Lovász ϑ-function in Quantum Mechanics

Transcription

1 The Lovász ϑ-function in Quantum Mechanics Zero-error Quantum Information and Noncontextuality Simone Severini UCL Oxford Jan 2013 Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

2 Message of the talk Apologies for changing (or better, restricting) the topic of this talk. The interplay between graph theory and fundamental questions in quantum information is rich. The Lovász ϑ-function has an operational interpretation in physics. Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

3 Summary Zero-error information theory The Lovász ϑ-function Quantum zero-error information theory A quantum Lovász ϑ-function A theory of noncommutative graphs? Noncontextuality and non-local games Based on work with Runyao Duan (Tsinghua), Adan Cabello (Sevilla), and Andreas Winter (ICREA-Barcelona) Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

4 One-shot zero-error capacity (from Shannon theory) Fact A discrete memoriless stationary channel (DMSC) with alphabet {1,..., n} is represented by a stochastic matrix M such that [M] i,j = Pr[i received when j transmitted]. Two symbols j and k are confusable if there is i such that [M] i,j [M] i,k > 0. The confusability graph G = (V, E ) of the DMSC has V = {1,..., n} and {j, k} E iff j and k are confusable. A set S V is an independent set of G = (V, E ) if {i, j} / E for every i, j S. The independence number α (G ) of G is the cardinality of the largest independence set in G. The one-shot zero-error capacity C s,0 (K) (or C s,0 (G )) of a DMSC K is the maximum number of symbols that can be transmitted without confusion in one use of the channel. C s,0 (G ) = log α (G ) and this is NP-hard to compute. Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

5 Kochen-Specker sets Fact An orthogonal representation 1 of a graph G = (V, E ) is an assignment of vectors 1,..., n to vertices V = {1,..., n} such that i j = 0 iff {i, j} E. Let G = (V, E ) be a graph such that V is partitioned into k d-cliques B 1,..., B k. We label each vertex of G by (q, i), where q [k] and i [d]. (A k-clique is a subgraph with k vertices and ( k 2 ) edges.) The graph G realizes a Kochen-Specker set 2 if it has an orthogonal representation { q, i : (q, i) V } of dimension d and G does not have a k-clique {v 1,..., v k }, where v i B i. If G realizes a Kochen-Specker set then C s,0 (G ) < log k. 1 Lovász (1979). 2 Kochen, Specker (1967). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

6 Quantum states and operations Systems with two parties are represented by a space H AB = C n A C m B. A Hermitian ρ AB on H AB such that Tr(ρ AB ) = 1 and ρ AB 0 represent the state of the system. Each state is a convex combination of one-dim projectors ψ ψ onto the subspaces C ψ. If we cannot write ρ AB = ω i ρ (i) A ρ(i) B, ω i = 1, then ρ AB is entangled. A projective measurement is implemented by projectors {P 1,..., P n } such that P i P j = δ ij P j and i P i = I. When a measurement if carried out on a state ψ the result i is obtained with probability p i = ψ P i ψ, and the state is forced into p 1/2 i P i ψ. A quantum channel is a map K(ρ) = j E j ρe j, where the E j s are (Kraus) Hermitian operators such that j E j E j = I. Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

7 Cubitt-Leung-Matthews-Winter protocol (2010) Assume G realizes a Kochen-Specker set. Suppose A(lice) wants to send a symbol q [k] to B(ob). Suppose A and B share a state ψ AB = d 1/2 d v =1 v A v B, where { v X {A,B } } is the standard basis of C d. 1 A measures her system with respect to the basis { q, i : i [d]} and obtains a state q, j, where j is randomly distributed in [d]. Given the structure of ψ AB, the system of B is forced to be in the state q, j as well. However, this information is conditioned on knowing (q, j). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

8 Cubitt-Leung-Matthews-Winter protocol (2010) Assume G realizes a Kochen-Specker set. Suppose A(lice) wants to send a symbol q [k] to B(ob). Suppose A and B share a state ψ AB = d 1/2 d v =1 v A v B, where { v X {A,B } } is the standard basis of C d. 1 A measures her system with respect to the basis { q, i : i [d]} and obtains a state q, j, where j is randomly distributed in [d]. Given the structure of ψ AB, the system of B is forced to be in the state q, j as well. However, this information is conditioned on knowing (q, j). 2 A sends the (classical) message (q, j) through the channel. B receives (x, y) S q,j = N[(q, j)] (q, j), where N[(q, j)] is the neighbourhood of (q, j). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

9 Cubitt-Leung-Matthews-Winter protocol (2010) Assume G realizes a Kochen-Specker set. Suppose A(lice) wants to send a symbol q [k] to B(ob). Suppose A and B share a state ψ AB = d 1/2 d v =1 v A v B, where { v X {A,B } } is the standard basis of C d. 1 A measures her system with respect to the basis { q, i : i [d]} and obtains a state q, j, where j is randomly distributed in [d]. Given the structure of ψ AB, the system of B is forced to be in the state q, j as well. However, this information is conditioned on knowing (q, j). 2 A sends the (classical) message (q, j) through the channel. B receives (x, y) S q,j = N[(q, j)] (q, j), where N[(q, j)] is the neighbourhood of (q, j). 3 B knows that the state of his system is in the set of orthogonal states { x, y : x, y S q,j }. B measures in a basis including these states to determine (q, j) with certainty. Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

10 Entanglement assisted one-shot zero-error capacity The entanglement assisted one-shot zero-error capacity C s,e,0 (K) (or C s,e,0 (G )) of a DMSC K is the maximum number of symbols that can be transmitted without confusion in one use of the channel. We can also write C s,e,0 (G ) = α(g ), for the entanglement-assisted independence number of G. Theorem If G realizes a Kochen-Specker set then C s,e,0 (G ) log k. Proof. By the Cubitt-Leung-Matthews-Winter protocol. Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

11 Strong graph product The strong product of G and H is the graph K = G H with V (G H) = V (G ) V (H) and {(i, j), (k, l)} E (K ) iff {i, k} E (G ) AND j = l or {j, l} E (H) AND i = k. Let G k := (G G ). }{{} k times Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

12 Zero-error capacity Fact The (asymptotic) zero-error capacity 3 of a DMSC K is defined as 1 C 0 (G ) = lim k k log α ( G k ) 1 = sup k k log ( G k ). C s,0 (G ) C 0 (G ). Proof. log α (C 5 ) = log 2 < 1 2 log α ( C 2 5 ) = 1 2 log 5 C 0 (C 5 ). 3 Shannon (1956). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

13 Lovász function Let { v 1,..., v n } range over all orthogonal representations of G and ψ over R d. The Lovász ϑ-function 4 is defined as ϑ (G ) = max n i=1 ψ v i 2. The chromatic number χ (G ) is the minimum number of colours needed to colour the vertices of G so that no two adjacent vertices are coloured the same. Theorem (Sandwich Theorem) α (G ) ϑ (G ) χ ( G ). a a See, Knuth (1994). 4 Lovász (1979). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

14 Example Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

15 Bounding the zero-error capacity Theorem (1) C 0 (G ) ϑ (G ), where G is the confusability graph of the channel K. (2) The ϑ-function can be approximated with arbitrary precision in polynomial time. Proof. (1) ϑ (G H) = ϑ (G ) ϑ (H). (2) ϑ (G ) is an SDP relaxation. Example C 0 (C 5 ) = 1 2 log 5. Take { v 1,..., v 5 R 3 } s.t. ( v i = cos φ cos 2πj ) 2πj T, cos φ sin 5 5, sin φ, where φ = tan 1 ( cos 4π/5) 1/2. Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

16 Perfect graphs The clique number ω (G ) of G is the cardinality of the largest complete subgraph in G. A graph G is perfect 5 if ω ( H ) = χ ( H ) = α (H), for every induced subgraph H. If G is perfect then C 0 (G ) = C s,0 (G ) = α (G ) = ϑ (G ) = χ ( G ). 5 defined by Berge (1973); characterized by Chudnovsky, Robertson, Seymour, Thomas (2002). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

17 Haemers bound Theorem There are graphs G such that C 0 (G ) < ϑ (G ). Proof. Define R (G ) = min M rank(m), where [M] i,j F with [M] i,j = 0 iff i = j or {i, j} E a. (1) C 0 (G ) R (G ). (2) There are graphs G such that R (G ) < ϑ (G ). For example, if G is the generalized quadrangle GQ (2, 4) then C 0 (G ) R (G ) 7 < ϑ (G ) = 9. a Haemers (1979). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

18 Bounding the entanglement assisted zero-error capacity The (asymptotic) entanglement assisted zero-error capacity 6 of a 1 DMSC K is defined as C e,0 (G ) = sup k k log ( G k ), but the parties A and B are allowed to share a quantum state. Theorem C e,0 (G ) ϑ (G ). a a Duan, Severini, Winter (2010); Beigi (2010). 6 Shannon (1956). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

19 Entanglement assisted zero-error capacity: gap Theorem There are graphs G such that C 0 (G ) < C e,0 (G ). a a Leung, Mancinska, Matthews, Ozols, Roy (2010). Proof. By the previous theorem, if G realizes a Kochen-Specker set with q bases then C 0 (G ) = log k = ϑ (G ). Let G = sp(6, F 2 ): the vertices are the 2 2m 1 points of F 2m 2 ; two vertices are adjacent if orthogonal in a certain symplectic space. By a, C 0 (G ) = log 7. The vectors of the root system E 7 are an orthogonal representation of G and V (G ) is partioned into 9 7-cliques. Thus, C e,0 (G ) = log 9. a Peeters (1995). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

20 Quantum generalization of confusability graphs The non-commutative (confusability) graph associated to a quantum channel K is the operator subspace S = span{e j E k : j, k}, where E j are the Kraus operators of K. A subspace S is associate to a channel iff I S and S = S. The non-commutative confusability graph of K 1 K 2 is S 1 S 2. The max number of one-shot zero-error distinguishable messages for K (the independence number α(s) of S) is the max size of a set of orthogonal vectors { φ m : m = 1,..., n} such that m = m, φ m φ m S : input states φ m and φ m lead to orthogonal output states iff 0 = TrK(φ m )K(φ m ) = j,k φ m E j E k φ m 2. Computing α(s) is QMA-complete and NP QMA. For a (classical) confusability graph G, S = {T : [T ] i,j = 0 iff i T j = 0, {i, j} / E (G )}. Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

21 A quantum Lovász function The Lovász ϑ-function of a graph G can be written as ϑ(g ) = max{ I + T : T i,j = 0 if {i, j} E (G ), T = T, I + T 0}. A naive quantum version is ϑ(s) = max{ I + T : T S, T = T, I + T 0}. Clearly, for classical graph G, ϑ(s) = ϑ(g ). ϑ(s) is in general not multiplicative under tensor product! Let L(C n ), be the space of Hermitian operators on C n. This is the operator space version of the complete graph. The quantum Lovász ϑ-function is ϑ(s) = sup n ϑ (S L(C n )) = sup n max{ I + T : T S L(C n ), T = T, I + T 0}. Denoting by α(s) the entanglement-assisted independence number of S, we have α(s) α(s) ϑ(s). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

22 SDP characterization and quantum bound Theorem There is an SDP characterization of ϑ(s), including a dual. Corollary ϑ(s 1 S 2 ) = ϑ(s 1 ) ϑ(s 2 ), for every S 1 < L(C n A ), S 2 < L(C n B ). Proof. From the dual of the SDP and by supermultiplicativity. Theorem C e,0 (S) ϑ(s) a. a Duan, Severini, Winter (2010). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

23 Open problems We saw that there are graphs G such that α(g ) < α(g ). This somehow "explains" why ϑ(s) is not tight. ϑ(s) bounds a quantity determined by the use of quantum resources. Problem Can we define a notion of entanglement-assisted perfectdness? We saw that there are graphs G such that C 0 (G ) < C e,0 (G )... Let S be a non-commutative graph. Is it true that C e,0 (S) = ϑ(s)? (I.e., is it true that C e,0 (S) is multiplicative? When G is a graph, this implies C e,0 (G ) = ϑ(g ). Why true? There are entanglement-assisted games (XOR), for which a SDP characterisation leads to multiplicativity of the optimal winning probability: val e ( n i=1 P i ) = n i=1val e (P n i ) = ( cos 2 (π/8) ) n. Without entanglement we only know upper bounds. 7 7 R. Cleve, W, Slofstra, F. Unger, S. Upadhyay (2006); Raz (1998). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

24 Interlude: a non-local game and the chromatic number Consider a game in which A and B receive two vertices i A and j B of a graph G from a referee, such that either i A = j B or {i A, j B } E (G ). Without communication, A and B send back to the referee two colours c (i A ) and c (j B ), such that if i A = j B then c (i A ) = c (j B ) and if {i A, j B } E (G ) then c (i A ) = c (j B ). A and B know the graph but can not communicate during the game. What is the minimum number of colours such that A and B answer correctly? The chromatic number χ (G ) of G is the minimum number of colours needed to colour the verticee of in such a way that adjacency vertices have different colours. A and B agree of a graph colouring before starting the game. Thus, val(g ) = χ (G ) ω (G ). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

25 Interlude: a non-local game and the quantum chromatic number If A and B share a state ψ AB then there are graph such that they can play the game with val e (G ) < χ (G ). For example, orthogonality graphs of large size: the vertices are {1, 1} 2n vectors and two vertices are adjacenct if their vectors are orthogonal. Definition The quantum chromatic number, val e (G ) = χ q (G ), is the min dim of unitary matrices U 1,..., U n associated to the vertices of G such that [U i U j ] k,k = 0 for each k. a a Cameron, Montanaro, Newmann, Severini, Winter (2006). How hard is to compute χ q (G )? What is the relation between χ q (G ) and other graph theoretic quantities? Theorem a Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

26 Klyachko-Can-Binicioğlu-Shumovsky (KCBS) inequality Consider 5 yes-no questions, P 0,..., P 4. Let P i and P i+1 be compatible (both questions can be jointly asked without mutual disturbance; when the questions are repeated, the same answers are obtained) and exclusive (not both can be true). What is the maximum number of yes answers one can get when asking the 5 questions to a physical system? If we ask the questions (0,1) to identically prepared systems, the average of yes answers is β = 4 i=0 P i 2. This is the KCBS inequality. 8 Physically, it means that we use observables that are commuting (compatible) and orthogonal (exclusive). Satisfied by any noncontextual hidden variable theory, but violated by quantum mechanics with a value β QM. 8 Klyachko, Can, Binicioğlu, Shumovsky (2009) Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

27 Noncontextual hidden variable theories In a noncontextual hidden variable theory the answer of P j is independent of whether we ask P j and P j 1 or P j and P j+1. In the example, the contexts are the edges of C 5, {P j, P j+1 mod 5 }. The assumption is that the answer P j will be the same for contexts containing P j. Physically, asking questions to a system means to measure its state. Here, we use projectors (see Slide 3). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

28 Maximum violation of the KCBS inequality Assume the system is in the state ψ = (0, 0, 1) T. Take the commuting and orthogonal questions/observables A i = 2 v i v i I, where v 0 = N 0 (1, 0, cos(π/5), v 1,4 = N 1 (cos(4π/5) ± ), and v 2,3 = N 2 (cos(2π/5) ± ). Notice that v i v i+1 mod 5 = 0 (compatibility). Then β QM = i mod 5 ψ ψ i 2. Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

29 Generalizing noncontextual inequalities Theorem Given a graph G, α (G ) is the max value obtained by classical mechanics. max n i=1 ψ ψ i 2 = ϑ (G ) is the max value obtained by quantum mechanics. Beyond quantum mechanics there are generalized theories whose max value is given by the fractional packing number. a a Cabello, Severini, Winter (2010). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

30 Conclusions Zero-error quantum information theory offers a new perspective on combinatorics (graph theory, Ramsey theory, etc.) and the possibility of extending important mathematical notions (independence number, perfectdness, etc.). The Lovász ϑ-function offers a new perspective on quantum mechanics (noncontextuality, Bell s inequality, etc.) and the possibility to better determine the boundaries between classical, quantum mechanics and general probabilistic theories. Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29

31 References Zero-error quantum information and the Lovász ϑ-function: R. Duan, S. Severini, A. Winter, Zero-error communication via quantum channels, noncommutative graphs and a quantum Lovász ϑ-function, arxiv: (2010). Non-contextuality and the Lovász ϑ-function: A. Cabello, S. Severini, A. Winter, (Non-)Contextuality of Physical Theories as an Axiom, arxiv: v1 (2010). Quantum chromatic number: P. J. Cameron, M. W. Newman, A. Montanaro, S. Severini, A. Winter, arxiv: v3 (2006). Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan / 29