The LovÆsz J-function in Quantum Mechanics · 2013-01-18 · Message of the talk Apologies for...

The Lovász ϑ-function in Quantum MechanicsZero-error Quantum Information and Noncontextuality

Simone Severini

UCL

Oxford Jan 2013

Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan 2013 1 / 29

Message of the talk

Apologies for changing (or better, restricting) the topic of this talk.

The interplay between graph theory and fundamental questions inquantum information is rich.

The Lovász ϑ-function has an operational interpretation in physics.


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), andAndreas Winter (ICREA-Barcelona)


One-shot zero-error capacity (from Shannon theory)

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 forevery i , j ∈ S . The independence number α (G ) of G is thecardinality of the largest independence set in G .The one-shot zero-error capacity Cs ,0 (K) (or Cs ,0 (G )) of a DMSC Kis the maximum number of symbols that can be transmitted withoutconfusion in one use of the channel.

FactCs ,0 (G ) = log α (G ) and this is NP-hard to compute.


Kochen-Specker sets

An orthogonal representation1 of a graph G = (V ,E ) is anassignment 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-cliquesB1, ...,Bk . We label each vertex of G by (q, i), where q ∈ [k ] andi ∈ [d ]. (A k-clique is a subgraph with k vertices and (k2) edges.)The graph G realizes a Kochen-Specker set2 if it has an orthogonalrepresentation {|q, i〉 : (q, i) ∈ V } of dimension d and G does nothave a k-clique {v1, ..., vk}, where vi ∈ Bi .

FactIf G realizes a Kochen-Specker set then Cs ,0 (G ) < log k.

1Lovász (1979).2Kochen, Specker (1967).Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan 2013 5 / 29

Quantum states and operations

Systems with two parties are represented by a spaceHAB ∼= CnA ⊗CmB .

A Hermitian ρAB on HAB such that Tr(ρAB ) = 1 and ρAB � 0represent the state of the system. Each state is a convex combinationof one-dim projectors |ψ〉〈ψ| onto the subspaces C|ψ〉. If we cannotwrite ρAB = ∑ ωiρ

(i )A ⊗ ρ

(i )B , ∑ ωi = 1, then ρAB is entangled.

A projective measurement is implemented by projectors {P1, ...,Pn}such that PiPj = δijPj and ∑i Pi = I . When a measurement ifcarried out on a state |ψ〉 the result i is obtained with probabilitypi = 〈ψ|Pi |ψ〉, and the state is forced into p−1/2

i Pi |ψ〉.A quantum channel is a map K(ρ) = ∑j EjρE

†j , where the Ej’s are

(Kraus) Hermitian operators such that ∑j EjE†j = I .


Cubitt-Leung-Matthews-Winter protocol (2010)

Assume G realizes a Kochen-Specker set. Suppose A(lice) wants to send asymbol 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 standardbasis of Cd .

1 A measures her system with respect to the basis {|q, i〉 : i ∈ [d ]} andobtains a state |q, j〉, where j is randomly distributed in [d ]. Given thestructure 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) ∈ Sq,j = N [(q, j)] ∪ (q, j), where N [(q, j)] is theneighbourhood of (q, j).

3 B knows that the state of his system is in the set of orthogonal states{|x , y〉 : |x , y〉 ∈ Sq,j}. B measures in a basis including these statesto determine (q, j) with certainty.


Entanglement assisted one-shot zero-error capacity

The entanglement assisted one-shot zero-error capacity Cs ,e ,0 (K) (orCs ,e ,0 (G )) of a DMSC K is the maximum number of symbols thatcan be transmitted without confusion in one use of the channel.

We can also write Cs ,e ,0 (G ) = α(G ), for the entanglement-assistedindependence number of G .

TheoremIf G realizes a Kochen-Specker set then Cs ,e ,0 (G ) ≥ log k.

Proof.By the Cubitt-Leung-Matthews-Winter protocol.


Strong graph product

The strong product of G and H is the graph K = G �H withV (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

.


Zero-error capacity

The (asymptotic) zero-error capacity3 of a DMSC K is defined asC0 (G ) = limk→∞

1k log α

(G�k

)= supk

1k log

(G�k

).

FactCs ,0 (G ) ≤ C0 (G ).

Proof.

log α (C5) = log 2 < 12 log α

(C�25

)= 1

2 log 5 ≤ C0 (C5).

3Shannon (1956).Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan 2013 10 / 29

Lovász function

Let {|v1〉, ..., |vn〉} range over all orthogonal representations of G and|ψ〉 over Rd . The Lovász ϑ-function4 is defined as

ϑ (G ) = maxn

∑i=1|〈ψ|vi 〉|2.

The chromatic number χ (G ) is the minimum number of coloursneeded to colour the vertices of G so that no two adjacent verticesare coloured the same.

Theorem (Sandwich Theorem)

α (G ) ≤ ϑ (G ) ≤ χ(G).a

aSee, Knuth (1994).

4Lovász (1979).Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan 2013 11 / 29

Example


Bounding the zero-error capacity

Theorem(1) C0 (G ) ≤ ϑ (G ), where G is the confusability graph of the channel K.(2) The ϑ-function can be approximated with arbitrary precision inpolynomial time.

Proof.(1) ϑ (G �H) = ϑ (G ) ϑ (H). (2) ϑ (G ) is an SDP relaxation.

Example

C0 (C5) = 12 log 5. Take {|v1〉, ..., |v5〉 ∈ R3} s.t.

vi =(cos φ cos

2πj5, cos φ sin

2πj5, sin φ

)T,

where φ = tan−1 (− cos 4π/5)1/2.


Perfect graphs

The clique number ω (G ) of G is the cardinality of the largestcomplete subgraph in G .

A graph G is perfect5 if ω(H)= χ

(H)= α (H), for every induced

subgraph H.

If G is perfect then C0 (G ) = Cs ,0 (G ) = α (G ) = ϑ (G ) = χ(G).

5defined by Berge (1973); characterized by Chudnovsky, Robertson, Seymour,Thomas (2002).


Haemers bound

TheoremThere are graphs G such that C0 (G ) < ϑ (G ).

Proof.Define R (G ) = minM rank(M), where [M ]i ,j ∈ F with [M ]i ,j 6= 0 iff i = jor {i , j} ∈ E a. (1) C0 (G ) ≤ R (G ). (2) There are graphs G such thatR (G ) < ϑ (G ). For example, if G is the generalized quadrangle GQ (2, 4)then C0 (G ) ≤ R (G ) ≤ 7 < ϑ (G ) = 9.

aHaemers (1979).


Bounding the entanglement assisted zero-error capacity

The (asymptotic) entanglement assisted zero-error capacity6 of aDMSC K is defined as Ce ,0 (G ) = supk 1k log

(G�k

), but the parties

A and B are allowed to share a quantum state.

TheoremCe ,0 (G ) ≤ ϑ (G ).a

aDuan, Severini, Winter (2010); Beigi (2010).

6Shannon (1956).Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan 2013 16 / 29

Entanglement assisted zero-error capacity: gap

TheoremThere are graphs G such that C0 (G ) < Ce ,0 (G ).a

aLeung, Mancinska, Matthews, Ozols, Roy (2010).

Proof.By the previous theorem, if G realizes a Kochen-Specker set with q basesthen C0 (G ) = log k = ϑ (G ). Let G = sp(6,F2): the vertices are the22m − 1 points of F2m2 ; two vertices are adjacent if orthogonal in a certainsymplectic space. By a, C0(G ) = log 7. The vectors of the root system E7are an orthogonal representation of G and V (G ) is partioned into 97-cliques. Thus, Ce ,0(G ) = log 9.

aPeeters (1995).


Quantum generalization of confusability graphs

The non-commutative (confusability) graph associated to a quantumchannel K is the operator subspace S = span{E †

j Ek : j , k}, where Ejare the Kraus operators of K. A subspace S is associate to a channeliff I ∈ S and S = S†.

The non-commutative confusability graph of K1 ⊗K2 is S1 ⊗ S2.The max number of one-shot zero-error distinguishable messages forK (the independence number α(S) of S) is the max size of a set oforthogonal vectors {|φm〉 : m = 1, ..., n} such that ∀m 6= m′,|φm〉〈φm ′ | ∈ S⊥: input states |φm〉 and |φm ′〉 lead to orthogonaloutput states iff 0 = TrK(φm)K(φm ′) = ∑j ,k |〈φm ′ |E †

j Ek |φm〉|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 )}.


A quantum Lovász function

The Lovász ϑ-function of a graph G can be written asϑ(G ) = max{‖I + T‖ : Ti ,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 notmultiplicative under tensor product!

Let L(Cn), be the space of Hermitian operators on Cn. This is theoperator space version of the complete graph. The quantum Lovászϑ-function is ϑ(S) = supn ϑ (S ⊗L(Cn)) = supn max{‖I + T‖ : T ∈S⊥ ⊗L(Cn), T = T †, I + T � 0}.Denoting by α(S) the entanglement-assisted independence number ofS , we have α(S) ≤ α(S) ≤ ϑ(S).


SDP characterization and quantum bound

Theorem

There is an SDP characterization of ϑ(S), including a dual.

Corollary

ϑ(S1 ⊗ S2) = ϑ(S1)ϑ(S2), for every S1 < L(CnA), S2 < L(CnB ).

Proof.From the dual of the SDP and by supermultiplicativity.

Theorem

Ce ,0 (S) ≤ ϑ(S)a.

aDuan, Severini, Winter (2010).


Open problems

We saw that there are graphs G such that α(G ) < α(G ). Thissomehow "explains" why ϑ(S) is not tight. ϑ(S) bounds a quantitydetermined by the use of quantum resources.

Can we define a notion of entanglement-assisted perfectdness?

We saw that there are graphs G such that C0 (G ) < Ce ,0 (G )...

Problem

Let S be a non-commutative graph. Is it true that Ce ,0 (S) = ϑ(S)? (I.e.,is it true that Ce ,0 (S) is multiplicative? When G is a graph, this impliesCe ,0 (G ) = ϑ(G ).

Why true? There are entanglement-assisted games (XOR), for which aSDP characterisation leads to multiplicativity of the optimal winningprobability: vale (∧ni=1Pi ) = ∏n

i=1vale (Pni ) =

(cos2 (π/8)

)n. Withoutentanglement we only know upper bounds.7

7R. Cleve, W, Slofstra, F. Unger, S. Upadhyay (2006); Raz (1998).Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan 2013 21 / 29

Interlude: a non-local game and the chromatic number

Consider a game in which A and B receive two vertices iA and jB of agraph G from a referee, such that either iA = jB or {iA, jB} ∈ E (G ).Without communication, A and B send back to the referee twocolours c (iA) and c (jB ), such that if iA = jB then c (iA) = c (jB )and if {iA, jB} ∈ E (G ) then c (iA) 6= c (jB ). A and B know thegraph but can not communicate during the game.

What is the minimum number of colours such that A and B answercorrectly? The chromatic number χ (G ) of G is the minimum numberof colours needed to colour the verticee of in such a way thatadjacency vertices have different colours. A and B agree of a graphcolouring before starting the game. Thus, val(G ) = χ (G ) ≥ ω (G ).


Interlude: a non-local game and the quantum chromaticnumber

If A and B share a state |ψ〉AB then there are graph such that theycan play the game with vale (G ) < χ (G ). For example, orthogonalitygraphs of large size: the vertices are {1,−1}2n vectors and twovertices are adjacenct if their vectors are orthogonal.

DefinitionThe quantum chromatic number, vale (G ) = χq (G ), is the min dim ofunitary matrices U1, ...,Un associated to the vertices of G such that[U†i Uj ]k ,k = 0 for each k.

a

aCameron, Montanaro, Newmann, Severini, Winter (2006).

How hard is to compute χq (G )? What is the relation betweenχq (G ) and other graph theoretic quantities?

Theorem

ϑ(G ) ≤ χq (G )a.

aMancinska, Roberson (2012).


Klyachko-Can-Binicioglu-Shumovsky (KCBS) inequality

Consider 5 yes-no questions, P0, ...,P4. Let Pi and Pi+1 becompatible (both questions can be jointly asked without mutualdisturbance; when the questions are repeated, the same answers areobtained) and exclusive (not both can be true). What is themaximum number of yes answers one can get when asking the 5questions to a physical system? If we ask the questions (0,1) toidentically prepared systems, the average of yes answers isβ = ∑4

i=0〈Pi 〉 ≤ 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 byquantum mechanics with a value βQM .

8Klyachko, Can, Binicioglu, Shumovsky (2009)Simone Severini (UCL) Lovász ϑ-function in Quantum Mechanics Oxford Jan 2013 24 / 29

Noncontextual hidden variable theories

In a noncontextual hidden variable theory the answer of Pj isindependent of whether we ask Pj and Pj−1 or Pj and Pj+1. In theexample, the contexts are the edges of C5, {Pj ,Pj+1mod 5}. Theassumption is that the answer Pj will be the same for contextscontaining Pj .

Physically, asking questions to a system means to measure its state.Here, we use projectors (see Slide 3).


Generalizing noncontextual inequalities

TheoremGiven a graph G, α (G ) is the max value obtained by classical mechanics.max∑n

i=1 |〈ψ|ψi 〉|2 = ϑ (G ) is the max value obtained by quantummechanics. Beyond quantum mechanics there are generalized theorieswhose max value is given by the fractional packing number.a

aCabello, Severini, Winter (2010).


Conclusions

Zero-error quantum information theory offers a new perspective oncombinatorics (graph theory, Ramsey theory, etc.) and the possibilityof extending important mathematical notions (independence number,perfectdness, etc.).

The Lovász ϑ-function offers a new perspective on quantummechanics (noncontextuality, Bell’s inequality, etc.) and thepossibility to better determine the boundaries between classical,quantum mechanics and general probabilistic theories.


References

Zero-error quantum information and the Lovász ϑ-function:

R. Duan, S. Severini, A. Winter, Zero-error communication viaquantum channels, noncommutative graphs and a quantum Lovászϑ-function, arXiv:1002.2514 (2010).

Non-contextuality and the Lovász ϑ-function:

A. Cabello, S. Severini, A. Winter, (Non-)Contextuality of PhysicalTheories as an Axiom, arXiv:1010.2163v1 (2010).

Quantum chromatic number:

P. J. Cameron, M. W. Newman, A. Montanaro, S. Severini, A.Winter, arXiv:0608016v3 (2006).


The LovÆsz J-function in Quantum Mechanics · 2013-01-18 · Message of the talk Apologies for...

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