A "quantum" Ramsey theorem for operator systems

0.17 MB·

by  Nik Weaver

A “QUANTUM” RAMSEY THEOREM FOR OPERATOR
SYSTEMS
6
1
NIKWEAVER
0
2
n Abstract. LetV bealinearsubspace ofMn(C) whichcontains theidentity
a matrixandisstableundertheformationofHermitianadjoints. Weprovethat
J ifnissufficientlylargethenthereexistsarankkorthogonalprojectionP such
3 thatdim(PVP)=1ork2.
1

]
A
1. Background
O
An operator system in finite dimensions is a linear subspace V of M (C) with
. n
h the properties
t
a • I ∈V
n
m
• A∈V ⇒A∗ ∈V
[ where I is the n×n identity matrix andA∗ is the Hermitianadjointof A. In this
n
2 paper the scalar field will be complex and we will write Mn =Mn(C).
v Operatorsystems play a role in the theory of quantumerrorcorrection. In clas-
9 sical information theory, the “confusability graph” is a bookkeeping device which
5
keeps track of possible ambiguity that can result when a message is transmitted
2
througha noisy channel. It is defined by taking as vertices allpossible source mes-
1
0 sages,andplacinganedgebetweentwomessagesiftheyaresufficientlysimilarthat
. data corruption could lead to them being indistinguishable on reception. Once the
1
0 confusability graph is known, one is able to overcome the problem of information
6 loss by using an independent subset of the confusability graph, which is known as
1 a “code”. If it is agreed that only code messages will be sent, then we can be sure
:
v that the intended message is recoverable.
i Wheninformationisstoredinquantummechanicalsystems,theproblemoferror
X
correctionchangesradically. Thebasictheoryofquantumerrorcorrectionwaslaid
r
downin [3]. In [2] it was suggestedthat in this setting the role of the confusability
a
graph is played by an operator system, and it was shown that for every operator
system a “quantum Lov´asz number” could be defined, in analogy to the classical
Lov´asznumber ofagraph. Thisis animportantparameterinclassicalinformation
theory. See also [5] for much more along these lines.
Theinterpretationofoperatorsystemsas“quantumgraphs”wasalsoproposedin
[8],basedonthemoregeneralideaofregardinglinearsubspacesofM as“quantum
n
relations”, and taking the conditions I ∈ V and A ∈ V ⇒A∗ ∈ V to respectively
n
express reflexivity and symmetry conditions. The idea is that the edge structure
of a classical graph can be encoded in an obvious way as a reflexive, symmetric
relationonaset. This pointofview wasexplicitly connectedto the quantumerror
correction literature in [9].
Date:Jan. 13, 2016.
1
2 NIKWEAVER
Ramsey’stheoremstatesthatforanykthereexistsnsuchthateverygraphwith
atleastnverticescontainseitherak-cliqueorak-anticlique,i.e.,asetofk vertices
among which either all edges are present or no edges are present. Simone Severini
askedtheauthorwhetherthereisa“quantum”versionofthistheoremforoperator
systems. The natural notions of k-clique and k-anticlique are the following.
Definition 1.1. Let V ⊆ M be an operator system. A quantum k-clique of V
n
is an orthogonal projection P ∈ M (i.e., a matrix satisfying P = P2 = P∗)
n
whose rank is k, such that PVP = {PAP : A ∈ V} is maximal; that is, such
that PVP = PM P ∼= M , or equivalently, dim(PVP) = k2. A quantum k-
n k
anticlique of V is a rank k projection P such that PVP is minimal; that is, such
that PVP =C·P ∼=M , or equivalently, dim(PVP)=1.
1
Thedefinitionofquantumk-anticliqueissupportedbythe factthatinquantum
error correction a code is taken to be the range of a projection satisfying just this
condition, PVP = C · P [3]. As mentioned earlier, classical codes are taken to
be independent sets, which is to say, anticliques. See also Section 4 of [9], where
intuition for why PVP is correctly thought of as a “restriction” of V is given.
ThemainresultofthispaperisaquantumRamseytheoremwhichstatesthatfor
everyk there existsnsuchthateveryoperatorsysteminM haseithera quantum
n
k-cliqueoraquantumk-anticlique. ThisanswersSeverini’squestionpositively. The
quantumRamseytheoremisnotmerelyanalogoustotheclassicalRamseytheorem;
using the bimodule formalism of [8], we can formulate a common generalization of
the two results. This will be done in the final section of the paper.
I especiallythank MichaelJuryforstimulating discussions,andinparticularfor
conjecturing Proposition 2.3 and improving Lemma 4.2.
Partof this workwasdone ata workshoponZero-errorinformation,Operators,
and Graphs at the Universitat Aut`onoma de Barcelona.
2. Examples
If G = (V,E) is any finite simple graph, without loss of generality suppose
V ={1,...,n} and define V to be the operator system
G
V =span{E :i=j or {i,j}∈E}⊆M .
G ij n
Here we use the notation E for the n×n matrix with a 1 in the (i,j) entry and
ij
0’s elsewhere. Also, let (e ) be the standard basis of Cn, so that E =e e∗.
i ij i j
The inclusionofthe diagonalE matricesinV correspondstoincluding aloop
ii G
at each vertex in G. In the error correction setting this is natural: we place an
edge between any two messages that might be indistinguishable on reception, and
this is certainly true of any message and itself. Once we adoptthe conventionthat
every graph has a loop at each vertex, an anticlique should no longer be a subset
S ⊆ V which contains no edges, it should be a subset which contains no edges
except loops. Such a set corresponds to the projection P onto span{e : i ∈ S},
S i
which has the property that P V P = span{E : i ∈ S}. Or course this is very
S G S ii
different from a quantum anticlique where PVP is one-dimensional.
To illustrate the dissimilarity between classical and quantum cliques and anti-
cliques, consider the diagonal operator system D ⊆M consisting of the diagonal
n n
n×n complex matrices. In the notation used above, this is just the operator sys-
tem V corresponding to the empty graph on n vertices. It might at first appear
G
to falsify the desired quantum Ramsey theorem, because of the following fact.
A QUANTUM RAMSEY THEOREM 3
Proposition 2.1. D has no quantum k-anticlique for k ≥2.
n
Proof. Let P ∈ M be a projection of rank k ≥ 2. Since rank(E ) = 1 for all
n ii
i, it follows that rank(PE P) = 0 or 1 for each i. If PE P = 0 for all i then
ii ii
P = n PE P = 0, contradiction. Thus we must have rank(PE P) = 1 for
i=1 ii ii
somePi, but then PEiiP cannot belong to C·P ={aP :a∈C}, since every matrix
in this set has rank 0 or k. So PD P 6=C·P. (cid:3)
n
SinceeveryoperatorsystemoftheformV containsthediagonalmatrices,none
G
ofthese operatorsystems hasnontrivialquantumanticliques. The surprisingthing
is that for n sufficiently large, they all have quantum k-cliques. This follows from
the next result.
Proposition 2.2. If n≥k2+k−1 then D has a quantum k-clique.
n
Proof. Withoutlossofgeneralityletn=k2+k−1. StartbyconsideringM acting
k
on Ck. Find k2 vectors v1,...,vk2 in Ck such that the rank 1 matrices vivi∗ are
linearly independent. (For example, we could take the k standard basis vectors e
i
plus the k2−k vectors e +e for i 6= j plus the k2−k vectors e +ie for i 6= j.
2 i j 2 i j
The corresponding rank 1 matrices span M and thus they must be independent
k
since dim(M ) = k2.) Making the identification Cn ∼= Ck⊕Ck2−1, we can extend
k
the v to orthogonal vectors w ∈ Cn as follows: take w = v ⊕ (1,0,...,0),
i i 1 1
w = v ⊕ (a ,1,0,...,0), w = v ⊕ (b ,b ,1,0,...,0), etc., with a ,b ,b ,...
2 2 1 3 3 1 2 1 1 2
successivelychosensothathw ,w i=0fori6=j. We needk2−1extradimensions
i j
toaccomplishthis. NowletP betherankkprojectionofCn ontoCk andletD be
n
the diagonaloperatorsystemrelativetoanyorthonormalbasisofCn thatcontains
the vectors wi for 1≤i≤k2. Then PD P contains Pw w∗P =v v∗ for all i, so
kwik n i i i i
dim(PD P)=k2. (cid:3)
n
A stronger version of this result will be proven in Lemma 4.3. The value n =
k2+k−1 may not be optimal, but note that in order for D to have a quantum
n
k-clique n must be at least k2, since dim(D )=n and we need dim(PD P)=k2.
n n
Next, we show that operator systems of arbitrarily large dimension may lack
quantum 3-cliques.
Proposition 2.3. Let V = span{I ,E ,E ,...,E ,E ,...,E } ⊆ M .
n n 11 12 1n 21 n1 n
Then V has no quantum 3-cliques.
n
Proof. Let P ∈ M be any projection. If Pe = 0 then PE P = PE P = 0 for
n 1 1i i1
alli,soP is aquantumanticlique. Otherwiseletk=rank(P)andletf ,...,f be
1 k
an orthonormal basis of ran(P) with f = Pe1 . Then PE P = Pe e∗P = f v∗
1 kPe1k 1i 1 i 1 i
where v = kPe kPe . The span of these matrices f v∗ is precisely span{f f∗},
i 1 i 1 i 1 i
since the projections of the e span ran(P). Similarly, the span of the matrices
i
PE P is precisely span{f f∗}. So PV P is just V ⊆ M ∼= PM P, relative to
i1 i 1 n k k n
the (f ) basis. If k ≥ 3 then dim(V ) = 2k < k2, so P cannot be a quantum
i k
clique. (cid:3)
3. Quantum 2-cliques
In contrast to Proposition 2.3, we will show in this section that any operator
systemwhose dimensionis atleastfour must havea quantum 2-clique. This result
isclearlysharp. Itissomewhatanalogoustothetrivialclassicalfactthatanygraph
that contains at least one edge must have a 2-clique.
4 NIKWEAVER
Define the Hilbert-Schmidt inner product of A,B ∈M to be Tr(AB∗). Denote
n
the set of Hermitian n×n matrices by Mh. Observe that any operator system is
n
spanned by its Hermitian part since any matrix A satisfies A = Re(A)+iIm(A)
where Re(A)= 1(A+A∗) and Im(A)= 1(A−A∗).
2 2i
Lemma 3.1. Let V ⊆ M be an operator system and suppose dim(V) ≤ 3. Then
n
its Hilbert-Schmidt orthocomplement is spanned by rank 2 Hermitian matrices.
Proof. WorkinMh. LetV =V∩Mh andletW betherealspanoftheHermitian
n 0 n 0
matrices in V⊥ whose rank is 2. We will show that W = V⊥ (in Mh); taking
0 0 0 n
complex spans then yields the desired result.
Suppose to the contrary that there exists a nonzero Hermitian matrix B ∈ V⊥
0
which is orthogonal to W . Say V = span{I ,A ,A }, where A and A are
0 0 n 1 2 1 2
not necessarily distinct from I . Since B ∈ V⊥, we have Tr(I B) = Tr(A B) =
n 0 n 1
Tr(A B) = 0, but Tr(B2) 6= 0. We will show that there is a rank 2 Hermitian
2
matrix C whose inner products against I , A , A , and B are the same as their
n 1 2
inner products against B. This will be a matrix in W which is not orthogonal to
0
B, a contradiction.
Since B is Hermitian, we can choose an orthonormal basis (f ) of Cn with
i
respect to which it is diagonal, say B = diag(b ,...,b ). We may assume
1 n
b ,...,b ≥ 0 and b ,...,b < 0. Let B+ = diag(b ,...,b ,0,...,0) and
1 j j+1 n 1 j
B− = diag(0,...,0,−b ,...,−b ) be the positive and negative parts of B,
j+1 n
so that B = B+ − B−. Let α = Tr(B+) = Tr(B−) (they are equal since
Tr(B) = Tr(I B) = 0). Then 1B+ is a convex combination of the rank 1 ma-
n α
trices f f∗, ..., f f∗; that is, the linear functional A 7→ 1Tr(AB+) is a convex
1 1 j j α
combinationofthe linearfunctionals A7→hAf ,f ifor1≤i≤j. By the convexity
i i
of the joint numerical range of three Hermitian matrices [1], there exists a unit
vector v ∈ Cn such that 1Tr(AB+) = hAv,vi for A = A , A , and B. Similarly,
α 1 2
there exists a unit vector w such that 1Tr(AB−) = hAw,wi for A = A , A , and
α 1 2
B. Then C = α(vv∗ −ww∗) is a rank 2 Hermitian matrix whose inner products
against I , A , A , and B are the same as their inner products against B. So C
n 1 2
has the desired properties. (cid:3)
Lemma 3.2. Let V ⊆ M be an operator system and suppose dim(V) = 4. Then
3
V has a quantum 2-clique.
Proof. The proof is computational. Say V = span{A ,A ,A ,A } where A = I
0 1 2 3 0 3
and the other A are Hermitian. It will suffice to find two vectors v,w ∈ C3 such
i
that the four vectors [hA v,vi,hA v,wi,hA w,vi,hA w,wi] ∈ C4 for 0 ≤ i ≤ 3 are
i i i i
independent. Thatis,weneedthe4×4matrixwhoserowsarethesevectorstohave
nonzerodeterminant. ThenlettingP betheorthogonalprojectionontospan{v,w}
will verify the lemma.
We can simplify by putting the A in a special form. First, by choosing a basis
i
of eigenvectors, we can assume A is diagonal. By subtracting a suitable multiple
1
of A from A , multiplying by a nonzero scalar, and possibly reordering the basis
0 1
vectors, we can arrange that A has the form diag(0,1,a). (Note that dim(V)=4
1
implies that A cannot be a scalar multiple of A .) These operations do not affect
1 0
span{A ,A ,A ,A }. Then,bysubtractingsuitable linearcombinationsofA and
0 1 2 3 0
A QUANTUM RAMSEY THEOREM 5
A , we can arrange that A and A have the forms
1 2 3
0 a a 0 b b
12 13 12 13
a¯12 0 a23 and ¯b12 0 b23.
a¯ a¯ a ¯b ¯b b
13 23 33 13 23 33
   
Let
1 1
v =α and w =0,
0 β
   
then evaluate the determinant of the 4×4 matrix described above and expand it
as a polynomial in α, β, α¯, and β¯. We just need this determinant to be nonzero
for some values of α and β; if this fails, then the polynomial coefficients must all
be zero, and direct computation shows that this forces one of A and A to be a
2 3
scalar multiple of the other. We omit the tedious but straightforwarddetails. (cid:3)
Theorem 3.3. Let V ⊆M be an operator system and suppose dim(V)≥4. Then
n
V has a quantum 2-clique.
Proof. Without loss of generality we can suppose that dim(V) = 4. Say V =
span{I ,A ,A ,A } where the A are Hermitian and linearly independent.
n 1 2 3 i
We first claim that there is a projection P of rank at most 3 such that PI P,
n
PA P,andPA P arelinearlyindependent. IfA andA arejointlydiagonalizable
1 2 1 2
then we can find three common eigenvectors v , v , and v such that the vectors
1 2 3
(1,1,1),(λ ,λ ,λ ),(µ ,µ ,µ )∈C3 arelinearlyindependent, whereλ andµ are
1 2 3 1 2 3 i i
the eigenvalues belonging to v for A and A , respectively. Then the projection
i 1 2
ontospan{v ,v ,v }verifiestheclaim. IfA andA arenotjointlydiagonalizable,
1 2 3 1 2
then we can find two eigenvectors v and v of A such that hA v ,v i 6= 0. Let-
1 2 1 2 1 2
ting v be a third eigenvector of A with the property that the eigenvalues of A
3 1 1
belonging to v , v , and v are not all equal, we can again use the projection onto
1 2 3
span{v ,v ,v }. This establishes the claim.
1 2 3
Now let P be as in the claim and find B ∈M such that PI P, PA P, PA P,
n n 1 2
and PBP are linearly independent. By Lemma 3.2 we can then find a rank 2 pro-
jectionQ≤P suchthatQI Q, QA Q, QA Q,andQBQarelinearlyindependent.
n 1 2
If QI Q, QA Q, QA Q, and QA Q are linearly independent then we are done.
n 1 2 3
Otherwise, let α, β, and γ be the unique scalars such that QA Q = αQI Q +
3 n
βQA Q+γQA Q. By Lemma 3.1 we can find a rank 2 Hermitian matrix C such
1 2
that Tr(I C)=Tr(A C)=Tr(A C)=0 but Tr(A C)6=0. Then C =vv∗−ww∗
n 1 2 3
for someorthogonalvectorsv andw. Thus, hAv,vi=hAw,wi forA=I , A , and
n 1
A , but not for A=A . It follows that the two conditions
2 3
hA v,vi=αhI v,vi+βhA v,vi+γhA v,vi
3 n 1 2
and
hA w,wi=αhI w,wi+βhA w,wi+γhA w,wi
3 n 1 2
cannotbothhold. Withoutlossofgeneralitysupposethefirstfails. ThenlettingQ′
be the projection onto span(ran(Q)∪{v}), we cannot have Q′A Q′ = αQ′I Q′+
3 n
βQ′A Q′+γQ′A Q′. Thus rank(Q′)=3 and dim(Q′VQ′)=4. The theorem now
1 2
follows by applying Lemma 3.2 to Q′VQ′. (cid:3)
Theorem3.3does notgeneralizeto arbitraryfour-dimensionalsubspacesof M .
n
Forinstance,letV =span{E ,E ,E ,E }⊂M ;byreasoningsimilartothatin
11 12 13 14 4
theproofofProposition2.3,ifP isanyrank2projectioninM thendim(PVP)≤2.
4
6 NIKWEAVER
4. The main theorem
The proof of our main theorem proceeds through a series of lemmas.
Lemma 4.1. Suppose the operator system V is contained in D . If dim(V) ≥
n
k2+k−1 then V has a quantum k-clique. If dim(V)≤ n−k then V has a quantum
k−1
k-anticlique. If n≥k3−k+1 then V has either a quantum k-clique or a quantum
k-anticlique.
Proof. If dim(V)≥k2+k−1=m then we can find a setof indices S ⊆{1,...,n}
of cardinality m such that dim(PVP) = m where P is the orthogonal projection
onto span{e : i ∈ S}. Then PVP ∼= D ⊆ M ∼= PM P and Proposition 2.2
i m m n
yieldsthatPVP,andhencealsoV,has aquantumk-clique. Ifdim(V)≤ n−k then
k−1
a result of Tverberg [6, 7] can be used to extract a quantum k-anticlique; this is
essentially Theorem 4 of [4]. Thus if k2+k−1 ≤ n−k then one of the two cases
k−1
mustobtain,i.e.,V musthaveeitheraquantumk-cliqueoraquantumk-anticlique.
A little algebra shows that this inequality is equivalent to n≥k3−k+1. (cid:3)
Lemma 4.2. Let v ,...,v be vectors in Cs. Then there are vectors w ,...,w ∈
1 r 1 r
Cr−1 such that the vectors v ⊕w ∈ Cs+r−1 are pairwise orthogonal and all have
i i
the same norm.
Proof. Let G be the Gramian matrix of the vectors v and let kGk be its operator
i
norm. Then rank(kGkI −G) ≤ r−1, so we can find vectors w ∈ Cr−1 whose
r i
Gramian matrix is kGkI −G. The Gramian matrix of the vectors v ⊕w is then
r i i
kGkI , as desired. (cid:3)
r
Then next lemma improves Proposition 2.2.
Lemma 4.3. Let n=k2+k−1 and suppose A1,...,Ak2 are Hermitian matrices
in M such that for each i we have hA e ,e i=1, and also hA e ,e i=0 whenever
n i i i i r s
max{r,s}>i. Then V =span{I,A1,...,Ak2} has a quantum k-clique.
Proof. LetAi havematrixentries(airs). Thegoalistofindvectorsv1,...,vk2 ∈Ck
such that the matrices
A′ = ai v v∗ ∈M
i rs r s k
1≤Xr,s≤k2
are linearly independent. Once we have done this, find vectors w ∈ Ck2−1 as in
i
Lemma 4.2 and let f = 1(v ⊕w ) ∈ Cn ∼= Ck ⊕Ck2−1 where N is the common
i N i i
norm of the v ⊕w . Then the f form an orthonormal set in Cn, so they can be
i i i
extended to an orthonormalbasis, and the operators whose matrices for this basis
are the A compress to the matrices 1 A′ on the initial Ck, which are linearly
i N2 i
independent. So PVP contains k2 linearly independent matrices, where P is the
orthogonalprojection onto Ck, showing that V has a quantum k-clique.
The vectors v are constructed inductively. Once v ,...,v are chosen so that
i 1 i
A′,...,A′ are independent, future choices of the v’s cannot change this since
1 i
A ,...,A all live on the initial i×i block. We can let v be any nonzero vec-
1 i 1
tor in Ck, since A = e e∗, so that A′ = v v∗ and this only has to be nonzero.
1 1 1 1 1 1
Now suppose v ,...,v have been chosen and we need to select v so that A′ is
1 i−1 i i
independentofA′,...,A′ . Afterchoosingv wewillhaveA′ = ai v v∗.
1 i−1 i i 1≤r,s≤i rs r s
P
A QUANTUM RAMSEY THEOREM 7
Let B be this sum restricted to 1 ≤ r,s ≤ i−1. That part is already determined
since v does not appear. Also let
i
u=ai v +···+ai v ;
1i 1 (i−1)i i−1
then we will have
A′ =B+uv∗+v u∗+v v∗
i i i i i
(using the assumption that ai =1). That is,
ii
A′ =(B−uu∗)+(u+v )(u+v )∗ =B′+u˜u˜∗
i i i
whereu˜=u+v is arbitrary,andthe questionis whether u˜ canbe chosento make
i
this matrix independent of A′,...,A′ . But the possible choices of A′ span M
1 i−1 i k
— there is no matrix which is Hilbert-Schmidt orthogonal to B′ +u˜u˜∗ for all u˜
— so there must be a choice of u˜ which makes A′ independent of A′,...,A′ , as
i 1 i−1
desired. (cid:3)
Next we prove a technical variation on Lemma 4.3.
Lemma 4.4. Let n = k4+k3+k−1 and let V be an operator system contained
in Mn. Suppose V contains matrices A1,...,Ak4+k3 such that for each i we have
hA e ,e i 6= 0, and also hA e ,e i = 0 whenever max{r,s} > i+1 and r 6= s.
i i i+1 i r s
Then V has a quantum k-clique.
Proof. Let A have matrix entries (ai ). Observe that for each i the compression
i rs
of A to span{e ,...,e } is diagonal. For each r > i +1 let the r-tail of A
i i+2 n i
be the vector (airr,...,ainn). Suppose there exist indices i1,...,ik2+k−1 such that
the r-tails of the A , 1 ≤ j ≤ k2 +k −1, are linearly independent, where r =
ij
max {i +2}. Then the compression of V to span{e ,...,e } contains k2+k−1
j j r n
linearly independent diagonal matrices, so it has a quantum k-clique by the first
assertion of Lemma 4.1. Thus, we may assume that for any k2 +k −1 distinct
indices i the matrices A have linearly dependent r-tails.
j ij
We construct an orthonormal sequence of vectors v and a sequence of Her-
i
mitian matrices B ∈ V, 1 ≤ i ≤ k2, such that the compressions of the B to
i i
span{v1,...,vk2,ek4+k3+1,...,ek4+k3+k−1} satisfy the hypotheses of Lemma 4.3.
This will ensure the existence of a quantum k-clique.
The firstk2+k−1 matricesA1,...,Ak2+k−1 havelinearlydependent r-tails for
r =k2+k+1. Thus there is a nontrivial linear combination B′ = k2+k−1α A
1 i=1 i i
whose r-tail is the zero vector. Letting j be the largest index suPch that αj is
nonzero, we have hB′e ,e i 6= 0 because hA e ,e i 6= 0 but hA e ,e i = 0
1 j j+1 j j j+1 i j j+1
for i < j. Thus the compression of B1′ to span{e1,...,ek2+k} is nonzero, so there
existsaunitvectorv inthisspansuchthathB′v ,v i6=0. ThenletB beascalar
1 1 1 1 1
multiple of either the real or imaginary part of B′ which satisfies hB v ,v i = 1.
1 1 1 1
Note that hB e ,e i = 0 for any r,s with max{r,s} > k2 +k. Apply the same
1 r s
reasoning to the next block of k2+k−1 matrices Ak2+k+1,...,A2k2+2k−1 to find
v and B , and proceed inductively. After k2 steps, k2(k2+k) = k4+k3 indices
2 2
will have been used up and k−1 (namely, ek4+k3+1,...,ek4+k3+k−1) will remain,
as needed. (cid:3)
Theorem 4.5. Every operator system in M8k11 has either a quantum k-clique or
a quantum k-anticlique.
8 NIKWEAVER
Proof. Set n = 8k11 and let V be an operator system in M . Find a unit vector
n
v ∈Cn, if one exists, such that the dimension of Vv ={Av :A∈V} is less than
1 1 1
8k8. Then find a unit vector v ∈(Vv )⊥, if one exists, such that the dimension of
2 1
(Vv )⊥∩(Vv ) is less than 8k8. Proceed in this fashion, at the rth step trying to
1 2
find a unit vector v in
r
(Vv )⊥∩···∩(Vv )⊥
1 r−1
such that the dimension of
(Vv )⊥∩···∩(Vv )⊥∩(Vv )
1 r−1 r
is less than8k8. Ifthis constructionlasts for k3 steps then the compressionof V to
span{v1,...,vk3} ∼= Mk3 is contained in Dk3, so this compression, and hence also
V, has either a quantum k-clique or a quantum k-anticlique by Lemma 4.1.
Otherwise,theconstructionfails atsomestaged. Thismeansthatthe compres-
sion V′ of V to F =(Vv )⊥∩···∩(Vv )⊥ has the property that the dimension of
1 d
V′v is at least 8k8, for every unit vector v ∈F.
WorkinF. Chooseanynonzerovectorw ∈F andfindA ∈V′ suchthatw =
1 1 2
A w is nonzero and orthogonal to w . Then find A ∈ V′ such that w = A w
1 1 1 2 3 2 2
is nonzero and orthogonal to span{w ,w ,A w ,A∗w ,A w ,A∗w }. Continue in
1 2 1 1 1 1 1 2 1 2
this way, at the rth step finding A ∈ V′ such that w = A w is nonzero and
r r+1 r r
orthogonalto span{w ,A w ,A∗w :i<r and j ≤r}. The dimension of this span
j i j i j
is at most2r2−r, so as long as r≤2k4 its dimension is less than 8k8 and a vector
w canbefound. Compressingtothespanofthew thenputsusinthesituation
r+1 i
ofLemma4.4withn=2k4,whichismorethanenough. Sothereexistsaquantum
k-clique by that lemma. (cid:3)
The constants in the proof could easily be improved, but only marginally. Very
likely the problem of determining optimal bounds on quantum Ramsey numbers is
open-ended, just as in the classical case.
5. A generalization
In this section we will present a result which simultaneously generalizes the
classical and quantum Ramsey theorems. This is less interesting than it sounds
because the proof involves little more than a reduction to these two special cases.
Perhaps the statement of the theorem is more significant than its proof.
At the beginning of Section 2 we showedhow any simple graph G on the vertex
set{1,...,n}givesriseto anoperatorsystemV ⊆M . This operatorsystemhas
G n
the special property that it is a bimodule over D , i.e., it is stable under left and
n
rightmultiplicationbydiagonalmatrices. Conversely,itisnothardtoseethatany
operator system in M which is also a D -D -bimodule must have the form V
n n n G
for some G ([8], Propositions2.2 and 2.5). The generaldefinition therefore goes as
follows:
Definition 5.1. ([8], Definition2.6 (d)) LetM be a unital ∗-subalgebraofM . A
n
quantum graph on M is an operator system V ⊆M which satisfies M′VM′ =V.
n
Here M′ ={A∈M :AB =BA for all B ∈M} is the commutant of M. This
n
definition is actually representation-independent: if M and N are ∗-isomorphic
unital ∗-subalgebras of two matrix algebras (possibly of different sizes), then the
quantum graphs on M naturally correspond to the quantum graphs on N ([8],
A QUANTUM RAMSEY THEOREM 9
Theorem 2.7). More properly, one could say that the pair (M,V) is the quantum
graph, just as a classical graph is a pair (V,E).
IfM=M thenitscommutantisC·I andthebimoduleconditioninDefinition
n n
5.1isvacuous: anyoperatorsysteminM isaquantumgraphonM . Ontheother
n n
hand, the commutant of M = D is itself, so that by the comment made above,
n
thequantumgraphsonD —theoperatorsystemswhichareD -D -bimodules—
n n n
correspond to simple graphs on the vertex set {1,...,n}. In this correspondence,
subsets of {1,...,n} give rise to orthogonalprojections P ∈D , and the k-cliques
n
andk-anticliquesofthegrapharerealizedinthematrixpictureasrankkorthogonal
projections P ∈ D which satisfy PV P = PM P or PD P, respectively. This
n G n n
suggests the following definition.
Definition 5.2. Let M be a unital ∗-subalgebra of M and let V ⊆ M be a
n n
quantum graph on M. A rank k projection P ∈ M is a quantum k-clique if it
satisfies PVP =PM P and a quantum k-anticlique if it satisfies PVP =PM′P.
n
Since every operator system contains the identity matrix, if V is a quantum
graph on M then M′ ⊆ V. So PVP = PM′P is the minimal possibility, as
PVP = PM P is the maximal possibility. Note the crucial requirement that P
n
must belong to M.
IfM=M thenM′ =C·I andtheprecedingdefinitionduplicatesthenotions
n n
ofquantumk-cliqueandquantumk-anticliqueusedearlierinthe paper,whereasif
M=D it effectively reproduces the classical notions of k-clique and k-anticlique
n
in a finite simple graph. In the classical setting fewer operator systems count as
graphs, but one also has less freedom in the choice of P when seeking cliques or
anticliques.
We require only the following simple lemma.
Lemma 5.3. Let V ⊆ M ∼= M ⊗ M be a quantum graph on M ⊗ I . If
nd n d n d
nd≥8k11 then there is a projection in M ⊗I whose rank is at least k, and which
n d
is either a quantum clique or a quantum anticlique of V.
Proof. SinceV isabimoduleover(M ⊗I )′ =I ⊗M ,ithastheformV =W⊗M
n d n d d
for some operator system W ⊆M . If d=1 then the desired result was proven in
n
Theorem 4.5, and if d ≥ k then any projection of the form P ⊗M , where P is a
d
rank 1 projection in M , will have rank at least k and be both a quantum clique
n
and a quantum anticlique. So assume 2≤d<k.
Now if d≥3 then d10/11 >2, so d<k(d10/11−1). Thus 1< k(d10/11−1), i.e.,
d
k +1< k ·d10/11 = k , which implies (k +1)11 < k11. So finally
d d d1/11 d d
8k11 k 11 k 11
n≥ >8 +1 >8 .
d (cid:18)d (cid:19) (cid:24)d(cid:25)
If d=2 then d<3(d10/11−1), so the same reasoning leads to the same inequality
n ≥ 8⌈k⌉11 provided k ≥ 3, and the inequality is immediate when k = d = 2. So
d
we conclude that in all cases n≥8⌈k⌉11. By Theorem4.5, W has a quantum ⌈k⌉-
d d
cliqueora quantum⌈k⌉-anticliqueQ∈M . ThenQ⊗I is correspondinglyeither
d n d
aquantum ⌈k⌉·d-clique ora quantum⌈k⌉·d-anticliqueofV, whichis enough. (cid:3)
d d
Notethatwecannotpromiseaquantumk-cliqueor-anticlique,onlya≥k-clique
or -anticlique, since the rank of any projection in M ⊗I is a multiple of d.
n d
10 NIKWEAVER
Theorem 5.4. For every k there exists n such that if M is a unital ∗-subalgebra
of M and V ⊆M is an operator system satisfying M′VM′ =V, then there is a
n n
projection P ∈M whose rankis at least k and suchthat PVP =PM P or PM′P.
n
Proof. Let R(k,k) be the classical Ramsey number and set n = 8k11 · R(k,k).
Now M has the form (M ⊗I )⊕···⊕(M ⊗I ) for some pair of sequences
n1 d1 nr dr
(n ,...,n ) and (d ,...,d ) such that n d +···+n d =n. Thus if r ≤ R(k,k)
1 r 1 r 1 1 r r
then for some i we must have n d ≥ 8k11, and compressing to that block then
i i
yieldsthe desiredconclusionby appealingtothe lemma. Otherwise,ifr >R(k,k),
then choose a sequence of rank 1 projections Q ∈M and work in QM Q where
i ni n
Q =(Q ⊗I )⊕···⊕(Q ⊗I ). Then QM Q ∼= M , QMQ∼= C·I ⊕
1 d1 r dr n d1+···+dr d1
···⊕C·I ∼=D ,andQVQisabimoduleoverthecommutantofQMQinQM Q,
dr r n
i.e., the ∗-algebra M ⊕···⊕M . It follows that there is a graph G= (V,E) on
d1 dr
the vertex set V ={1,...,r} such that QVQ has the form
QVQ= E ⊗M ,
ij didj
X
taking the sum over the set of pairs {(i,j):i=j or {i,j}∈E} ([8], Theorem 2.7).
Since r > R(k,k), there exists either a k-clique or a k-anticlique in G, and this
gives rise to a diagonal projection in QMQ whose rank is at least k and which is
either a quantum k-clique or a quantum k-anticlique of V. (cid:3)
Again, whenM=M Theorem5.4recoversthe quantumRamsey theoremand
n
whenM=D itrecoverstheclassicalRamseytheorem(thoughinbothcaseswith
n
worse constants).
Theorem5.4couldalsobeprovenbymimickingthe proofofTheorem4.5. How-
ever,inordertoaccomodatetherequirementthatP belongtoMweneedtomodify
thelastpartoftheproofsoastobesurethateachw belongstoWw ∩···∩Ww .
r 1 r−1
ThismeansthatinsteadofneedingWvtohavesufficientlylargedimensionforeach
v, we need it to have sufficiently small codimension. Ensuring that this must be
the case if the construction in the first part of the proof fails then requires that
construction to take place in a space whose dimension is exponential in k. This
explains the dramatic difference between classical and quantum Ramsey numbers
(the first grows exponentially, the second polynomially).
References
[1] Y.H.Au-YeungandY.T.Poon,Aremarkontheconvexityandpositivedefiniteness
concerningHermitianmatrices,Southeast Asian Bull. Math. 3(1979), 85-92.
[2] R.Duan,S.Severini,andA.Winter,Zero-errorcommunicationviaquantumchannels,
noncommutativegraphs,andaquantumLova´sznumber,IEEETrans.Inform.Theory
59(2013), 1164-1174.
[3] E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A
55(1997), 900-911.
[4] E. Knill,R.Laflamme, and L. Viola,Theory of quantum errorcorrection for general
noise,Phys. Rev. Lett. 84(2000), 2525-2528.
[5] D. Stahlke, Quantum source-channel coding and non-commutative graph theory,
arXiv:1405.5254.
[6] H. Tverberg, A generalization of Radon’s theorem, J. London Math. Soc. 41(1966),
123-128.
[7] ———,AgeneralizationofRadon’stheorem,II,Bull.Austral. Math.Soc. 24(1981),
321-325.
[8] N.Weaver, Quantum relations,Mem. Amer. Math. Soc. 215(2012), v-vi,81-140.
[9] ———,Quantum graphsasquantum relations,arXiv:1506.03892.