Tensor rank is not multiplicative under the tensor product

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Linear

Algebra

and

its

Applications

www.elsevier.com/locate/laa

Tensor

rank

is

not

multiplicative

under

the

tensor product

Matthias Christandla_{, Asger Kjærulff Jensen}a_,

Jeroen Zuiddamb,∗

a_{DepartmentofMathematicalSciences,Universityof Copenhagen,}

Universitetsparken5,2100Copenhagen Ø,Denmark

b_{CentrumWiskunde&Informatica,SciencePark123,1098XGAmsterdam,}

Netherlands

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received29August2017 Accepted19December2017 Availableonline21December2017 SubmittedbyJ.M.Landsberg MSC: 15A69 Keywords: Tensorrank Borderrank Degeneration Youngflattening

Algebraiccomplexitytheory Quantuminformationtheory

Thetensorrankofatensort is thesmallestnumberr such

thatt canbedecomposedasasumofr simple tensors.Let

s beak-tensorandlett be an-tensor.Thetensorproduct of s and t isa(k + )-tensor.Tensorrankissub-multiplicative underthetensorproduct.Werevisittheconnectionbetween restrictionsanddegenerations. Aresult ofour studyisthat tensorrankisnotingeneralmultiplicativeunderthetensor product. Thisanswers a question of Draismaand Sapthar-ishi.Specifically,ifatensort hasborderrankstrictlysmaller thanitsrank,thenthetensorrankoft isnot multiplicative undertakinga sufficientlyhighttensorproductpower. The “tensorKroneckerproduct”fromalgebraiccomplexitytheory isrelatedtoourtensorproductbutdifferent,namelyit mul-tipliestwok-tensorstogetak-tensor.Nonmultiplicativityof thetensorKroneckerproducthasbeenknownsincethework ofStrassen.

Itremainsanopenquestionwhetherborderrankand asymp-toticrankaremultiplicativeunderthetensorproduct. Inter-estingly,lowerboundsonborderrankobtainedfrom

general-* Correspondingauthor.

E-mailaddresses:[email protected](M. Christandl),[email protected](A.K. Jensen),

[email protected](J. Zuiddam).

https://doi.org/10.1016/j.laa.2017.12.020

ized flattenings(includingYoungflattenings)multiplyunder thetensorproduct.

©2017ElsevierInc.Allrightsreserved.

1. Introduction

Let Ui,Vi be finite-dimensional vector spaces over afield F. Let t be a k-tensor in

U1⊗· · ·⊗Uk.Thetensorrank oft isthesmallestnumber r suchthat t canbewrittenas

asumofr simpletensorsu1⊗ · · · ⊗ uk inU1⊗ · · · ⊗ Uk,andisdenotedbyR(t).Letting

F bethecomplexnumbersC,theborder rank of t isthesmallestnumberr suchthatt

is alimitpoint (intheEuclidean topology)ofasequenceof tensorsinU1⊗ · · · ⊗ Uk of

rankatmostr, andisdenotedby R(t).

Lett∈ U1⊗ · · · ⊗ Uk ands∈ V1⊗ · · · ⊗ Vbeak-tensorandan-tensorrespectively.

Define thetensor product oft ands asthe(k + )-tensor

t⊗ s ∈ U1⊗ · · · ⊗ Uk⊗ V1⊗ · · · ⊗ V.

Ifk = ,thendefine thetensorKroneckerproduct oft ands asthek-tensor t s ∈ (U1⊗ V1)⊗ · · · ⊗ (Uk⊗ Vk)

obtained fromt⊗ s by groupingUi andVi together foreach i.Inalgebraic complexity

theory, the tensor Kronecker product is usually justdenoted by ‘⊗’. Using the tensor Kronecker productonedefinestheasymptotic rank oft asthelimitlim_n→∞R(tn)1/n_. (This limit exists and equals the infimum infnR(tn)1/n, see for example Lemma 1.1

in[1].)Asymptoticrankisdenotedby  R(t).

This paper isabouttherelationship betweentensorrankand thetensor product.It follows fromthedefinitionthatrankissub-multiplicativeunderthetensorproduct. Proposition 1.Lett,s be anytensors. Then, R(t⊗ s)≤ R(t)R(s).

Theresultofthispaper isthattheaboveinequalitycanbestrict.

Theorem. Tensorrankisnotingeneralmultiplicativeundertensor product.Specifically, if a tensor t has border rankstrictly smaller than itstensor rank, thenthe tensor rank of t isnotmultiplicative under ataking asufficientlyhigh tensorpower.

Thetheorem answersaquestionposedinthelecture notesofJanDraisma[2, Chap-ter 6]andaquestionofRamprasadSaptharishi (personalcommunication,relatedtoan earlier version of the survey [3]). The theorem was stated as a fact in [4, page 1097], referring to [5]fortheproof;however,[5]studiesonlythetensor Kroneckerproduct .

IthasbeenknownsincetheworkofStrassenthattensorrankisnotmultiplicativeunder thetensorKroneckerproduct ,see Example 3.

Weconstructthreeinstancesofthisphenomenon(Proposition 13,Proposition 17and

Proposition 18) to provethe theorem. Explicitly, one of our examples is the following strictinequality(Proposition 14).

Example2.Letb1,b2 be thestandardbasisof C2. Define the3-tensorW3 as b2⊗ b1⊗

b1+ b1⊗ b2⊗ b1+ b1⊗ b1⊗ b2∈ (C2)⊗3.ThenwehavethestrictinequalityR(W3⊗2)≤ 8< 9= R(W3)2.

In Section5 we will prove thatExample 2 is essentially minimal over the complex numbers, inthe sense that if s ∈ C⊗ C2_{⊗ C}2 _{and t ∈ C}2_{⊗ C}n _{⊗ C}m_{, then one has}

R(s t)= R(s⊗ t)= R(s)R(t). This weprove using thetheory of canonical forms of matrixpencilsandaformulafortheirtensorrank.

Ourgeneralapproachistostudyapproximatedecompositions(orborderrank decom-positions)oftensors.Itturnsoutthataborderrankdecompositionofatensort canbe transformedintoatensorrankdecompositionoftensorpowersoft with apenaltythat dependson theso-called errordegree ofthe approximation. Moreprecisely, thenotion of border rank R(t) has amore precise variant Re(t) thatallows only approximations witherrordegreeatmoste (seeSection2fordefinitions).Thisvariantgoesback to[6]

and[7].WeproveinCorollary 11(1)that

R(s⊗n)≤ (ne + 1)Re(s)n, (1) which we use to construct nonmultiplicativity examples. In particular, we see that as soonas Re_{(s)< R(s), the quantityR(s)}n _{grows faster than theright-hand side of (1)}

andthusleadstononmultiplicativityexamplesforlargeenough n.

Itfollowsfromthedefinitionsthatalsoborderrankandasymptoticrankare submul-tiplicativeunderthetensorproduct:R(t⊗ s)≤ R(t)R(s),and

 R(t⊗ s)≤  R(t)  R(s).We leaveitasanopenquestionwhethertheseinequalitiescanbestrict.InSection4wewill see thatlower bounds on borderrank obtainedfrom generalized flattenings(including Youngflattenings)areinfactmultiplicativeunderthetensorproduct.

It follows from R(t s) ≤ R(t⊗ s) that tensor rank, border rank and asymptotic rank are submultiplicative underthe tensor Kronecker product: R(t s) ≤ R(t)R(s), R(t s)≤ R(t)R(s),and  R(t s)≤  R(t) 

R(s).Ift and s are2-tensors(matrices), then tensor rank, border rank and asymptotic rank are equal and multiplicative under the tensor Kronecker product.However, for k≥ 3, it is well-known thateach of thethree inequalitiescanbe strict,seethefollowingexample.

Example3.Considerthefollowingtensors T = 

i∈{1,2}

T =  i∈{1,2} bi⊗ 1 ⊗ bi ∈ F2⊗ F ⊗ F2, T   =  i∈{1,2} 1⊗ bi⊗ bi ∈ F ⊗ F2⊗ F2.

(This graphical notationis borrowedfrom [8].) Eachtensor hasrank, borderrank and asymptotic rank equal to 2, since they are essentially identity matrices. However the tensor Kroneckerproductisthe2× 2 matrixmultiplication tensor

2, 2, 2 = T = 

i,j,k∈{1,2}

(bi⊗ bj)⊗ (bj⊗ bk)⊗ (bk⊗ bi)

whose tensor rankandborder rankisat most7[9]and whoseasymptotic rankisthus at most 7, whichis strictly lessthat23_{= 8. (The tensorrank of2,2,2 equals 7over} any field[10]and theborderrankof2,2,2 equals7overthecomplexnumbersC[11]. Both statementsareinfacttrueforany tensorwiththesamesupportas2,2,2[12].) 2. Degenerationandrestriction

Werevisitthetheoryofdegenerationsandrestrictionsoftensorsandhowtotransform degenerations into restrictions.Ournon-multiplicativityresultsrelyonthese ideas.Let

t ∈ U1⊗ · · · ⊗ Uk and s ∈ V1⊗ · · · ⊗ Vk be k-tensors. We sayt restricts to s, written

t≥ s,ifthere arelinearmapsAi: Ui→ Vi suchthat(A1⊗ · · · ⊗ Ak)t= s.Letd,e∈ N.

We say t degenerates to s with approximation degree d and error degree e, written

te_ds,if thereare linearmaps Ai(ε): Ui→ Vi depending polynomiallyon ε such that

(A1(ε)⊗· · ·⊗Ak(ε))t= εds+ εd+1s1+· · ·+εd+eseforsometensorss1,. . . ,se.Naturally,

te_{s means∃d: t}e

ds,andtds means∃e: teds,andt s means∃d∃e: teds.(We

note that our notation tds corresponds to td+1s in [5].) Clearly, degeneration is

multiplicativeinthefollowing sense.

Proposition 4.Lett1,t2,s1,s2 betensors. Ift1ed11s1 andt2

e2

d2s2,thent1⊗ t2

e1+e2

d1+d2 s1⊗ s2 andt1 t2de11+e+d22s1 s2.

Theerrordegreee is upperboundedbytheapproximation degreed inthefollowing way.

Proposition 5.Lett,s be k-tensors.If tds,then tkd−d_d s.

Proof. Suppose (A1(ε)⊗ · · · ⊗ Ak(ε))t = εds+ εd+1s1+· · · + εd+ese. For every i let

Bi(ε) bethematrixobtainedfromAi(ε) by truncatingeachentryinAi(ε) todegreeat

most d.Then (B1(ε)⊗ · · · ⊗ Bk(ε))t= εds+ εd+1u1+· · · + εkdukd for somek-tensors

Foranyr∈ N, letb1,. . . br denotethestandardbasisofFr.Letr,k∈ N and let Tr(k) := r  i=1 (bi)⊗k ∈ (Fr)⊗k

be the rank-r order-k unit tensor. Let s ∈ V1⊗ · · · ⊗ Vk. The tensor rank of s is the

smallestnumber r suchthatTr(k)≥ s,andisdenotedbyR(s).Thisdefinitionoftensor

rankiseasilyseentobeequivalenttothedefinitiongivenintheintroduction.Theborder rankof s is the smallestnumber r suchthat Tr(k) s,and is denoted byR(s). Note

thatthis definition worksover any fieldF. WhenF equals C, this definition of border rankis equivalenttothedefinitiongivenintheintroduction[13–15,5].Define

R_de(s) := min{r ∈ N | Tr(k)eds}

Rd(s) := min{r ∈ N | Tr(k)ds}

Re(s) := min{r ∈ N | Tr(k)es}.

(OurnotationRd(s) correspondstoRd+1(s) in[5].)Errordegreeinthecontextofborder

rankwasalreadystudiedin[6]and [7].The followingpropositions followdirectlyfrom

Proposition 4andProposition 5. Proposition6.Re1+e2 d1+d2(s1⊗ s2)≤ R e1 d1(s1)R e2 d2(s2).

Proposition7.Lets be ak-tensor.ThenRd(s)= Rkd−d_d (s).

The following theorem is our main technical result on which the rest of the paper rests.WenotethatforthetensorKroneckerproductthestatementiswell-knowninthe contextofalgebraic complexitytheory[6,7,16,17,8].

Theorem8. Lett,s bek-tensors.Ifte_{s and|F|≥ e+ 2,thenwehavet T}

e+1(k)≥ s.

Proof. Byassumptiontherearematrices Ai(ε) withentriespolynomialinε suchthat



A1(ε)⊗ · · · ⊗ Ak(ε)



t = εds + εd+1s1+· · · + εd+ese

forsometensorss1,. . . ,se.Multiplybothsidesbyε−dandcalltheright-handsideq(ε),



ε−dA1(ε)⊗ · · · ⊗ Ak(ε)



t = s + εs1+· · · + εese=: q(ε).

Let α0,. . . ,αe be distinct nonzero elements of the ground fieldF (by assumption our

groundfield islarge enoughto do this). Viewq(ε) as apolynomialinε. Write q(ε) as

q(ε) = e  j=0 q(αj)  0≤m≤e: m=j ε− αm αj− αm .

Wenow seehowtowrite q(0) asalinearcombinationoftheq(αj),namely

q(0) = e  j=0 q(αj)  0≤m≤e: m=j αm αm− αj , thatis, q(0) = e  j=0 βjq(αj) with βj:=  0≤m≤e: m=j αm αm− αj .

Now we wantto write s as arestriction of t Te+1(k). Define the linear maps B1 := e

j=0βj α−dj A1(αj)⊗ b∗j and Bi := _j=0e βj Ai(αj)⊗ bj∗ for i∈ {2,. . . ,k}.Then t

Te+1(k)≥ s because (B1⊗ · · · ⊗ Bk)(t Te+1(k)) = e  j=0 βj  α−d_j A1(αj)⊗ · · · ⊗ Ak(αj)  t = e  j=0 βjq(αj) = q(0) = s.

This finishestheproof. 2

Remark9.InthestatementofTheorem 8weassumethat|F| islargeenough.Forsmall fields onecandothefollowing.Fork,d∈ N, let[0..d] denote theset{0,1,2,. . . ,d} and

define thek-tensor

χd(k) :=  a∈[0..d]k_: a1+···+ak=d ba1⊗ · · · ⊗ bak∈ (F d₎⊗k_.

Lett,s bek-tensors.Itisnothardtoshowthat,iftds,thentχd(k)≥ s.Bydefinition

of χd(k) wehaveR(χd(k))≤

_k+d−1

k−1



.Wemaythusconcludethatt Tk+d−1

k−1 (k)≥ s.

Wecollectseveralalmost immediatecorollaries.

Corollary 10.Letti,si beki-tensorsfori∈ [n]. AssumeF is largeenough.

1. If∀i: tieisi,then(t1⊗ · · · ⊗ tn) T _iei+1( iki)≥ s1⊗ · · · ⊗ sn. 2. If∀i: tidisi,then(t1⊗ · · · ⊗ tn) T i(ki−1)di+1( iki)≥ s1⊗ · · · ⊗ sn.

Proof. To prove the first statement, apply Proposition 4 to obtain the degeneration

t1⊗· · ·⊗tn

iei_s

1⊗· · ·⊗sn.Theorem 8yieldstheresult.Toprovethesecondstatement,

Proposition 5givestikidi−disi.ByProposition 4,t1⊗ · · · ⊗ tn

ikidi−di_s

1⊗ · · · ⊗ sn.

Theorem 8provesthestatement. 2

Corollary11. Lets be ak-tensor.Assume F islargeenough.

1. R(s⊗n)≤ (ne+ 1)Re(s)n.

2. R(s⊗n)≤ ((k − 1)nd+ 1)Rd(s)n.

Proof. Thisfollows fromCorollary 10. 2 Corollary12. Lets be ak-tensor.

1. limn→∞R(s⊗n)1/n≤ R(s).

2. limn→∞R(s⊗n)1/n= limn→∞R(s⊗n)1/n.

3. IfR(s)< R(s), thenforsome n∈ N,R(s⊗n)< R(s)n_.

3. Tensorrankisnotmultiplicativeunderthetensor product

Because of Corollary 11, in order to find nonmultiplicativity examples,it is enough to find a tensor t for which Re_{(t) < R(t). We will give three families of examples of}

nonmultiplicativity.Fork≥ 3,definethek-tensor Wk :=  i∈{1,2}k_: type(i)=(k−1,1) bi1⊗ · · · ⊗ bik ∈ (F 2₎⊗k_,

wheretype(i)= (k− 1,1) means thati isapermutationof(1,1,. . . ,1,2).

Proposition 13.Let |F| be large enough. Let k ≥ 3. For n large enough, we have a strict inequalityR(W_k⊗n)< R(Wk)n. Forexample, R(W3⊗7)< R(W3)7 and R(W8⊗2)< R(W8)2.

Proof. The rank of Wk equals k. This canbe shownwith the substitution methodas

explainedinforexample[18]. However,Rk−1(Wk)≤ 2,namely

_{1 1} ε 0  ⊗ · · · ⊗1 1 ε 0  ⊗1−1 ε 0  T2(k) = εWk+ ε2(· · ·) + · · · + εk(b2⊗ · · · ⊗ b2). ApplyingCorollary 11(1)tothisdegenerationgivesR(W_k⊗n)≤ (n(k − 1)+ 1)2n_.

There-fore,forn largeenough,R(W_k⊗n)≤ 2n_{(n(k− 1)+ 1)< k}n_{= R(W}

k)n. 2

Infact,ifchar(F)= 2 and√2∈ F, thenwe candirectlyshowastrict inequalityfor

Proposition 14.R(W₃⊗2)≤ 8< 9= R(W3)2 if charF= 2 and

√

2∈ F.

Proof. As mentionedintheproofofProposition 13,R(W3)= 3.Ifc∈ F\ {0} suchthat

√ c∈ F,thenR(W3+ cb2⊗ b2⊗ b2)≤ 2.Namely, W3+ c b2⊗ b2⊗ b2= 1 2√c  (b1+ √ c b2)⊗3− (b1− √ c b2)⊗3  .

(Over C thisalso follows from thefact thattheCayley hyperdeterminant evaluated at

W3+ cb2⊗b2⊗b2isanonzeroconstanttimes c.Onemayalsoseethisbynotingthatthe imageofW3+ cb2⊗b2⊗b2underthemomentmap liesoutsidetheimageofthemoment

polytope associatedto theorbitGL2× GL2× GL2· W [19,20].)WeexpandW3⊗ W3 as

W3⊗ W3=  W3+ b2⊗ b2⊗ b2 ⊗2₋ W3+1₂b2⊗ b2⊗ b2  ⊗ b2⊗ b2⊗ b2 − b2⊗ b2⊗ b2⊗  W3+1₂b2⊗ b2⊗ b2  .

Bytheabove,weknowthattherankofW3+b2⊗b2⊗b2andtherankofW3+1₂b2⊗b2⊗b2 are atmost2.Therefore,therankofW3⊗ W3 isat most22+ 2+ 2= 8. 2

Remark 15.LetSk be thesymmetric groupof order k. Clearly thetensor W3⊗ W3 is invariant underthe action of the subgroup S3× S3 ⊆ S6 and under the action of the permutation (14)(25)(36) ∈ S6 that swaps thetwo copies of W3. Remarkably, the de-compositionofW3⊗ W3givenintheproofofProposition 14alsohasthissymmetry,in thesensethattheaboveactionsleavethesetofsimpletermsappearinginthe decompo-sitioninvariant.Thedecompositionissaidtobe partiallysymmetric.Infact,eachterm is itselfinvariantunderS3× S3.

Remark 16.Itis stated in[21] thatR(W3 W3)= 7, whichimplies thatR(W3⊗ W3) equals 7 or 8. We obtained numerical evidence pointing to 8. After the first version of our manuscript appeared on the arXiv, Chen and Friedland delivered a proof that R(W3⊗ W3)≥ 8[22].Forthethirdpower,itisknownthatR(W3 W3 W3)= 16[23]. A similar constructionas intheproofof Proposition 14givesR(W3⊗ W3⊗ W3)≤ 21. This upperboundisimprovedto 20in[22].

In Proposition 13,wetook thenth powerofatensorin(F2)⊗k withn large enough depending on k. In our next example, we takethe square of a tensor in (Fd₎⊗k _with

d≥ 8.Fork≥ 3 andq≥ 1,definethetensor

Strk_q :=

q+1



i=2

bi⊗ bi⊗ b1⊗ b⊗k−31 + b1⊗ bi⊗ bi⊗ b⊗k−31 ∈ (Fq+1)⊗k.

This tensorisnamedafter Strassen,whousedStr3q toderive theupperboundω≤ 2.48

Proposition 17. Assume that F is large enough. For q ≥ 7 and any k ≥ 3, we have a strictinequalityR((Strk_q)⊗2)< R(Strk_q)2.

Proof. TherankofStrk_qequals2q,againbythesubstitutionmethod.WehaveR1_(Strk q)≤

q+1,seetheproofofProposition 31in[24].ApplyingCorollary 11(1)tothisdegeneration givesR((Strk_q)⊗n)≤ (n+ 1)(q + 1)n.Therefore, forq≥ 7 andn= 2,wehavethestrict inequalityR((Strk_q)⊗2)≤ 3(q + 1)2_{< (2q)}2_{= R(Str}k

q)2. 2

Ourthirdexampleusesmatrixmultiplicationtensors. Let n1,n2,n3∈ N.Define the 3-tensor n1, n2, n3 :=  i∈[n1]×[n2]×[n3] (bi1⊗ bi2)⊗ (bi2⊗ bi3)⊗ (bi3⊗ bi1) ∈ (Fn1⊗ Fn2₎⊗ (Fn2⊗ Fn3₎⊗ (Fn3⊗ Fn1_).

Proposition 18.Assume that F islarge enough. Forn≥ 78, wehave a strictinequality

R(2,2,4⊗n)< R(2,2,4)n.

Proof. The rank of 2,2,4 equals 14 over any field [25, Theorem 2]. On the other hand,R4_{(2,2,4) ≤ 13 over any field [26, Theorem 1]. Thus, when F is large enough}

Corollary 11(1) implies,forn≥ 78,thestrict inequalityR(2,2,4⊗n)≤ 13n(4n+ 1)<

14n_{= R(2,2,4)}n_{. 2}

In the language of graph tensors [8], Proposition 18 says that tensor rank is not multiplicativeundertakingdisjoint unionsofgraphs.

4. Generalized flatteningsaremultiplicative

Intheprevioussectionwehaveseenthattensorrankcanbestrictlysubmultiplicative underthetensorproduct.Wedonotknowwhetherthesameistrueforborderrank.In fact,inthissectionwe observethatlower boundsonborderrankobtainedfrom gener-alized flatteningsaremultiplicative.Inthissection wefocuson3-tensorsfornotational convenience.Theideasdirectlyextendtok-tensors foranyk.

Lett beatensor inV1⊗ V2⊗ V3.Wecantransformt intoamatrixbygroupingthe tensorlegsinto twogroups

V1⊗ V2⊗ V3→ V1⊗ (V2⊗ V3)

v1⊗ v2⊗ v3→ v1⊗ (v2⊗ v3).

(There are three ways to do this for a3-tensor.) This is called flattening. Therank of aflatteningof t is alower bound for theborder rank oft. (Rankand border rankare equalformatrices.)

Wenowdefine generalized flattenings.Lett beatensorinV1⊗ V2⊗ V3.Insteadofa basicflatteningV1⊗ V2⊗ V3→ V1⊗ (V2⊗ V3),wechoosevectorspacesV1 andV2 and apply somelinearmap F : V1⊗ V2⊗ V3→ V1 ⊗ V2 to t.Toobtainaborderranklower bound usingF wehavetocompensateforthefactthatF possiblyincreasestheborder rankofasimpletensor. Thefollowinglemmadescribestheresultinglowerbound. Lemma 19.Lett∈ V1⊗ V2⊗ V3 be atensor.Let

F : V1⊗ V2⊗ V3→ V1 ⊗ V2

be alinearmap. The borderrankof t is atleast

R(t)≥ R(F (t))

max R(F (v1⊗ v2⊗ v3))

, (2)

where themaximumisover allsimple tensorsv1⊗ v2⊗ v3 inV1⊗ V2⊗ V3.

Proof. Suppose R(t) = r. Then there is a sequence of tensors ti converging to t

with R(ti) ≤ r for each i. Each ti thus has a decomposition into simple tensors

ti =

r

j=1ti,j. Since F (ti)→ F (t), there exists an i0 suchthat for alli ≥ i0 we have R(F (ti))≥ R(F (t)). Moreover, wehavethe inequalitiesR(F (ti))≤

r

j=1R(F (ti,j))≤

r· maxsR(F (s)), where the maximum is over all simple tensors s. We conclude that

R(t)≥ R(F (t))/maxsR(F (s)). 2

Note thattheright-handsideof(2)mightnotbeaninteger.Thelowerboundin(2)

is multiplicativeunderthetensorproductinthefollowing sense.

Proposition 20. Let s ∈ V1⊗ V2 ⊗ V3 and t ∈ W1 ⊗ W2 ⊗ W3 be tensors. Let F1 :

V1⊗ V2⊗ V3→ V1 ⊗ V2 andF2: W1⊗ W2⊗ W3→ W1 ⊗ W2 belinearmaps. Theborder

rankof s⊗ t∈ V1⊗ V2⊗ V3⊗ W1⊗ W2⊗ W3 isatleast R(s⊗ t) ≥ R(F1(s))

max R(F1(v1⊗ v2⊗ v3))

R(F2(t))

max R(F2(w1⊗ w2⊗ w3))

where themaximizations areoversimple tensorsinV1⊗ V2⊗ V3 andinW1⊗ W2⊗ W3

respectively.

Proof. Combine F1 andF2 intoasinglelinearmap

F : V1⊗ V2⊗ V3⊗ W1⊗ W2⊗ W3→ (V1 ⊗ W1 )⊗ (V2 ⊗ W2 ).

One then follows the proof of Lemma 19 and usesthe fact thatmatrix rank is multi-plicativeunderthetensorKroneckerproduct. 2

Youngflattenings [27,28]are aspecial caseof generalized flattenings. For complete-ness, we finish with a concise description of Young flattenings and the corresponding multiplicativitystatement.Weworkover thecomplexnumbersC. LetSλV bean

irre-ducibleGLV-module of type λ. Considerthe space V ⊗ SλV as a GLV-module under

thediagonalaction.ThePierirule saysthatwehaveaGLV-decomposition

V ⊗ SλV ∼=

μ

SμV,

where thedirect sumis over partitionsμ oflength at mostdim V obtained from λ by

adding a box in the Young diagram of λ. This decomposition yields GLV-equivariant

embeddingsSμV → V ⊗SλV ,calledPieriinclusions orpartialpolarizationmaps.These

mapsareuniqueuptoscaling.SuchaPieriinclusioncorrespondstoaGLV-equivariant

mapφμ,λ: V∗→ SμV∗⊗ SλV . Everyelementφμ,λ(v) iscalledaPierimap.The Young

flattening Fμ,λonV1⊗ V2∗⊗ V3isobtainedbyfirstapplyingthemap φμ,λtoonetensor

leg,

V1⊗ V2∗⊗ V3→ V1⊗ SμV2∗⊗ SλV2⊗ V3, andthenflatteningintoamatrix,

V1⊗ SμV2∗⊗ SλV2⊗ V3→ (V1⊗ SμV2∗)⊗ (SλV2⊗ V3).

Notethatfor anysimple tensorv1⊗ v2⊗ v3,therankof Fμ,λ(v1⊗ v2⊗ v3) equalsthe rankof φμ,λ(v2).Proposition 20thusspecializes asfollows.

Proposition21.Lets∈ V1⊗ V2⊗ V3andt∈ W1⊗ W2⊗ W3.Letλ,μ andν,κ bepairsof

partitionsasabove. Theborder rankofs⊗ t∈ V1⊗ V2⊗ V3⊗ W1⊗ W2⊗ W3 isatleast R(s⊗ t) ≥ R(Fμ,λ(s))

max R(φμ,λ(v2))

R(Fν,κ(t))

max R(φν,κ(w2))

wherethemaximizations are overv2∈ V2 andw2∈ W2 respectively. Wereferto [29]foranoverviewof theapplications ofYoungflattenings. 5. Multiplicativityforcomplexmatrix pencilsand2-tensors

Inthissectionallvectorspacesareoverthecomplexnumbers.Thegoalofthissection istoprovethefollowingproposition.

Proposition22. Lets∈ C⊗ Cd⊗ Cd andt∈ C2⊗ Cn⊗ Cm.Then R(t s) = R(t ⊗ s) = R(t)R(s).

Remark23.Proposition 22showsthatExample 2isessentiallyminimaloverthecomplex numbers.Namely,anyexampleofnon-multiplicativityoftensorrankunder⊗ musteither be with a5-tensorin(Cd_{⊗ C}d_{)⊗ (C}d1⊗ Cd2 ⊗ Cd3_{) with d}

1,d2,d3 ≥ 3,d≥ 2 or ina tensor space oforder 6or more.Moreover, onecanshow usingProposition 22 and the well-knownclassificationoftheGL×3₂ -orbitsinC2_{⊗ C}2_{⊗ C}2_{thatifs,t∈ C}2_{⊗ C}2_{⊗ C}2 and R(s⊗ t)< R(s)R(t),thens and t arebothisomorphicto thetensorW3.

TheelementsofC2_⊗Cn_⊗Cm_{areoftencalledmatrixpencils.Thetensorrankofmatrix}

pencilsiscompletelyunderstood,inthesensethateverymatrixpencilisequivalentunder localisomorphismstoapencilincanonicalform,forwhichtherankisgivenbyasimple formula. Thisformulawillallowusto giveashortproof ofProposition 22.

We begin with introducing the canonical form for matrix pencils. For a proof we referto[30, Chapter XII].RecallthatthestandardbasiselementsofCnaredenotedby

b1,. . . ,bn.

Definition 24.Given ti ∈ U ⊗ Vi⊗ Wi, define diagU(t1,. . . ,tn) astheimage of

n i=1ti

under the natural inclusion _i(U ⊗ Vi⊗ Wi) → U ⊗

 iVi  ⊗iWi  . For ε ∈ N

define thetensorLε∈ C2⊗ Cε⊗ Cε+1 by

Lε:= b1⊗ ε  i=1 bi⊗ bi + b2⊗ ε  i=1 bi⊗ bi+1 = b1⊗ ⎛ ⎝ 1 0 1 0 . .. ... 1 0 ⎞ ⎠ + b2⊗ ⎛ ⎝ 0 1 0 1 .. . . .. 0 1 ⎞ ⎠ and forη∈ N definethetensorNη ∈ C2⊗ Cη+1⊗ Cη by

Nη:= b1⊗ η  i=1 bi⊗ bi + b2⊗ η  i=1 bi+1⊗ bi = b1⊗ ⎛ ⎜ ⎜ ⎝ 1 1 . .. 1 0 0 ··· 0 ⎞ ⎟ ⎟ ⎠ + b2⊗ ⎛ ⎜ ⎜ ⎝ 0 0 ··· 0 1 1 . .. 1 ⎞ ⎟ ⎟ ⎠ .

Theorem25(Canonicalform).Lett∈ C2⊗ Cn⊗ Cm.Thereexistinvertiblelinearmaps A ∈ GL2,B ∈ GLn and C ∈ GLm and natural numbers ε1,. . . ,εp,η1,. . . ,ηq ∈ N and

an ×  Jordanmatrix F suchthat,with M = b1⊗ I+ b2⊗ F ,wehave

(A⊗ B ⊗ C)t = diag_C2(0, L_ε1, . . . , L_εp, N_η1, . . . , N_ηq, M ), (3) where the0 standsforsome0-tensor of appropriatedimensions. The right-hand sideof (3) iscalled thecanonicalformof t.

Nextwegiveaformulaforthetensorrankofmatrixpencilsincanonicalform( The-orem 27).Theorem 27 isdue to Grigoriev [31], JáJá [32] and Teichert[33],see also [5, Theorem 19.4]or[29, Theorem 3.11.1.1].

Definition26.LetF beaJordanmatrixwitheigenvaluesλ1,λ2,. . . ,λp.Letd(λi) bethe

numberof Jordan blocks inF of size at leasttwo with eigenvalue λi. Define m(F ):=

maxid(λi).

Theorem 27.Let t = diag_C2(0,Lε1,. . . ,Lεp,Nη1,. . . ,Nηq,b1⊗ I+ b2⊗ F ) be atensor incanonicalformas in(3).The tensor rankof t equals

R(t) = p  i=1 (εi+ 1) + q  i=1 (ηi+ 1) +  + m(F ).

Example28.LetW3= b2⊗ b1⊗ b1+ b1⊗ b2⊗ b1+ b1⊗ b1⊗ b2∈ (C2)⊗3 asinExample 2. ThecanonicalformofW3 is

W3∼= b1⊗  1 0 0 1  + b2⊗  0 1 0 0  ,

sointhenotationofTheorem 25wehavep= q = 0 andF =0 1_{0 0}.Wecanthus apply

Theorem 27with= 2 andm(F )= 1 togetR(W3)= 2+ 1= 3.

Wearenowreadyto givetheshort proofofProposition 22.

Proof ofProposition 22. Lets∈ C⊗ Cd_{⊗ C}d_{,t∈ C}2_{⊗ C}n_{⊗ C}m_{. Wemayassumethat}

s= 1⊗ r_i=1bi⊗ bi withr = R(s).ByTheorem 25wemayassumethatt isincanonical

form,t = diag_C2(0,Lε1,. . . ,Lεp,Nη1,. . . ,Nηq,M ). The tensorKronecker product t s

isisomorphicto

t s ∼= diag_C2(t, . . . , t

  

r

).

Byanappropriatelocal basistransformation weputthis incanonicalform

t s ∼= diag_C2(L⊕r_ε1, . . . , L⊕r_εp, N_η⊕r₁ , . . . , N_η⊕r_q , M⊕r),

whichbyTheorem 27hasrankr· R(t)= R(s)R(t). 2

Remark29.Proposition 22isalsotrueoverthefinitefieldFqwhenq≥ n,m.Toseethis

onemayuse theformulafrom [5, Section 19.5]fortherankofpencilsover finitefields, whichforq≥ n,m isasfollows:

R(t) = p  i=1 (εi+ 1) + q  i=1 (ηi+ 1) +  + δ(B).

HereB istheregularpart ofthepencilt andδ(B) isthenumberofinvariantdivisors of B thatdonotdecompose intoaproduct ofunassociated linearfactors. (Wereferto [5]

for definitions.)The invariant divisorsof diag(B,. . . ,B) arejustthe invariant divisors of B countedforeachcopyofB andsoProposition 22follows.

Wenote thatpartoftheresultsinthissectionhavebeen independentlyobtainedin Section 2 of[22].

Acknowledgements

WethankJonathanSkowerafordiscussion,FulvioGesmundoforsuggestions regard-ing Section 4, and Nick Vannieuwenhoven for discussion regarding the literature. We acknowledgefinancialsupportfromtheEuropeanResearchCouncil(ERCGrant Agree-ment no. 337603), the Danish Council for Independent Research (Sapere Aude), and VILLUM FONDENviatheQMATHCentreofExcellence(Grantno. 10059).JZis sup-portedbyNWO(617.023.116)andtheQuSoftResearchCenterforQuantumSoftware. References

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