Induced Metric And Matrix Inequalities On Unitary Matrices

0.15 MB·English

by  H. F. Chau

Induced Metric And Matrix Inequalities On Unitary Matrices
H. F. Chau,1,∗ Chi-Kwong Li,2,† Yiu-Tung Poon,3,‡ and Nung-Sing Sze4,§
1Department of Physics and Center of Theoretical and Computational Physics,
University of Hong Kong, Pokfulam Road, Hong Kong
2Department of Mathematics, College of William & Mary, Williamsburg, VA 23187-8795, USA¶
3Department of Mathematics, Iowa State University, Ames, IA 50011, USA
4Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
(Dated: January 24, 2012)
Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] showed that one can define certain
metricsandpseudo-metricsonU(n),thegroupofalln×nunitarymatrices,basedonthearguments
oftheeigenvaluesoftheunitarymatrices. Moreimportantly,thesemetricsandpseudo-metricshave
2 quantum information theoretical meanings. So it is instructive to study this kind of metrics and
1 pseudo-metrics on U(n). Here we show that any symmetric norm on Rn induces a metric on
0 U(n). Furthermore, using the same technique, we prove an inequality concerning the eigenvalues
2 of a product of two unitary matrices which generalizes a few inequalities obtained earlier by Chau

n [arXiv:1006.3614v1].
a
J PACSnumbers: 02.10.Yn,03.65.Aa,03.67.Mn

1
2
I. INTRODUCTION 3. f(Y−1XY) = f(X) for all X,Y ∈ U(n). The ra-

] tionaleisthatthecosttoevolveaquantumsystem
h
should be eigenbasis independent. Although this
In quantum information science, it is instructive to
p
assumption is questionable for bipartite systems,
- measure the cost needed to evolve a quantum system [1]
h we will stick to it in this paper for the evolution
as well as to quantify the difference between two quan-
t cost for monopartite system is already a worthy
a tum evolutions on a system [2]. To some extent, the
m solutions of both problems are closely related to cer- topics to investigate.
[ tain pseudo-metric functions on unitary operators. To 4. f(XY) ≤ f(X)+f(Y) for all X,Y ∈ U(n). The
2 see this, suppose we are given a certain quantifiable cost reason behind is that a possible way to implement
required to implement a unitary operation acting on an
v XY is to first apply Y then follow by X. If we
n-dimensionalHilbertspace. We mayrepresentthis cost
7 further demand that the cost is additive (in the
by a non-negative function f: U(n)→R, where U(n) is
4 sense that the cost of applying Y and then X is
0 the group of all n×n unitary matrices. The larger the equal to the cost of applying Y plus the cost of
1 value of f(X), the higher the cost of implementing the
applyingX),whichisnotanunreasonabledemand
. unitary operation X. Besides, f(X) = 0 if it is costless
7 after all, then the inequality follows.
0 to performX. We may further requirethis costfunction
1 f to satisfy the following constraints. Acostfunctionf inducesafunctiond: U(n)×U(n)→R
1 bytheequationd(X,Y)=f(XY−1)forallX,Y ∈U(n).
: Constraints for the cost function f:
v Surely, d(X,Y) can be regarded as the cost needed to
Xi 1. f(eirX) = f(X) for all r ∈ R and X ∈ U(n). In transform Y to X. In this respect, the induced function
dprovidesapartialanswertotheproblemofquantifying
addition, f(I) = 0. The underlying reason is that
r the difference between two quantumevolutions on a sys-
a changing the global phase of X has no effect on
tem. The larger the value of d(X,Y), the more different
the quantum system. Besides, the identity opera-
the quantum operations X and Y is. More importantly,
tiondoesnotchangeany quantumstate andhence
since f obeys the above four constraints, d(·,·) must be
should be costless.
pseudo-metric on U(n) because it satisfies d(X,Y) ≥ 0,
2. f(X−1)= f(X) for all X ∈U(n). This is because d(X,Y) = d(Y,X) and d(X,Z) ≤ d(X,Y)+d(Y,Z) for
X−1 can be implemented by running the quantum all X,Y,Z ∈ U(n). Nevertheless, d(·,·) is not a metric
for d(X,Y)=0 does not imply X =Y. We remark that
circuitfor X backwardin time with the same cost.
the induced d also obeys d(ZX,ZY) = d(X,Y) for all
X,Y,Z ∈U(n).
Conversely,supposethereisapseudo-metricdonU(n)
quantifying the difference between two unitary opera-
∗ Correspondingauthor,[email protected]
tionsactingonan-dimensionalquantumsystem. Surely,
† [email protected]
‡ [email protected] it should satisfy d(X,X)=0, d(eirX,Y)=d(X,Y) and
§ [email protected] d(X,Y)=d(ZX,ZY)forallr ∈R,X,Y,Z ∈U(n). The
¶ (in the spring of 2012) Department of Mathematics, University reasonisthatthedifficultyindistinguishingbetweentwo
ofHongKong,PokfulamRoad,HongKong unitary operations is unchanged by varying the global
2
phase in one of the operations and by applying a com- ber of quantum information science questions are re-
mon quantum operation to them. (Again, this reason is lated to the costfunction f (or equivalently, the pseudo-
valid as we restrict our study to monopartite systems.) metric or its “un-optimize” metric version d). Besides,
Moreimportantly,dinducesthefunctionf(X)=d(X,I) the third constraint for f, namely, f(Y−1XY) = f(X)
on U(n) which obeys the four constraints on f. (The for all X,Y ∈ U(n), implies that the cost function f
second constraint follows from f(X−1) = d(X−1,I) = depends on the eigenvalues of its input argument only.
d(XX−1,X) = d(I,X) = d(X,I) = f(X). And the Equivalently,itmeansthatthecorrespondingmetricand
other three constraints can be proven in a similar way.) pseudo-metric d(X,Y)’s on U(n) are functions of the
To summarize, we have argued that the cost function f eigenvalues of XY−1 only.
describing the resources required to evolve a (monopar- Inthispaper,weadoptthefollowingstrategytoinves-
tite)quantumsystemisequivalenttoquantifyingthedif- tigate the problem of metrics, pseudo-metrics and their
ferencebetweentwoquantumevolutionsona(monopar- relation with quantum information science. We begin
tite) systemthroughthe inducedpseudo-metricfunction byfindingmetricsandpseudo-metricsd(X,Y)’sonU(n)
d. And we remark on passing that our discussions so that are functions of the eigenvalues of XY−1 only by
fararevalidforinfinite-dimensionalquantumsystemsas means of Proposition 2. More precisely, we prove that a
well. symmetric norm of Rn induces a metric and a pseudo-
Recently, Chau [3, 4] introduced a family of costfunc- metric on U(n) of the required type. We then show in
tionsonU(n)basedonatightquantumspeedlimitlower Example 4 that some of the new metrics and pseudo-
boundontheevolutiontimeofaquantumsystemhedis- metrics discovered in this way indeed have quantum in-
coveredearlier[5]. Inquantuminformationscience,these formation science meanings. Interestingly, Proposition 2
cost functions can be interpreted as the least amount of has merit on its own for we can adapt its proof to show
resources (measured in terms of the product of the evo- an inequality concerning the eigenvalues of a product of
lution time and the average absolute deviation from the two unitary matrices. This inequality is a generalization
medianoftheenergy)neededtoperformaunitaryopera- ofseveralinequalities first proveninRef. [3] using eigen-
tionX ∈U(n)[4]. Withtheabovequantuminformation value perturbation technique. Finally, we briefly discuss
science meaning in mind, it is not surprising that each theconnectionofourfindingsandtheHorn’sproblemon
costfunctioninthisfamilydependsonlyontheeigenval- eigenvalueinequalitiesforthesumofHermitianmatrices.
ues of its input argument X. Actually, it can be written
as a certain weighted sum of the absolute value of the
argument of the eigenvalues of X [3, 4]. II. METRIC AND PSEUDO-METRIC INDUCED
BY A SYMMETRIC NORM
Byeigenvalueperturbationmethod,Chau[3,4]proved
thatfor eachcostfunction inthe family, the correspond-
ing induced function d is indeed a pseudo-metrics on To show that a symmetric normon Rn induces a met-
U(n) (and hence the cost function really satisfies the ric and a pseudo-metric on U(n), we make use of the
four constraints listed earlier). In fact, he proved some- following result by Thompson [6]:
thingmore. Inadditiontothisinducedfamilyofpseudo-
Theorem1(Thompson). IfAandB areHermitianma-
metrics,healsodiscoveredafamilyofcloselyrelatedmet-
trices, there exist unitary matrices X and Y (depending
rics on U(n). The only difference between them is that
on A and B) such that
the family of metrics is an“un-optimized” versionof the
family of metrics in the sense that it does not take into
exp(iA)exp(iB)=exp iXAX−1+iYBY−1 . (1)
account the fact that altering the global phase of a uni- (cid:0) (cid:1)
tary operationdoes notaffect the costatall[3, 4]. More
precisely, the underlying cost functions for the family of
metricsobeythefourconstraintslistedaboveexceptthat
Note that Thompson proved his result by assuming
thefirstoneisreplacedbyf(X)=0ifandonlyifX =I.
the validity of the Horn’s conjecture concerning the re-
Note that given X,Y ∈ U(n), the family of metrics can lation of the eigenvalues of the Hermitian matrices A,
also be expressed as certain weighted sums of the ab- B, and C = A + B. The Horn’s conjecture was con-
solute value of the argument of the eigenvalues of the
firmed based on the works of Klyachko [7] and Knutson
matrix XY−1 [3, 4].
andTao [8]; see Ref. [9] for anexcellentsurvey ofthe re-
Interestingly,the family of metricsonU(n)discovered sults. Later,AgnihotriandWoodward[10]improvedthe
by Chau provides another partialanswer to the problem result of Thompson and gave a necessary and sufficient
of quantifying the difference between two quantum evo- condition for the eigenvalues of (special) unitary matri-
lutionsonasystem. Specifically,Chau[3,4]showedthat cesX,Y andZ =XY usingquantumSchubertcalculus.
the metric functions he discovered can be used to give a Theproofistechnicalandthe statementoftheresultin-
quantitativemeasureonthedegreeofnon-commutativity volve a large set of inequalities on the arguments of the
between two unitary matrices X and Y in terms of cer- eigenvalues of the unitary matrices X,Y and Z = XY
tain resources needed to transform XY to YX. by putting them in suitable interval [r,r+2π). So, it is
The abovebackgroundinformationshowsthata num- not easy to use. In fact, it suffices (and is actually more
3
practical)touseTheorem1toderiveourresults. Wewill Note that to arrive at the second inequality above, we
further discuss the connection between our results with have used the fact that
the Horn’s problem in Section IV. We first present our
results in the following. kM +Nkk ≤kMkk+kNkk (5)
Recall that a symmetric norm g: Rn → [0,∞) is a
normfunctionsuchthatg(v)=g(vP)foranyv∈R1×n, foranyn×ncomplex-valuedmatricesM,N andfork =
1,...,n. Here kMk is the Ky Fan k-norm, which is
and any permutation matrix or diagonalorthogonalma- k
defined as the sum of the k largest singular values of
trix P.
M [11].
Proposition 2. Let g : Rn → [0,∞) be a symmetric Since g(u) ≤ g(v) for any u,v ∈ R1×n if and only if
norm. We may define a metric on U(n) as follows: kukk ≤kvkk for k =1,...,n [12, 13], it follows that
dg(X,Y)=g(|a1|,...,|an|), (2) dg(X,Z)≤g(|c1|,...,|cn|)
≤g(|a |+|b |,...,|a |+|b |)
1 1 n n
where XY−1 has eigenvalues eiaj’s with π ≥ a1 ≥ ··· ≥ ≤g(|a |,...,|a |)+g(|b |,...,|b |)
1 n 1 n
a > −π. Furthermore, we may define a pseudo-metric
n =d (X,Y)+d (Y,Z). (6)
on U(n) by g g
▽ Since the infimum in Eq. (3) is actually a minimum,
dg(X,Y)=rin∈fRg(|a1(r)|,...,|an(r)|), (3) there exist r(X,Y),s(Y,Z) ∈ R such that d▽g(X,Y) =
d (eirX,Y) and d▽(Y,Z) = d (eisY,Z) = d (Y,e−isZ).
g g g g
where eirXY−1 has eigenvalues eiaj(r)’s with π ≥ From Eq. (6),
a (r)≥···≥a (r)>−π.
1 n
Note that the infimum above is actually a minimum d▽g(X,Y)+d▽g(Y,Z)=dg(eirX,Y)+dg(Y,e−isZ)
as we can search the infimum in any compact interval of ≥d (eirX,e−isZ)
g
the form [r ,r +2π].
0 0
=d (ei(r+s)X,Z)
g
▽
≥d (X,Z). (7)
g
Proof. Surely d (X,Y), d▽(X,Y) ≥ 0 for all X,Y ∈
g g The proof is complete.
U(n). Besides, d (X,X) = g(0,0,...,0) = 0. And if
g
X 6=Y,atleastoneeigenvalueofXY−1mustbedifferent
from1. Sincegisanorm,weconcludethatdg(X,Y)>0. Example 3. For any µ = (µ ,...,µ ) ∈ Rn, define the
Suppose XY−1 has eigenvalues eiaj’s with π ≥ a1 ≥ µ-norm by 1 n
··· ≥ a > −π. Clearly, the eigenvalues of YX−1 are
n
e−iaj’s. As g is a symmetric norm, g(|a1|,...,|an|) = n
g(| − an|,...,| − a1|). Hence, dg(X,Y) = dg(Y,X). kvk =max |µ v |: {i ,...,i }={1,...,n}.
By applying the same argument to eirXY−1, we get µ X j ij 1 n 
j=1
dg(eirX,Y) = dg(Y,eirX) for all r ∈ R. From Eqs. (2)  (8)
and (3), we know that d▽g(X,Y) = infr∈Rdg(eirX,Y) = Clearly this is a family of symmetric norms; and the in-
infr∈Rdg(X,e−irY). Hence, d▽g(X,Y)=d▽g(Y,X). duced metrics and pseudo-metric on U(n) are the fam-
Finally, we verify the triangle inequalities for dg(·,·) ilies of metrics and pseudo-metrics introduced by Chau
and d▽g(·,·). Let X,Y,Z ∈ U(n). Suppose dg(X,Y) = in Refs. [3, 4].
g(|a |,...,|a |) and d (Y,Z) = g(|b |,...,|b |) where
1 n g 1 n
eia1,...,eian are the eigenvalues of XY−1, and Example 4. One may pick g to be the ℓp norm defined
eib1,...,eibn are the eigenvalues of YZ−1. Suppose by ℓ (v) = n |v |p 1/p for any p ∈ [1,∞]. The
XZ−1 has eigenvalues eicj’s with π ≥ c1 ≥ ··· ≥ cn > p (cid:16)Pj=1 j (cid:17)
−π. Then by Theorem 1, there exist Hermitian matri- induced metric on U(n) has some interesting quantum
ces A,B,C = A+B with eigenvalues a ≥ ··· ≥ a , information science meanings. In fact, it will be shown
1 n
b ≥···≥b andc˜ ≥···≥c˜ suchthatifwereplacec˜ in Ref. [14] that this induced metric is a new family of
1 n 1 n j
by c˜ −2π if c˜ >π andreplacec˜ by c˜ +2π if c˜ ≤−π, indicator functions on the minimum resources needed to
j j j j j
thenthe resultingnentrieswillbe thesameasc ,...,c perform a unitary transformation. Moreover,these indi-
1 n
if they are arranged in descending order. Consequently, cator functions are closely related to a new set of quan-
if kvk is the sum of the k largest entries of v ∈ R1×n tum speed limit bounds on time-independent Hamilto-
k
for k =1,...,n, then nians [14] generalizing the earlier results by Chau [3–5].
k(|c |,...,|c |)k ≤k(|c˜ |,...,|c˜ |)k
1 n k 1 n k Remark 5. Inthe perturbationtheorycontext,wecon-
≤k(|a1|,...,|an|)kk+k(|b1|,...,|bn|)kk sider X˜ = XE, where E is very close to the identity.
=k(|a |+|b |,...,|a |+|b |)k . (4) Suppose X =eiA, where A has eigenvalues π−ε>a ≥
1 1 n n k 1
4
···≥a >−π+ε,andE =eiB suchthattheeigenvalues Corollary 7. Let X,Y ∈U(n) and that X, Y and XY
n
of B lie in [−ε,ε] for an ε > 0. Then we may conclude haveeigenvalueseiaj’s,eibj’sandeicj’s,respectivelywith
that X˜ has eigenvalues π > c1 ≥ ··· ≥ cn > −π such π ≥ |a1| ≥ ··· ≥ |an| ≥ 0, π ≥ |b1| ≥ ··· ≥ |bn| ≥ 0 and
that |cj −aj|≤ε. π ≥|c1|≥···≥|cn|≥0. Then
p p
|c |≤ (|a |+|b |), (12)
III. SEVERAL INEQUALITIES ON PRODUCTS X jℓ+kℓ−ℓ X jℓ kℓ
ℓ=1 ℓ=1
OF TWO UNITARY MATRICES
for any 1≤j <···<j ≤n and 1≤k <···<k ≤n
1 p 1 p
TheprooftechniqueusedinProposition2canbeused with jp+kp−p≤n.
toshowaninequalitygeneralizingafewsimilaronesorig-
Proof. Eq. (12) is the direct consequences of Proposi-
inally reported by Chau in Ref. [3].
tion 6 and the inequality
First,recallthatgiventwonon-increasingsequencesof
real numbers u=(u1,...,un) and u′ =(u′1,...,u′n), we p p
say that u is weakly sub-majorized by u′ if k u ≤ λ↓ (A+B)≤ λ↓ (A)+λ↓ (B) (13)
Pj=1 j X jℓ+kℓ−ℓ Xh jℓ kℓ i
k u′ for 1≤k ≤n. Furthermore,a real-valuedfunc- ℓ=1 ℓ=1
Pj=1 j
tionh(u)issaidtobeSchur-convexifh(u)≤h(u′)when-
reported in Ref. [15]. Here λ↓(A) denotes the jth eigen-
ever u is weakly sub-majorized by u′. j
value of the Hermitian matrix A arranged in descending
order.
Proposition 6. Let
Remark 8. Actually, Eq. (13) belongs to a class of ma-
h(s↓(A+B),s↓(A),s↓(B))≤0 (9)
trix inequalities in the form
be an inequality valid for all n-dimensional Hermitian
matrices A and B, where s↓(A) denotes the sequence of Xλ↓k(A+B)≤Xλ↓i(A)+Xλ↓j(B), (14)
singular values of A arranged in descending order. Sup- k∈K i∈I j∈J
pose further that h is a Schur-convexfunction of its first
whereA,B aren×nHermitianmatricesandI,J,Kare
argument whenever the second and third arguments are
subsets of {1,2,...,n} with equal cardinality. This class
kept fixed. Then,
of matrix inequalities is sometimes called the Lidskii-
type inequalities. Thus, Proposition 6 implies that ev-
h(AAE↓(XY),AAE↓(X),AAE↓(Y))≤0 (10)
ery Lidskii-type inequality for Hermitian matrix induces
a corresponding inequality for unitary matrix.
where AAE↓(X) denotes the sequence of absolute value
of the principal value of argument of the eigenvalues of
ann×nunitarymatrixX arrangedindescendingorder.
IV. RELATION TO THE HORN’S PROBLEM
In other words, if the eigenvalues of the unitary matrix
X are eia1,...eian with aj ∈(−π,π] for all j and |a1|≥
In fact, Lidskii-type inequalities are closely related to
|a |≥···≥|a |, then AAE↓(X)=(|a |,|a |,...,|a |).
2 n 1 2 n the Horn’s problem in matrix theory. Horn [16] conjec-
turedthateigenvaluesofthen×nHermitianmatricesA,
Proof. Let X,Y ∈ U(n). And write X = exp(iA), Y =
B andA+B arecompletelycharacterizedbyinequalities
exp(iB)andXY =exp(iC)wherethe eigenvaluesofthe
in the form Eq. (14) and the equality
Hermitian matrices A,B,C are all in the range (−π,π].
By Theorem 1, we can find a Hermitian matrix C˜ and n n
XsomYe=We1x,Wp(2iC˜∈),Uw(nh)e.reHCe˜nc=e,Wh(1sA↓(WC˜)1−,1s↓+(AW),2sB↓(WB)2−)1≤fo0r. Xj=1λ↓j(A+B)=Xj=1hλ↓j(A)+λ↓j(B)i. (15)
Note that the eigenvalues of C˜ need not lie on the in-
terval(−π,π]. Yet,wecantransformC˜ toC byreplacing (That is to say,he believed that eigenvalues of A, B and
those eigenvalues a ’s of C˜ by a +2π if a ≤ −π and A+B obey Eq. (15) and certain Lidskii-type inequal-
j j j
ities. Furthermore, given three decreasing sequences
replacing them by a −2π if a >π. Obviously, s↓(C) is
j j of real numbers (a )n , (b )n and (c )n satisfy-
weakly sub-majorized by s↓(C˜). Therefore, j j=1 j j=1 j j=1
ing n c = n (a +b ) and the corresponding
Pj=1 j Pj=1 j j
h(AAE↓(XY),AAE↓(X),AAE↓(Y)) Lidskii-likeinequalitiesintheformPk∈Kck ≤Pi∈Iai+
b , then there exist Hermitian matrices A, B and
=h(s↓(C),s↓(A),s↓(B))≤h(s↓(C˜),s↓(A),s↓(B))≤0. APj+∈JB jwhose eigenvalues equal a ’s, b ’s and c ’s, re-
j j j
(11) spectively.) Horn also wrote down a highly inefficient
inductive algorithm to find the subsets I, J and K [16].
So, we are done. The Horn’s problem was proven by combined works of
5
Klyashko [7] and Knutson and Tao [8]. Besides, the ex- see how to deduce our induced inequalities from those
istence of a minimal set of Lidskii-type inequalities for completely characterizing the multiplicative version of
the Horn’s problem was also shown [7, 8, 17, 18]. In the Horn’s problem as this problem seems to be non-
this regard, Remark 8 can be restated as follow: each of trivial. In fact, a major difficulty of this approach is the
theminimalsetofLidskii-typeinequalitiesfortheHorn’s different ways to order the eigenvalues eiaj’s — ours are
probleminduces an inequality for the eigenvalues ofuni- orderedbythevaluesof|a |’swhilethosearisingfromthe
j
tary matrices X, Y and XY. multiplicative version of the Horn’s problem are ordered
by the values of a ’s. Note that in applications of ma-
Naturally, one asks if these corresponding inequali- j
trix inequalities to practical problems such as numerical
tiescompletelycharacterizestheeigenvaluesoftheprod-
analysisandperturbationtheory,itisoftenthecasethat
uct of unitary matrices. This problem, which is some-
onecandeduce the useful resultsusing the basic Lidskii-
timescalledthemultiplicativeversionoftheHorn’sprob-
type inequalities in the form of Eqs. (12) or (13), and
lem, was solvedby the combined worksofAgnihorti and
rarely would one use the full generalizations in Eq. (14).
Woodward[10]andBelkale[17,19]bymeansofquantum
In fact, specializing the general results in (14) to deduce
Schubert calculus. Phrased in the content of our cur-
well knownmatrix inequalities may actually be quite in-
rent discussion, they proved the following. Let e2πiαj’s,
volved. Forexample,seeTheorem3.4andthediscussion
e2πiβj’s and e2πiγj’s be eigenvalues of the n × n spe-
afterit inRef.[20]. Inthatpaper,we obtainedourmain
cial unitary matrices X, Y and Z, respectively. Surely,
results using Thompson’s theorem efficiently. As men-
one may constrain the phases of the eigenvalues by
n α = 0 and α ≥ α ≥ ··· ≥ α > α −1. And tioned before, it will be instructive to use the general
Pj=1 j 1 2 n 1 inequalities of the multiplicative version of Horn’s prob-
β ’s and γ ’s are similarly constrained. Then, the eigen-
j j
lem to deduce our results, but it may not be easy and
valuesofX,Y andZ satisfyingXYZ =I arecompletely
not very practical.
characterized in the sense of the Horn’s problem by in-
ACKNOWLEDGMENTS
equalities in the form
We like to thank K.-Y. Lee for pointing out a mistake
α + β + γ ≤d (16) in our draft. H.F.C. is supported in part by the RGC
X i X j X k
i∈I˜ j∈J˜ k∈K˜ grant HKU 700709P of the HKSAR Government. Re-
search of C.K.L. is supported by a USA NSF grant, a
HK RGC grant, and the 2011 Shanxi 100 Talent Pro-
forsomed(I˜,J˜,K˜)∈NknownastheGromov-Wittenin- gram. He is an honoraryprofessorof University of Hong
variant,wherethesubsetsI˜,J˜ andK˜ of{1,2,...,n}are Kong, Taiyuan University of Technology, and Shanghai
of the same cardinality. Similar to the Horn’s problem, University. Research of Y.T.P. is supported by a USA
only a highly inefficient recursive algorithm is known to NSF grant and a HK RGC grant. Research of N.S.S. is
date to find these inequalities. Thus, it is instructive to supported by a HK RGC grant.
[1] S.Lloyd,“Ultimatephysicallimitstocomputation,”Na- (1998).
ture406, 1047–1054 (2000). [8] A. Knutson and T. Tao, “The honeycomb model of
[2] A. Chefles, A. Kitagawa, M. Takeoka, M. Sasaki, and GLn(C) tensor products I: Proof of the saturation con-
J. Twamley, “Unambiguous discrimination among ora- jecture,” J. Amer.Math. Soc. 12, 1055–1090 (1999).
cle operators,” J. Phys. A: Math. Gen. 40, 10183–10213 [9] W. Fulton, “Eigenvalues, invariant factors, highest
(2007). weights, andSchubertcalculus,” Bull.Amer.Math.Soc.
[3] H. F. Chau, “Metrics on unitary matrices, bounds on (N. S.) 37, 209–249 (2000).
eigenvaluesofproductofunitarymatrices,andmeasures [10] S. Agnihotri and C. Woodward, “Eigenvalues of prod-
of non-commutativity between two unitary matrices,” ucts of unitary matrices and quantum Schubert calcu-
(2010), arXiv:1006.3614v1. lus,” Math. Res. Lett. 5, 817–836 (1998).
[4] H. F. Chau, “Metrics on unitary matrices and their ap- [11] K. Fan, “Maximum properties and inequalities for the
plicationtoquantifyingthedegreeofnon-commutativity eigenvalues of completely continuous operators,” Proc.
betweenunitarymatrices,”Quant.Inform.&Comp.11, Nat. Acad. Sci. U.S.A.37, 760–766 (1951).
721–740 (2011). [12] K. Fan and A. J. Hoffman, “Some metric inequalities in
[5] H.F.Chau,“Tightupperboundonthemaximumspeed the space of matrices,” Proc. Amer. Math. Soc. 6, 111–
of evolution of a quantum state,” Phys. Rev. A 81, 116 (1955).
062133:1–4 (2010). [13] C.-K. Li and N.-K. Tsing, “On the unitarily invariant
[6] R. C. Thompson, “Proof of a conjectured exponential norms andsome related results,” Linearand Multilinear
formula,” Linear and Multilinear Algebra 19, 187–197 Algebra 20, 107–119 (1987).
(1986). [14] K.-Y. Lee and H.F. Chau, (2011), in preparation.
[7] A. A. Klyachko, “Stable bundles, representation the- [15] R.C. Thompson, “Singular valueinequalities for matrix
oryandHermitian operators,” SelectaMath.4, 419–445 sums and minors,” Linear Algebra Appl. 11, 251–269
6
(1975). minefacetsoftheLittlewood-Richardsoncone,”J.Amer.
[16] A. Horn, “Eigenvalues of sums of Hermitian matrices,” Math. Soc. 17, 19–48 (2004).
Pacific J. Math. 12, 225–241 (1962). [19] P.Belkale, “Quantumgeneralization oftheHornconjec-
[17] P. Belkale, “Local systems on P1−S for S a finite set,” ture,” J. Amer. Math. Soc. 21, 365–408 (2008).
Compositio Math. 129, 67–86 (2001). [20] C.-K. Li and Y.-T. Poon, “Principal submatrices of a
[18] A. Knutson, T. Tao, and C. Woodward, “The honey- Hermitian matrix,” Linear and Multilinear Algebra 51,
combmodelofGLn(C)tensorproductsII:Puzzlesdeter- 199–208 (2003).