diff --git "a/1001.0004.txt" "b/1001.0004.txt"
new file mode 100644--- /dev/null
+++ "b/1001.0004.txt"
@@ -0,0 +1,3227 @@
+arXiv:1001.0004v1  [quant-ph]  31 Dec 2009The Lie Algebraic Significance of
+Symmetric Informationally Complete Measurements
+D.M. Appleby, Steven T. Flammia and Christopher A. Fuchs
+Perimeter Institute for Theoretical Physics
+Waterloo, Ontario N2L 2Y5, Canada
+December 30, 2009
+Abstract
+Examplesofsymmetric informationallycomplete positiveoperatorva lued mea-
+sures (SIC-POVMs) have been constructed in every dimension ≤67. However,
+it remains an open question whether they exist in all finite dimensions. A SIC-
+POVM is usually thought of as a highly symmetric structure in quantum state
+space. However, its elements can equally well be regarded as a basis for the Lie
+algebra gl(d,C). In this paper we examine the resulting structure constants,
+which are calculated from the traces of the triple products of the S IC-POVM
+elements and which, it turns out, characterize the SIC-POVM up to unitary
+equivalence. We show that the structure constants have numero us remarkable
+properties. In particular we show that the existence of a SIC-POV M in di-
+mensiondis equivalent to the existence of a certain structure in the adjoint
+representation of gl( d,C). We hope that transforming the problem in this way,
+from a question about quantum state space to a question about Lie algebras,
+may help to make the existence problem tractable.
+Contents
+1. Introduction 1
+2. The Angle Tensors 7
+3. Spectral Decompositions 14
+4. TheQ-QTProperty 18
+5. Lie Algebraic Formulation of the Existence Problem 21
+6. The Algebra sl( d,C) 31
+7. Further Identities 33
+8. Geometrical Considerations 36
+9. TheP-PTProperty 49
+10. Conclusion 52
+11. Acknowledgements 53
+References 531
+1.Introduction
+Symmetric informationally complete positive operator-valued measu res (SIC-
+POVMs) present us with what is, simultaneously, one of the most inte resting, and
+one of the most difficult and tantalizing problems in quantum informatio n [1–46].
+SIC-POVMs are important practically, with applications to quantum t omography
+and cryptography [ 4,8,12,15,20,29], and to classical signal processing [ 24,36].
+However, without in any way wishing to impugn the significance of the a pplications
+which have so far been proposed, it appears to us that the interes t of SIC-POVMs
+stems less from these particular proposed uses than from rather broader, more gen-
+eral considerations: the sense one gets that SICs are telling us so mething deep,
+and hitherto unsuspected about the structure of quantum stat e space. In spite of
+its being the central object about which the rest of quantum mech anics rotates,
+and notwithstanding the efforts of numerous investigators [ 47], the geometry of
+quantum state space continues to be surprisingly ill-understood. T he hope which
+inspires our efforts is that a solution to the SIC problem will prove to b e the key,
+not just to SIC-POVMs narrowly conceived, but to the geometry o f state space in
+general. Such things are, by nature, unpredictable. However, it is not unreasonable
+to speculate that a better theoretical understanding of the geo metry of quantum
+state space might have important practical consequences: not o nly the applica-
+tions listed above, but perhaps other applications which have yet to be conceived.
+On a more foundational level one may hope that it will lead to a much imp roved
+understanding of the conceptual message of quantum mechanics [7,43,45,48].
+Having said why we describe the problem as interesting, let us now exp lain why
+we describe it as tantalizing. The trouble is that, although there is an abundance of
+reasons for suspecting that SIC-POVMs exist in every finite dimens ion (exact and
+high-precision numerical examples [ 1,2,5,11,16,19,28,39,46] having now been
+constructed in every dimension up to 67), and in spite of the intense efforts of many
+people [1–46] extending over a period of more than ten years, a general existe nce
+proof continues to elude us. In their seminal paper on the subject , published in
+2004, Renes et al[5] say “A rigorous proof of existence of SIC-POVMs in all finite
+dimensions seems tantalizingly close, yet remains somehow distant.” T hey could
+have said the same if they were writing today.
+The purposeofthis paperis totryto takeourunderstandingofSI C mathematics
+(as it might be called) a little further forward. The research we repo rt began with
+a chance numerical discovery made while we were working on a differen t problem.
+Pursuing that initial numerical hint we uncovered a rich and interest ing set of
+connections between SIC-POVMs in dimension dand the Lie Algebra gl( d,C). The
+existence of these connections came as a surprise to us. However , in retrospect it
+is, perhaps, not so surprising. Interest in SIC-POVMs has, to dat e, focused on the
+fact that an arbitrary density matrix can be expanded in terms of a SIC-POVM.
+However, a SIC-POVM in dimension ddoes in fact provide a basis, not just for the
+space of density matrices, but for the space of all d×dcomplex matrices— i.e.the
+Liealgebragl( d,C). Boykin et al[49] haverecentlyshownthatthere isaconnection
+betweentheexistenceproblemformaximalsetsofMUBs(mutuallyu nbiasedbases)
+and the theory of Lie algebras. Since SIC-POVMs share with MUBs th e property
+of being highly symmetrical structures in quantum state space it mig ht have been
+anticipated that there are also some interesting connections betw een SIC-POVMs
+and Lie algebras.2
+Our main result (proved in Sections 3,4and5) is that the proposition, that a
+SIC-POVM exists in dimension d, is equivalent to a proposition about the adjoint
+representation of gl( d,C). Our hope is that transforming the problem in this way,
+from a question about quantum state space to a question about Lie algebras, may
+help to make the SIC-existence problem tractable. But even if this h ope fails to
+materialize we feel that this result, along with the many other result s we obtain,
+provides some additional insight into these structures.
+Inddimensional Hilbert space Hda SIC-POVM is a set of d2operatorsE1,
+...,Ed2of the form
+Er=1
+dΠr (1)
+where the Π rare rank-1 projectors with the property
+Tr(ΠrΠs) =/braceleftigg
+1r=s
+1
+d+1r/ne}ationslash=s(2)
+We will refer to the Π ras SIC projectors, and we will say that {Πr:r= 1,...,d2}
+is a SIC set.
+It follows from this definition that the Ersatisfy
+d2/summationdisplay
+r=1Er=I (3)
+(sotheyconstitute aPOVM),andthattheyarelinearlyindependen t (sothePOVM
+is informationally complete).
+It is an open question whether SIC-POVMs exist for all values of d. However,
+examples have been constructed analytically in dimensions 2–15 inclus ive [1,2,11,
+16,19,28,39,46], and in dimensions 19, 24, 35 and 48 [ 16,46]. Moreover, high
+precisionnumerical solutionshave been constructed in dimensions 2 –67inclusive [ 5,
+46]. Thislendssomeplausibilitytothe speculationthat theyexistinalldime nsions.
+For a comprehensive account of the current state of knowledge in this regard, and
+many new results, see the recent study by Scott and Grassl [ 46].
+All known SIC-POVMs have a group covariance property. In other words, there
+exists
+(1) a group Ghavingd2elements
+(2) a projective unitary representation of GonHd:i.e.a mapg→UgfromG
+to the set of unitaries such that Ug1Ug2∼Ug1g2for allg1,g2(where the
+notation “ ∼” means “equals up to a phase”)
+(3) a normalized vector |ψ/an}bracketri}ht(the fiducial vector)
+such that the SIC-projectors are given by
+Πg=Ug|ψ/an}bracketri}ht/an}bracketle{tψ|U†
+g (4)
+(where we label the projector by the group element g, rather than the integer ras
+above).
+Most known SIC-POVMs are covariant under the action of the Weyl- Heisenberg
+group (though not all—see Renes et al[5] and, for an explicit example of a non
+Weyl-Heisenberg SIC-POVM, Grassl [ 19]). Here the group is Zd×Zd, and the
+projective representation is p→Dp, wherep= (p1,p2)∈Zd×ZdandDpis the3
+corresponding Weyl-Heisenberg displacement operator
+Dp=d−1/summationdisplay
+rτ(2r+p1)p2|r+p1/an}bracketri}ht/an}bracketle{tr| (5)
+In this expression τ=eiπ(d+1)
+d, the vectors |0/an}bracketri}ht,...|d−1/an}bracketri}htare an orthonormal basis,
+and the addition in |r+p1/an}bracketri}htis modulod. For more details see, for example, ref. [ 16].
+One should not attach too much weight to the fact that all known SI C-POVMs
+have a group covariance property as this may only reflect the fact that group co-
+variant SIC-POVMs are much easier to construct. So in this paper w e will try to
+prove as much as we can without assuming such a property. One pot ential benefit
+ofthis attitude is that, by accumulatingenough facts about SIC-P OVMsin general,
+we may eventually get to the point where we can answer the question , whether all
+SIC-POVMs actually do have a group covariance property.
+The fact that the d2operatorsΠ rare linearly independent means that they form
+a basis for the complex Lie algebra gl( d,C) (the set of all operators acting on Hd).
+Since the Π rare Hermitian, then iΠrforms a basis also for the real Lie algebra
+u(d) (the set of all anti-Hermitian operators acting on Hd). So for any operator
+A∈gl(d,C) there is a unique set of expansion coefficients arsuch that
+A=d2/summationdisplay
+r=1arΠr (6)
+To find the expansion coefficients we can use the fact that
+d2/summationdisplay
+s=1Tr(ΠrΠs)/parenleftbiggd+1
+dδst−1
+d2/parenrightbigg
+=δrt (7)
+from which it follows
+ar=d+1
+dTr(ΠrA)−1
+dTr(A) (8)
+Specializing to the case A= ΠrΠswe find
+ΠrΠs=d+1
+d
+d2/summationdisplay
+t=1TrstΠt
+−dδrs+1
+d+1I (9)
+where
+Trst= Tr(Π rΠsΠt) (10)
+To a large extent this paper consists in an exploration of the proper ties of these
+important quantities, which we will refer to as the triple products. T hey are inti-
+mately related to the geometric phase, in which context they are us ually referred
+to as 3-vertex Bargmann invariants (see Mukunda et al[50], and references cited
+therein). We have, as an immediate consequence of the definition,
+Trst=Ttrs=Tstr=T∗
+rts=T∗
+tsr=T∗
+srt (11)
+It is convenient to define
+Jrst=d+1
+d(Trst−T∗
+rst) (12)
+Rrst=d+1
+d(Trst+T∗
+rst) (13)4
+SoJrstis imaginary and completely anti-symmetric; Rrstis real and completely
+symmetric. Both these quantities play a significant role in the theory . It follows
+from Eq. ( 9) that
+[Πr,Πs] =d2/summationdisplay
+t=1JrstΠt (14)
+So theJrstare structure constants for the Lie algebra gl( d,C). As an immediate
+consequence of this they satisfy the Jacobi identity:
+d2/summationdisplay
+b=1/parenleftbig
+JrsbJtba+JstbJrba+JtrbJsba/parenrightbig
+= 0 (15)
+for allr,s,t,a. The Jacobi identity holds for any representation of the structu re
+constants. In the following sections we will derive many other identit ies which are
+specific to this particular representation.
+Turning to the quantities Rrst, it follows from Eq. ( 9) that they feature in the
+expression for the anti-commutator
+{Πr,Πs}=/summationdisplay
+tRrstΠt−2(dδrs+1)
+d+1I (16)
+They also play an important role in the description of quantum state s pace. Let
+ρbe any density matrix and let pr=1
+dTr(Πrρ) be the probability of obtaining
+outcomerin the measurement described by the POVM with elements1
+dΠr. Then
+it follows from Eq. ( 8) thatρcan be reconstructed from the probabilities by
+ρ=d2/summationdisplay
+r=1/parenleftbigg
+(d+1)pr−1
+d/parenrightbigg
+Πr (17)
+Suppose, now, that the prareanyset ofd2real numbers. So we do not assume
+that theprare even probabilities, let alone the probabilities coming from a density
+matrix according to the prescription pr=1
+dTr(Πrρ). Then it is shown in ref. [ 34]
+that theprare in fact the probabilities coming from a pure state if and only if they
+satisfy the two conditions
+d2/summationdisplay
+r=1p2
+r=2
+d(d+1)(18)
+d2/summationdisplay
+r,s,t=1Rrstprpspt=2(d+7)
+d(d+1)2(19)
+Let us look at the quantities JrstandRrstin a little more detail. For each r
+choose a unit vector |ψr/an}bracketri}htsuch that Π r=|ψr/an}bracketri}ht/an}bracketle{tψr|. Then the Gram matrix for these
+vectors is of the form
+Grs=/an}bracketle{tψr|ψs/an}bracketri}ht=Krseiθrs(20)
+where the matrix θrsis anti-symmetric and
+Krs=/radicalbigg
+dδrs+1
+d+1(21)
+Note that the SIC-POVM does not determine the angles θrsuniquely since making
+the replacements |ψr/an}bracketri}ht →eiφr|ψr/an}bracketri}htleaves the SIC-POVM unaltered, but changes5
+the angles θrsaccording to the prescription θrs→θrs−φr+φs. This freedom
+to rephase the vectors |ψr/an}bracketri}htis not usually important. However, it sometimes has
+interesting consequences (see Section 9). It can be thought of as a kind of gauge
+freedom.
+The Gram matrix satisfies an important identity. Every SIC-POVM ha s the
+2-design property [ 5,17]
+d2/summationdisplay
+r=1Πr⊗Πr=2d
+d+1Psym (22)
+wherePsymis the projector onto the symmetric subspace of Hd⊗Hd. Expressed
+in terms of the Gram matrix this becomes
+d2/summationdisplay
+r=1Gs1rGs2rGrt1Grt2=d
+d+1/parenleftbig
+Gs1t1Gs2t2+Gs1t2Gs2t1/parenrightbig
+(23)
+Turning to the triple products we have
+Trst=GrsGstGtr=KrsKstKtreiθrst(24)
+where
+θrst=θrs+θst+θtr (25)
+Note that the tensor θrstis completely anti-symmetric. In particular θrst= 0 if any
+two of the indices are the same. Note also that re-phasing the vect ors|ψr/an}bracketri}htleaves
+the tensors Trstandθrstunchanged. They are in that sense gauge invariant.
+Finally, we have the following expressions for JrstandRrst:
+Jrst=2i
+d√
+d+1sinθrst (26)
+Rrst=2(d+1)
+dKrsKstKtrcosθrst (27)
+Like the triple products, JrstandRrstare gauge invariant.
+For later reference let us note that the matrix Jr, with matrix elements
+(Jr)st=Jrst (28)
+is the adjoint representative of Π rin the SIC-projector basis:
+adΠrΠs= [Πr,Πs] =d2/summationdisplay
+t=1JrstΠt (29)
+It can be seen that all the interesting features of the tensor Grs(respectively,
+the tensors Trst,JrstandRrst) are contained in the order-2 angle tensor θrs(re-
+spectively, the order-3 angle tensor θrst). It is also easy to see that, for any unitary
+U, the transformation
+Πr→UΠrU†(30)
+leaves the angle tensors invariant. This suggests that we shift our focus from indi-
+vidual SIC-POVMs to families of unitarily equivalent SIC-POVMs—SIC- families,
+as we will call them for short.
+We begin our investigation in Section 2by giving necessary and sufficient con-
+ditions for an arbitrary tensor θrs(respectively θrst) to be the rank-2 (respectively
+rank-3) angle tensor corresponding to a SIC-family. We also show t hat either angle
+tensor uniquely determines the corresponding SIC-family. Finally we describe a6
+method for reconstructing the SIC-family, starting from a knowle dge of either of
+the two angle tensors.
+In Sections 3,4and5we prove the central result of this paper: namely, that
+the existence of a SIC-POVM in dimension dis equivalent to the existence of a
+certain very special set of matrices in the adjoint representation of gl(d,C). In
+Section3we show that, for any SIC-POVM, the adjoint matrices Jrhave the
+spectral decomposition
+Jr=Qr−QT
+r (31)
+whereQris a rankd−1 projector which has the remarkable property of being
+orthogonal to its own transpose:
+QrQT
+r= 0 (32)
+We refer to this feature of the adjoint matrices as the Q-QTproperty. In Section 3
+we also show that from a knowledge of the Jmatrices it is possible to reconstruct
+the corresponding SIC-family. In Section 4we characterize the general class of
+projectors which have the property of being orthogonal to their own transpose.
+Then, in Section 5, we prove a converse of the result established in Section 3. The
+Q-QTproperty is not completely equivalent to the property of being a SIC set.
+However, it turns out that it is, in a certain sense, very nearly equiv alent. To be
+more specific: let Lrbe any set of d2Hermitian operators which constitute a basis
+for gl(d,C) and letCrbe the adjoint representative of Lrin this basis. Then the
+necessary and sufficient condition for the Crto have the spectral decomposition
+Cr=Qr−QT
+r (33)
+whereQris a rankd−1 projector such that QrQT
+r= 0 is that there exists a
+SIC set Π rsuch thatLr=ǫr(Πr+αI) for some fixed number α∈Rand signs
+ǫr=±1. In particular, the existence of an Hermitian basis for gl( d,C) having the
+Q-QTproperty is both necesary and sufficient for the existence of a SIC -POVM in
+dimensiond.
+In Section 6we digress briefly, and consider sl( d,C) (the Lie algebra consisting
+of all trace-zero d×dcomplex matrices). As we have explained, this paper is
+motivated by the hope that a Lie algebraic perspective will cast light o n the SIC-
+existence problem, rather than by an interest in Lie algebras as suc h. We focus on
+gl(d,C) because that is the casewherethe connection with SIC-POVMsse ems most
+straightforward. However a SIC-POVM also gives rise to an interes ting geometrical
+structure in sl( d,C), as we show in Section 6.
+In Section 7we derive a number of additional identities satisfied by the Jand
+Qmatrices.
+The complex projectors Qr,QT
+rand the real projector Qr+QT
+rdefine three
+families of subspaces. It turns out that there are some interestin g geometrical
+relationships between these subspaces, which we study in Section 8.
+Finally, in Section 9we show that, with the appropriate choice of gauge, the
+Gram matrix corresponding to a Weyl-Heisenberg covariant SIC-fa mily has a fea-
+ture analogous to the Q-QTproperty, which we call the P-PTproperty. It is an
+open question whether this result generalizes to other SIC-families , not covariant
+with respect to the Weyl-Heisenberg group.7
+2.The Angle Tensors
+The purpose of this section is to establish the necessary and sufficie nt conditions
+for an arbitrary tensor θrs(respectively θrst) to be the order-2 (respectively order-
+3) angle tensor for a SIC-family. We will also show that either one of t he angle
+tensors is enough to uniquely determine the SIC-family. Moreover, we will describe
+explicit procedures for reconstructing the family, starting from a knowledge of one
+of the angle tensors.
+We begin by considering the general class of POVMs (not just SIC-P OVMs)
+which consist of d2rank-1 elements. A POVM of this type is thus defined by a set
+ofd2vectors|ξ1/an}bracketri}ht,...,|ξd2/an}bracketri}htwith the property
+d2/summationdisplay
+r=1|ξr/an}bracketri}ht/an}bracketle{tξr|=I (34)
+Note that/summationtextd2
+r=1/vextenddouble/vextenddouble|ξr/an}bracketri}ht/vextenddouble/vextenddouble2=d, so the vectors |ξr/an}bracketri}htcannot all be normalized. In the
+particular case of a SIC-POVM the vectors all have the same norm/vextenddouble/vextenddouble|ξr/an}bracketri}ht/vextenddouble/vextenddouble=1√
+d.
+However in the general case they may have different norms.
+Given a set of such vectors consider the Gram matrix
+Prs=/an}bracketle{tξr|ξs/an}bracketri}ht (35)
+Clearly the Gram matrix cannot determine the POVM uniquely since if Uis any
+unitary operator then the vectors U|ξr/an}bracketri}htwill define another POVM having the same
+Gram matrix. However, the theorem we now prove shows that this is the only free-
+dom. In other words, the Gram matrix fixes the POVM up to unitary e quivalence.
+The theorem also provides us with a criterion for deciding whether an arbitrary
+d2×d2matrixPis the Gram matrix corresponding to a POVM of the specified
+type. As a corollary this will give us a criterion for deciding whether an arbitrary
+tensorθrsis specifically the order-2 angle tensor for a SIC-family.
+Theorem 1. LetPbe anyd2×d2Hermitian matrix. Then the following conditions
+are equivalent:
+(1)Pis a rankdprojector.
+(2)Psatisfies the trace identities
+Tr(P) = Tr(P2) = Tr(P3) = Tr(P4) =d (36)
+(3)Pis the Gram matrix for a set of d2vectors|ξr/an}bracketri}ht(not all normalized) such
+that|ξr/an}bracketri}ht/an}bracketle{tξr|is a POVM:
+/an}bracketle{tξr|ξs/an}bracketri}ht=Prs (37)
+d2/summationdisplay
+r=1|ξr/an}bracketri}ht/an}bracketle{tξr|=I (38)
+SupposePsatisfies these conditions. To construct a POVM correspondi ng toP
+let thedcolumn vectors
+ξ11
+ξ12
+...
+ξ1d2
+,
+ξ21
+ξ22
+...
+ξ2d2
+,...,
+ξd1
+ξd2
+...
+ξdd2
+(39)8
+be any orthonormal basis for the subspace onto which Pprojects. Define
+|ξr/an}bracketri}ht=d/summationdisplay
+a=1ξ∗
+ar|a/an}bracketri}ht (40)
+where the vectors |a/an}bracketri}htare any orthonormal basis for Hd. ThenPis the Gram matrix
+for the vectors |ξ1/an}bracketri}ht,...,|ξd2/an}bracketri}ht. Moreover, the necessary and sufficient condition for
+any other set of vectors |η1/an}bracketri}ht,...,|ηd2/an}bracketri}htto have Gram matrix Pis that there exist a
+unitary operator Usuch that
+|ηr/an}bracketri}ht=U|ξr/an}bracketri}ht (41)
+for allr.
+Proof.We begin by showing that (3) = ⇒(1). Suppose |ξ1/an}bracketri}ht,...|ξd2/an}bracketri}htis any set of
+d2vectors such that |ξr/an}bracketri}ht/an}bracketle{tξr|is a POVM. So
+d2/summationdisplay
+r=1|ξr/an}bracketri}ht/an}bracketle{tξr|=I (42)
+Let
+Prs=/an}bracketle{tξr|ξs/an}bracketri}ht (43)
+be the Gram matrix. Then Pis Hermitian. Moreover, P2=Psince
+d2/summationdisplay
+t=1PrtPts=/an}bracketle{tξr|
+d2/summationdisplay
+t=1|ξt/an}bracketri}ht/an}bracketle{tξs|
+|ξr/an}bracketri}ht
+=/an}bracketle{tξr|ξs/an}bracketri}ht
+=Prs (44)
+Also
+Tr(P) =d2/summationdisplay
+r=1/an}bracketle{tξr|ξr/an}bracketri}ht=d (45)
+(as can be seen by taking the trace on both sides of Eq. ( 42)). SoPis a rank-d
+projector.
+We next show that (1) = ⇒(3). LetPbe a rank-dprojector, and let the d
+column vectors
+ξ11
+ξ12
+...
+ξ1d2
+,
+ξ21
+ξ22
+...
+ξ2d2
+,...,
+ξd1
+ξd2
+...
+ξdd2
+(46)
+be an orthonormal basis for the subspace onto which it projects. So
+d2/summationdisplay
+r=1ξ∗
+arξbr=δab (47)
+for alla,b, and
+d2/summationdisplay
+a=1ξarξ∗
+as=Prs (48)9
+for allr,s. Now let |ξ1/an}bracketri}ht,...|ξd2/an}bracketri}htbe the vectors defined by Eq. ( 40). Then it follows
+from Eq. ( 47) that
+d2/summationdisplay
+r=1|ξr/an}bracketri}ht/an}bracketle{tξr|=d/summationdisplay
+a,b=1
+d2/summationdisplay
+r=1ξ∗
+arξbr
+|a/an}bracketri}ht/an}bracketle{tb|
+=d/summationdisplay
+a=1|a/an}bracketri}ht/an}bracketle{ta|
+=I (49)
+implying that |ξr/an}bracketri}ht/an}bracketle{tξr|is POVM. Also, it follows from Eq. ( 48) that
+/an}bracketle{tξr|ξs/an}bracketri}ht=d/summationdisplay
+a=1ξarξ∗
+as=Prs (50)
+implying that the |ξr/an}bracketri}hthave Gram matrix P.
+We next turn to condition (2). The fact that (1) = ⇒(2) is immediate. To
+prove the reverse implication observe that condition (2) implies
+Tr(P4)−2Tr(P3)+Tr(P2) = 0 (51)
+Letλ1,...,λ d2be the eigenvalues of P. Then Eq. ( 51) implies
+d2/summationdisplay
+r=1λ2
+r(λr−1)2= 0 (52)
+It follows that each eigenvalue is either 0 or 1. Since Tr( P) =dwe must have d
+eigenvalues = 1 and the rest all = 0. So Pis a rank-dprojector.
+It remains to show that the POVM corresponding to a given rank- dprojector
+is unique up to unitary equivalence. To prove this let Pbe a rank-dprojector, let
+|ξr/an}bracketri}htbe the vectors defined by Eq. ( 40), and let |η1/an}bracketri}ht,...,|ηd2/an}bracketri}htbe any other set of
+vectors such that
+/an}bracketle{tηr|ηs/an}bracketri}ht=Prs (53)
+for allr,s. Define
+ηar=/an}bracketle{tηr|a/an}bracketri}ht (54)
+Then
+d2/summationdisplay
+r=1η∗
+arηbr=/an}bracketle{ta|
+d2/summationdisplay
+r=1|ηr/an}bracketri}ht/an}bracketle{tηr|
+|b/an}bracketri}ht=δab (55)
+(because |ηr/an}bracketri}ht/an}bracketle{tηr|is a POVM) and
+d/summationdisplay
+a=1ηarη∗
+as=Prs (56)
+(because the |ηr/an}bracketri}hthave Gram matrix P). So thedcolumn vectors
+
+η11
+η12
+...
+η1d2
+,
+η21
+η22
+...
+η2d2
+,...,
+ηd1
+ηd2
+...
+ηdd2
+(57)10
+are an orthonormal basis for the subspace onto which Pprojects. But the column
+vectors 
+ξ11
+ξ12
+...
+ξ1d2
+,
+ξ21
+ξ22
+...
+ξ2d2
+,...,
+ξd1
+ξd2
+...
+ξdd2
+(58)
+are also an orthonormal basis for this subspace. So there must ex ist ad×dunitary
+matrixUabsuch that
+ηar=d/summationdisplay
+b=1Uabξbr (59)
+for alla,r. Define
+U=d/summationdisplay
+a,b=1U∗
+ab|a/an}bracketri}ht/an}bracketle{tb| (60)
+Then
+|ηr/an}bracketri}ht=U|ξr/an}bracketri}ht (61)
+for allr. /square
+In the case of a SIC-POVM we have
+|ξr/an}bracketri}ht=1√
+d|ψr/an}bracketri}ht (62)
+where the vectors |ψr/an}bracketri}htare normalized, and
+Prs=1
+dGrs=1
+dKrseiθrs(63)
+whereGis the Gram matrix of the vectors |ψr/an}bracketri}htandθrsis the order-2 angle tensor.
+In the sequel we will distinguish these matrices by referring to Gas the Gram
+matrix and Pas the Gram projector.
+We have
+Corollary 2. Letθrsbe a real anti-symmetric tensor. Then the following state-
+ments are equivalent:
+(1)θrsis an order- 2angle tensor corresponding to a SIC-family.
+(2)θrssatisfies
+d2/summationdisplay
+t=1KrtKtsei(θrt+θts)=dKrseiθrs(64)
+for allr,s.
+(3)θrssatisfies
+d2/summationdisplay
+r,s,t=1KrsKstKtrei(θrs+θst+θtr)=d4(65)
+and
+d2/summationdisplay
+r,s,t,u=1KrsKstKtuKurei(θrs+θst+θtu+θur)=d5(66)11
+LetΠr,Π′
+rbe two different SIC-sets, and let θrs,θ′
+rsbe corresponding order- 2
+angle tensors. Then there exists a unitary Usuch that
+Π′
+r=UΠrU†(67)
+for allrif and only if
+θ′
+rs=θrs−φr+φs (68)
+for some arbitrary set of phase angles φr(in other words two SIC-sets are unitarily
+equivalent if and only if their order- 2angle tensors are gauge equivalent).
+A SIC-family can be reconstructed from its order- 2angle tensor θrsby calculating
+an orthonormal basis for the subspace onto which the Gram pro jector
+Prs=1
+dKrseiθrs(69)
+projects, as described in Theorem 1.
+Remark. The sense in which we areusing the term “gaugeequivalence”is explain ed
+in the passage immediately following Eq. ( 21).
+Note that condition (2) imposes d2(d2−1)/2 independent constraints (taking
+account of the anti-symmetry of θrs). Condition (3), by contrast, only imposes 2
+independent constraints. It is to be observed, however, that th e price we pay for
+the reduction in the number of equations is that Eqs. ( 65) and (65) are respectively
+cubic and quartic in the phases, whereas Eq. ( 64) is only quadratic.
+Proof.Letθrsbe an arbitrary anti-symmetric tensor, and define
+Prs=1
+dKrseiθrs(70)
+The anti-symmetry of θrsmeans that Pis automatically Hermitian. So it follows
+from Theorem 1that a necessary and sufficient condition for Prsto be a rank- d
+projector, and for θrsto be the order-2 angle tensor of a SIC-family, is that
+d2/summationdisplay
+t=1KrtKtsei(θrt+θts)=dKrseiθrs(71)
+for allr,s.
+To prove the equivalence of conditions (1) and (3) note that the co nditions
+Tr(P) = Tr(P2) =dare an automatic consequence of Phaving the specified form.
+So it follows from Theorem 1thatθrsis the order-2 angle tensor of a SIC-family if
+and only if Eqs. ( 65) and (66) are satisfied.
+Now let Πr, Π′
+rbe two SIC-sets and let θrs,θ′
+rsbe order-2 angle tensors corre-
+sponding to them. Then there exist normalized vectors |ψr/an}bracketri}ht,|ψ′
+r/an}bracketri}htsuch that
+Πr=|ψr/an}bracketri}ht/an}bracketle{tψr| Π′
+r=|ψ′
+r/an}bracketri}ht/an}bracketle{tψ′
+r| (72)
+for allr, and
+/an}bracketle{tψr|ψs/an}bracketri}ht=Krseiθrs/an}bracketle{tψ′
+r|ψ′
+s/an}bracketri}ht=Krseiθ′
+rs (73)
+for allr,s.
+Suppose, first of all, that there exists a unitary Usuch that
+Π′
+r=UΠrU†(74)12
+Then there exist phase angles φrsuch that
+|ψ′
+r/an}bracketri}ht=eiφrU|ψr/an}bracketri}ht (75)
+for allr, which is easily seen to imply that
+θ′
+rs=θrs−φr+φs (76)
+for allr,s. Soθrs,θ′
+rsare gauge equivalent.
+Conversely, suppose there exist phase angles φrsuch that
+θ′
+rs=θrs−φr+φs (77)
+Define
+|ψ′′
+r/an}bracketri}ht=e−iφr|ψ′
+r/an}bracketri}ht (78)
+Then
+/an}bracketle{tψ′′
+r|ψ′′
+s/an}bracketri}ht=Krseiθrs=/an}bracketle{tψr|ψs/an}bracketri}ht (79)
+for allr,s. So it follows from Theorem 1that there exists a unitary Usuch that
+|ψ′′
+r/an}bracketri}ht=U|ψr/an}bracketri}ht (80)
+for allr. Consequently
+Π′
+r=|ψ′′
+r/an}bracketri}ht/an}bracketle{tψ′′
+r|=UΠrU†(81)
+for allr. So Πrand Π′
+rare unitarily equivalent. /square
+We now turn to the order-3 angle tensors. We have
+Theorem 3. Letθrstbe a real completely anti-symmetric tensor. Then the follow -
+ing conditions are equivalent:
+(1)θrstis the order- 3angle tensor for a SIC-family
+(2)For some fixed aand allr,s,t
+θars+θast+θatr=θrst (82)
+and for all r,s
+d2/summationdisplay
+t=1KrtKtseiθrst=dKrs (83)
+(3)For some fixed aand allr,s,t
+θars+θast+θatr=θrst (84)
+and
+d2/summationdisplay
+r,s,t=1KrsKstKtreiθrst=d4(85)
+d2/summationdisplay
+r,s,t,u=1KrsKstKtuKurei(θrst+θtur)=d5(86)13
+LetΠr,Π′
+rbe two different SIC-sets and let θrst,θ′
+rstbe the corresponding order-
+3angle tensors. Then the necessary and sufficient condition fo r there to exist a
+unitaryUsuch that
+Π′
+r=UΠrU†(87)
+for allris thatθ′
+rst=θrstfor allr,s,t(in other words two SIC-sets are unitarily
+equivalent if and only if their order- 3angle tensors are identical).
+Letθrstbe the order- 3angle tensor corresponding to a SIC-family. Then the
+order-2angle tensor is given by (up to gauge freedom)
+θrs=θars (88)
+for any fixed a, from which the SIC-family can be reconstructed using the me thod
+described in Theorem 1.
+Remark. Unlike the order-2tensor, the order-3angletensoris gaugeinvar iant. This
+means that it provides what is, in many ways, a more useful charact erization of
+the SIC-family. For that reason we will be almost exclusively concern ed with the
+order-3 tensor in the remainder of this paper.
+Proof.The fact that (1) = ⇒(2) is an immediate consequence of the definition of
+theorder-3angletensorandcondition(2)ofCorollary 2. Toprovethat(2) = ⇒(1)
+letθrstbe a completely anti-symmetric tensor such that condition (2) holds . Define
+θrs=θars (89)
+for allr,s. Then Eq. ( 83) implies
+d2/summationdisplay
+t=1KrtKtsei(θrt+θts)=eiθrs
+d2/summationdisplay
+t=1KrtKtseiθrst
+∗
+=dKrseiθrs(90)
+for allr,s. It follows from Corollary 2thatθrsis the order-2 and θrstthe order-3
+angle tensor of a SIC-family.
+The equivalence of conditions (1) and (3) is proved similarly.
+It remains to show that two SIC-sets are unitarily equivalent if and o nly if
+their order-3 angle tensors are identical. To see this let Πr=|ψr/an}bracketri}ht/an}bracketle{tψr|and Π′
+r=
+|ψ′
+r/an}bracketri}ht/an}bracketle{tψ′
+r|be two different SIC-sets having the same order-3 angle tensor θrst. Let
+θrs(respectively θ′
+rs) be the order-2 angle tensor corresponding to the vectors |ψr/an}bracketri}ht
+(respectively |ψ′
+r/an}bracketri}ht). Choose some fixed index a. We have
+θ′
+ar+θ′
+sa+θ′
+rs=θar+θsa+θrs (91)
+for allr,s. Consequently
+θ′
+rs=θrs+φr−φs (92)
+for allr,s, where
+φr=θar−θ′
+ar (93)
+Soθ′
+rsandθrsare gauge equivalent. It follows from Corollary 2that Πrand Π′
+rare
+unitarily equivalent. Conversely, suppose that Πrand Π′
+rare unitarily equivalent,
+and letθrs,θ′
+rsbe order-2 angle tensors corresponding to them. It follows from
+Corollary 2thatθrsandθ′
+rsare gauge equivalent. It is then immediate that the
+order-3 angle tensors are identical. /square14
+Finally, let us note that when expressed in terms of the triple produc ts Eq. (83)
+reads
+d2/summationdisplay
+t=1Trst=dK2
+rs (94)
+while Eq. ( 85) reads
+d2/summationdisplay
+r,s,t=1Trst=d4(95)
+For Eq. ( 86) we have to work a little harder. We have
+d2/summationdisplay
+r,s,t,u=11
+K2
+rtTrstTtur=d5(96)
+from which it follows
+d5=d2/summationdisplay
+r,s,t,u=1/parenleftbig
+−dδrt+d+1/parenrightbig
+TrstTtur
+= (d+1)d2/summationdisplay
+r,s,t,u=1TrstTtur−dd2/summationdisplay
+r,s,u=1K2
+rsK2
+ru
+= (d+1)d2/summationdisplay
+r,s,t,u=1TrstTtur−d5(97)
+Consequently
+d2/summationdisplay
+s,u=1Tr/parenleftbig
+TsTu/parenrightbig
+=d2/summationdisplay
+r,s,t,u=1TrstTtur=2d5
+d+1(98)
+This equation be alternatively written
+d2/summationdisplay
+r,s=1Tr/parenleftbig
+TrTs/parenrightbig
+=2d5
+d+1(99)
+whereTris the matrix with matrix elements ( Tr)uv=Truv.
+When they are written like this, in terms of the triple products, the f act that
+Eq. (94) implies Eqs. ( 95) and (98) becomes almost obvious. The reverse implica-
+tion, by contrast, is rather less obvious.
+3.Spectral Decompositions
+LetTr,Jr,Rrbe thed2×d2matrices whose matrix elements are
+(Tr)st=Trst (Jr)st=Jrst (Rr)st=Rrst(100)
+whereJrst,RrstarethequantitiesdefinedbyEqs.( 12)and(13). SoJristheadjoint
+representation matrix of Π r. In this section we derive the spectral decompositions
+of these matrices. To avoid confusion we will use the notation |ψ/an}bracketri}htto denote a ket in
+ddimensional Hilbert space Hd, and/bardblψ/an}bracketri}ht/an}bracketri}htto denote a ket in d2dimensional Hilbert15
+spaceHd2. In terms of this notation the spectral decompositions will turn ou t to
+be:
+Tr=d
+d+1Qr+2d
+d+1/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (101)
+Jr=Qr−QT
+r (102)
+Rr=Qr+QT
+r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (103)
+In these expressions the vector /bardbler/an}bracketri}ht/an}bracketri}htis normalized, and its components in the stan-
+dard basis are all real. Qris a rankd−1 projector such that
+Qr/bardbler/an}bracketri}ht/an}bracketri}ht=QT
+r/bardbler/an}bracketri}ht/an}bracketri}ht= 0 (104)
+and which has, in addition, the remarkable property of being orthog onal to its own
+transpose (also a rank d−1 projector):
+QrQT
+r= 0 (105)
+Explicit expressions for /bardbler/an}bracketri}ht/an}bracketri}htandQrwill be given below.
+It will be convenient to define the rank 2( d−1) projector
+¯Rr=Qr+QT
+r (106)
+We have
+¯Rr=J2
+r (107)
+and
+Rr=¯Rr+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (108)
+SinceQris Hermitian we have
+QT
+r=Q∗
+r (109)
+whereQ∗
+ris the matrix whose elements are the complex conjugates of the cor re-
+sponding elements of Qr. So¯Rris twice the real part of Qrand−iJris twice the
+imaginary part.
+In Section 5we will show that Eq. ( 102) is essentially definitive of a SIC-POVM.
+To be more specific, let Lrbe any set of d2Hermitian matrices which constitute a
+basis for gl( d,C), and letCrbe the adjoint representative of Lrin that basis. Then
+we will show that Crhas the spectral decomposition
+Cr=Qr−QT
+r (110)
+whereQris a rankd−1 projector which is orthogonal to its own transpose if and
+only if the Lrare a family of SIC projectors up to multiplication by a sign and
+shifting by a multiple of the identity.
+Having stated our results let us now turn to the task of proving the m. We begin
+byderivingthespectraldecompositionof Tr. Multiplyingboth sidesoftheequation
+ΠrΠs=d+1
+dd2/summationdisplay
+t=1TrstΠt−K2
+rsI (111)
+by Πrwe find
+ΠrΠs=d+1
+dd2/summationdisplay
+t=1TrstΠrΠt−K2
+rsΠr16
+=(d+1)2
+d2d2/summationdisplay
+t=1(Tr)2
+stΠt−d+1
+dd2/summationdisplay
+t=1TrstK2
+rtI−K2
+rsΠr(112)
+We have
+d2/summationdisplay
+t=1TrstK2
+rt=1
+d+1d2/summationdisplay
+t=1Trst(dδrt+1)
+=1
+d+1
+dTrsr+d2/summationdisplay
+t=1Trst
+
+=2d
+d+1Tsrr
+=2d
+d+1K2
+rs (113)
+Consequently
+ΠrΠs=d+1
+dd2/summationdisplay
+t=1/parenleftbiggd+1
+d(Tr)2
+st−K2
+rsK2
+rt/parenrightbigg
+Πt−K2
+rsI (114)
+Comparing with Eq. ( 111) we deduce
+(Tr)2
+rs=d
+d+1Trst+d
+d+1K2
+rsK2
+rt (115)
+Now define
+/bardbler/an}bracketri}ht/an}bracketri}ht=/radicalbigg
+d+1
+2dd2/summationdisplay
+s=1K2
+rs/bardbls/an}bracketri}ht/an}bracketri}ht (116)
+where the basis kets /bardbls/an}bracketri}ht/an}bracketri}htare given by (in column vector form)
+/bardbl1/an}bracketri}ht/an}bracketri}ht=
+1
+0
+...
+0
+,/bardbl2/an}bracketri}ht/an}bracketri}ht=
+0
+1
+...
+0
+,...,/bardbld2/an}bracketri}ht/an}bracketri}ht=
+0
+0
+...
+1
+(117)
+It is easily verified that /bardbler/an}bracketri}ht/an}bracketri}htis normalized. Eq. ( 115) then becomes
+T2
+r=d
+d+1Tr+2d2
+(d+1)2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (118)
+Using Eq. ( 113) we find
+/an}bracketle{t/an}bracketle{ts/bardblTr/bardbler/an}bracketri}ht/an}bracketri}ht=/radicalbigg
+d+1
+2dd2/summationdisplay
+t=1TrstK2
+rt
+=/radicalbigg
+2d
+d+1K2
+rs
+=2d
+d+1/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht (119)
+So/bardbler/an}bracketri}ht/an}bracketri}htis an eigenvector of Trwith eigenvalue2d
+d+1.17
+Also define
+Qr=d+1
+dTr−2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (120)
+So in terms of the order-3 angle tensor the matrix elements of Qrare
+Qrst=d+1
+dKrsKrt/parenleftbig
+Ksteiθrst−KrsKrt/parenrightbig
+(121)
+Qris Hermitian (because Tris Hermitian). Moreover
+Q2
+r=(d+1)2
+d2T2
+r−4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl=Qr (122)
+SoQris a projection operator. Since
+Tr(Tr) =/summationdisplay
+uTruu=d2/summationdisplay
+u=1K2
+ru=d (123)
+we have
+Tr(Qr) =d−1 (124)
+We have thus proved that the spectral decomposition of Tris
+Tr=d
+d+1Qr+2d
+d+1/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (125)
+whereQris a rankd−1 projector, as claimed.
+We next prove that QT
+r/bardbler/an}bracketri}ht/an}bracketri}ht= 0. The fact that the components of /bardbler/an}bracketri}ht/an}bracketri}htin the
+standard basis are all real means
+/an}bracketle{t/an}bracketle{ts/bardblTT
+r/bardbler/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{ter/bardblTr/bardbls/an}bracketri}ht/an}bracketri}ht=2d
+d+1/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht (126)
+So/bardbler/an}bracketri}ht/an}bracketri}htis an eigenvector of TT
+ras well asTr, again with the eigenvalue2d
+d+1. In
+view of Eq. ( 120) it follows that QT
+r/bardbler/an}bracketri}ht/an}bracketri}ht= 0.
+Turning to the problem of showing that Qris orthogonal to its own transpose.
+We have
+QrQT
+r=/parenleftbiggd+1
+dTr−2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightbigg/parenleftbiggd+1
+dTT
+r−2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightbigg
+=(d+1)2
+d2TrTT
+r−4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (127)
+It follows from Eq. ( 24) that
+/an}bracketle{t/an}bracketle{ts/bardblTrTT
+r/bardblt/an}bracketri}ht/an}bracketri}ht=d2/summationdisplay
+u=1TrsuTrtu
+=GrsGrtd2/summationdisplay
+u=1GsuGtuGurGur (128)
+In view of Eq. ( 23) (i.e.the fact that every SIC-POVM is a 2-design) this implies
+/an}bracketle{t/an}bracketle{ts/bardblTrTT
+r/bardblt/an}bracketri}ht/an}bracketri}ht=2d
+d+1|Grs|2|Grt|2
+=2d
+d+1K2
+rsK2
+rt18
+=4d2
+(d+1)2/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardblt/an}bracketri}ht/an}bracketri}ht (129)
+So
+TrTT
+r=4d2
+(d+1)2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (130)
+and consequently
+QrQT
+r= 0 (131)
+Eqs. (102) and (103) are immediate consequences of the results already proved
+and the definitions of Jr,Rr.
+We definedthe Jmatricestobe theadjointrepresentativesofthe SIC-projecto rs,
+considered as a basis for the Lie algebra gl( d,C), and that is certainly a most
+important fact about them. However, the results of this section s how that, along
+with the vectors /bardbler/an}bracketri}ht/an}bracketri}ht, they actually determine the whole structure. Specifically,
+we have
+Qr=1
+2/parenleftbig
+Jr+J2
+r/parenrightbig
+(132)
+Rr=J2
+r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (133)
+Tr=d
+2(d+1)/parenleftig
+Jr+J2
+r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightig
+(134)
+Moreover, if we know the Tmatrices then we know the order-3 angle tensor, which
+in view of Theorem 3means we can reconstruct the SIC-projectors. Since the
+vectors/bardbler/an}bracketri}ht/an}bracketri}htare given, once and for all, this means that the problem of proving th e
+existenceofa SIC-POVMin dimension dis equivalent to the problem ofprovingthe
+existence of a certain remarkable structure in the adjoint repres entation of gl( d,C)
+(as we will see in more detail in Section 5).
+In the Introduction webegan with the concept ofa SIC-POVM,and then defined
+theJmatrices in terms of it. However, one could, if one wished, go in the op posite
+direction, and take the Lie algebraic structure to be primary, with t he SIC-POVM
+being the secondary, derivative entity.
+4.TheQ-QTProperty
+The next five sections are devoted to a study of the Jmatrices which, as we will
+see, have numerous interesting properties. We begin our investiga tion by trying to
+get some additional insight into what we will call the Q-QTproperty: namely, the
+fact that the Jmatrices have the spectral decomposition
+Jr=Qr−QT
+r (135)
+whereQrisarankd−1projectorwhichisorthogonaltoits owntranspose. We wish
+to characterize the general class of matrices which are of this typ e. The following
+theorem provides one such characterization.
+Theorem 4. LetAbe a Hermitian matrix. Then the following statements are
+equivalent:19
+(1)Ahas the spectral decomposition
+A=P−PT(136)
+wherePis a projector which is orthogonal to its own transpose.
+(2)Ais pure imaginary and A2is a projector.
+Proof.To show that (1) = ⇒(2) observe that the fact that Pis Hermitian means
+PT=P∗(137)
+whereP∗is the matrix whose elements are the complex conjugates of the cor re-
+sponding elements of P. So Eq. ( 136) implies that the components of Aare pure
+imaginary. Since PPT= 0 it also implies that A2is a projector.
+To show that (2) = ⇒(1) observe that the fact that A2is a projector means
+that the eigenvalues of A=±1 or 0. So
+A=P−P′(138)
+whereP,P′are orthogonal projectors. Since Ais pure imaginary we must have
+PT−(P′)T=AT=A∗=−A=P′−P (139)
+PTand (P′)Tare also orthogonal projectors. So if PT|ψ/an}bracketri}ht=|ψ/an}bracketri}ht, and|ψ/an}bracketri}htis nor-
+malized, we must have
+1 =/an}bracketle{tψ|PT|ψ/an}bracketri}ht
+=/angbracketleftbig
+ψ/vextendsingle/vextendsingle/parenleftbig
+PT−(P′)T/parenrightbig/vextendsingle/vextendsingleψ/angbracketrightbig
+=/an}bracketle{tψ|P′|ψ/an}bracketri}ht−/an}bracketle{tψ|P|ψ/an}bracketri}ht (140)
+Since
+0≤ /an}bracketle{tψ|P′|ψ/an}bracketri}ht ≤1 (141)
+0≤ /an}bracketle{tψ|P|ψ/an}bracketri}ht ≤1 (142)
+we must have /an}bracketle{tψ|P′|ψ/an}bracketri}ht= 1, implying P′|ψ/an}bracketri}ht=|ψ/an}bracketri}ht. Similarly P′|ψ/an}bracketri}ht=|ψ/an}bracketri}htimplies
+PT|ψ/an}bracketri}ht=|ψ/an}bracketri}ht. So
+P′=PT(143)
+/square
+We also have the following statement, inspired in part by Ref. [ 51],
+Theorem 5. The necessary and sufficient condition for a matrix Pto be a projector
+which is orthogonal to its own transpose is that
+P=SDST(144)
+whereSis an any real orthogonal matrix and Dhas the block-diagonal form
+D=
+σ ... 0 0...0
+............
+0... σ 0...0
+0...0 0...0
+............
+0...0 0...0
+(145)20
+with
+σ=1
+2/parenleftbigg
+1−i
+i1/parenrightbigg
+(146)
+In other words Dhasncopies ofσon the diagonal, where n= rank(P), and0
+everywhere else.
+Proof.Sufficiency is an immediate consequence of the fact that σis a rank 1 pro-
+jector such that σσT= 0.
+To prove necessity let dbe the dimension of the space and nthe rank of P. It
+will be convenient to define
+|1/an}bracketri}ht=
+1
+0
+...
+0
+,|2/an}bracketri}ht=
+0
+1
+...
+0
+, ... |d/an}bracketri}ht=
+0
+0
+...
+1
+(147)
+In terms of these basis vectors we have
+P=d/summationdisplay
+r,s=1Prs|r/an}bracketri}ht/an}bracketle{ts| (148)
+Now let |a1/an}bracketri}ht,...,|an/an}bracketri}htbe an orthonormal basis for the subspace onto which P
+projects, and let |a∗
+r/an}bracketri}htbe the column vector which is obtained from |ar/an}bracketri}htby tak-
+ing the complex conjugate of each of its components. Taking comple x conjugates
+on each side of the equation
+P|ar/an}bracketri}ht=|ar/an}bracketri}ht (149)
+gives
+P∗|a∗
+r/an}bracketri}ht=|a∗
+r/an}bracketri}ht (150)
+So|a∗
+1/an}bracketri}ht,...,|a∗
+n/an}bracketri}htis an orthonormal basis for the subspace onto which PT=P∗
+projects. Since PTis orthogonal to Pwe conclude that
+/an}bracketle{tar|a∗
+s/an}bracketri}ht= 0 (151)
+for allr,s.
+Next define vectors |b1/an}bracketri}ht,...,|b2n/an}bracketri}htby
+|b2r−1/an}bracketri}ht=1√
+2/parenleftbig
+|a∗
+r/an}bracketri}ht−|ar/an}bracketri}ht/parenrightbig
+(152)
+|b2r/an}bracketri}ht=i√
+2/parenleftbig
+|a∗
+r/an}bracketri}ht+|ar/an}bracketri}ht/parenrightbig
+(153)
+By construction these vectors are orthonormal and real. So we c an extend them
+to an orthonormal basis for the full space by adding a further d−2nvectors
+|b2n+1/an}bracketri}ht,...,|bd/an}bracketri}ht, which can also be chosen to be real. We have
+P=n/summationdisplay
+r=1|ar/an}bracketri}ht/an}bracketle{tar|
+=1
+2n/summationdisplay
+r=1/parenleftig
+|b2r−1/an}bracketri}ht/an}bracketle{tb2r−1|−i|b2r−1/an}bracketri}ht/an}bracketle{tb2r|+i|b2r/an}bracketri}ht/an}bracketle{tb2r−1|+|b2r/an}bracketri}ht/an}bracketle{tb2r|/parenrightig
+(154)21
+So if we define
+S=d/summationdisplay
+r=1|br/an}bracketri}ht/an}bracketle{tr| (155)
+thenSis a real orthogonal matrix such that
+P=SDST(156)
+where
+D=1
+2n/summationdisplay
+r=1/parenleftig
+|2r−1/an}bracketri}ht/an}bracketle{t2r−1|−i|2r−1/an}bracketri}ht/an}bracketle{t2r|+i|2r/an}bracketri}ht/an}bracketle{t2r−1|+|2r/an}bracketri}ht/an}bracketle{t2r|/parenrightig
+(157)
+is the matrix defined by Eq. ( 145). /square
+This result implies the following alternative characterization of the cla ss of ma-
+trices to which the Jmatrices belong
+Corollary 6. LetAbe a Hermitian matrix. Then the following statements are
+equivalent:
+(1)Ahas the spectral decomposition
+A=P−PT(158)
+wherePis a projector which is orthogonal to its own transpose.
+(2)There exists a real orthogonal matrix Ssuch that
+A=SDST(159)
+whereDhas the block diagonal form
+D=
+σy...0 0...0
+............
+0... σ y0...0
+0...0 0...0
+............
+0...0 0...0
+(160)
+σybeing the Pauli matrix
+σy=/parenleftbigg0−i
+i0/parenrightbigg
+(161)
+In other words Dhasncopies ofσyon the diagonal, where n=1
+2rank(A),
+and0everywhere else (note that a matrix of this type must have eve n rank).
+Proof.Immediate consequence of Theorem 5. /square
+5.Lie Algebraic Formulation of the Existence Problem
+This section is the core of the paper. We show that the problem of pr oving the
+existence of a SIC-POVM in dimension dis equivalent to the problem of proving
+the existence of an Hermitian basis for gl( d,C) all of whose elements have the Q-QT
+property. We hope that this new way of thinking will help make the SIC -existence
+problem more amenable to solution.
+The result we prove is the following:22
+Theorem 7. LetLrbe a set ofd2Hermitian matrices forming a basis for gl(d,C).
+LetCrstbe the structure constants relative to this basis, so that
+[Lr,Ls] =d2/summationdisplay
+t=1CrstLt (162)
+and letCrbe the matrix with matrix elements (Cr)st=Crst. Then the following
+statements are equivalent
+(1)EachCrhas the spectral decomposition
+Cr=Pr−PT
+r (163)
+wherePris a rankd−1projector which is orthogonal to its own transpose.
+(2)There exists a SIC-set Πr, a set of signs ǫr=±1and a real constant
+α/ne}ationslash=−1
+dsuch that
+Lr=ǫr(Πr+αI) (164)
+Remark. The restriction to values of α/ne}ationslash=−1
+dis needed to ensure that the matrices
+Lrare linearly independent, and therefore constitute a basis for gl( d,C) (otherwise
+they would all have trace = 0). The Q-QTproperty continues to hold even if α
+does =−1
+d.
+It will be seen that it is not only SIC-sets which have the Q-QTproperty, but
+also any set of operators obtained from a SIC-set by shifting by a c onstant and
+multiplying by an r-dependent sign. Sothe Q-QTpropertyis not strictly equivalent
+to the property of being a SIC-set. However, it could be said that t he properties
+are almost equivalent. In particular, the existence of an Hermitian b asis for gl(d,C)
+having the Q-QTproperty implies the existence of a SIC-POVM in dimension d,
+and conversely.
+Proof that (2) =⇒(1).Taking the trace on both sides of
+[Πr,Πs] =d2/summationdisplay
+t=1JrstΠt (165)
+we deduce that
+d2/summationdisplay
+t=1Jrst= 0 (166)
+Then from the definition of Lrin terms of Π rwe find
+Crst=ǫrǫsǫtJrst (167)
+Consequently
+Cr=Pr−PT
+r (168)
+where
+Pr=ǫrSQrS (169)
+Sbeing the symmetric orthogonal matrix
+S=
+ǫ10...0
+0ǫ2...0
+.........
+0 0... ǫ d2
+(170)
+The claim is now immediate.23
+Proof that (1) =⇒(2).Forthis we need to workharder. Since the proofis rather
+lengthy we will break it into a number of lemmas. We first collect a few ele mentary
+facts which will be needed in the sequel:
+Lemma 8. LetLrbe any Hermitian basis for gl(d,C), and letCrstandCrbe
+the structure constants and adjoint representatives as defi ned in the statement of
+Theorem 7. Letlr= Tr(Lr). Then
+(1)Thelrare not all zero.
+(2)TheCrstare pure imaginary and antisymmetric in the first pair of indi ces.
+(3)TheCrstare completely antisymmetric if and only if the Crare Hermitian.
+(4)In every case
+d2/summationdisplay
+t=1Crstlt= 0 (171)
+for allr,s.
+(5)In the special case that the Crare Hermitian
+d2/summationdisplay
+r=1lrLr=κI (172)
+where
+κ=1
+d
+d2/summationdisplay
+r=1l2
+r
+>0 (173)
+Proof.To prove (1) observe that if the lrwere all zero it would mean that the
+identity was not in the span of the Lr—contrary to the assumption that they form
+a basis.
+To prove(2) observethat taking Hermitian conjugates on both sid es of Eq. ( 162)
+gives
+−[Lr,Ls] =d2/summationdisplay
+t=1C∗
+rstLt (174)
+from which it follows that C∗
+rst=−Crst. The fact that Csrt=−Crstis an imme-
+diate consequence of the definition.
+(3) is now immediate.
+(4) is proved in the same way as Eq. ( 166).
+To prove (5) observe that if the Crare Hermitian it follows from (2) and (3) that
+d2/summationdisplay
+r=1lrCrst= 0 (175)
+for alls,t. Consequently the matrix
+d2/summationdisplay
+r=1lrLr (176)24
+commutes with everything. But the only matrices for which that is tr ue are multi-
+ples of the identity. It follows that
+d2/summationdisplay
+r=1lrLr=κI (177)
+for some real κ. Taking the trace on both sides of this equation we deduce
+d2/summationdisplay
+r=1l2
+r=dκ (178)
+The fact that κ>0 is a consequence of this and statement (1). /square
+We next observe that if the Crhave theQ-QTproperty they must, in particular,
+be Hermitian. It turns out that that is, by itself, already a very str ong constraint.
+Before stating the result it may be helpful if we explain the essential idea on
+which it depends. Although we have not done so before, and will not d o so again, it
+will be convenient to make use of the covariant/contravariantinde x notation which
+is often used to describe the structure constants. Define the me tric tensor
+Mrs= Tr(LrLs) (179)
+and letMrsbe its inverse. So
+d2/summationdisplay
+t=1MrtMts=Mr
+s=/braceleftigg
+1r=s
+0r/ne}ationslash=s(180)
+We can use these tensors to raise and lower indices (we use the Hilber t-Schmidt
+inner product for this purpose because the fact that gl( d,C) is not semi-simple
+means that its Killing form is degenerate [ 52–55]). In particular, the matrices
+Lr=d2/summationdisplay
+t=1MrsLs (181)
+are the basis dual to the Lr:
+Tr(LrLs) =Mr
+s (182)
+Suppose we now define structure constants ˜Crstby
+[Lr,Ls] =d2/summationdisplay
+t=1˜CrstLt(183)
+(so in terms of the Crstwe have ˜Ct
+rs=Crst). It follows from the relation
+˜Crst= Tr/parenleftbig
+[Lr,Ls]Lt/parenrightbig
+= Tr/parenleftbig
+Lr[Ls,Lt]/parenrightbig
+(184)
+that the ˜Crstare completely antisymmetric for any choice of the Lr. If we now
+require that the matrices Crbe Hermitian it means that, not only the ˜Crst, but
+also theCrstmust be completely antisymmetric. Since the two quantities are
+related by
+˜Crst=d2/summationdisplay
+u=1CrsuMut (185)25
+this is a very strong requirement. It means that the Lrmust, in a certain sense,
+be close to orthonormal (relative to the Hilbert-Schmidt inner prod uct). More
+precisely, it means we have the following lemma:
+Lemma 9. LetLr,CrstandCrbe defined as in the statement of Theorem 7, and
+letlr= Tr(Lr). Then the Crare Hermitian if and only if
+Tr(LrLs) =βδrs+γlrls (186)
+whereβ,γare real constants such that β >0andγ <1
+d.
+If this condition is satisfied we also have
+d2/summationdisplay
+r=1lrLr=β
+1−dγI (187)
+d2/summationdisplay
+r=1l2
+r=dβ
+1−dγ(188)
+Proof.To prove sufficiency observe that, in view of Eq. ( 185), the condition means
+˜Crst=βCrst+γltd2/summationdisplay
+u=1Crsulu (189)
+In view of Lemma 8, and the fact that β/ne}ationslash= 0, this implies
+Crst=1
+β˜Crst (190)
+Since the ˜Crstare completely antisymmetric we conclude that the Crstmust be
+also. It follows that the Crare Hermitian.
+To prove necessity let ˜Cr(respectively M) be the matrix whose matrix elements
+are˜Crst(respectively Mst). Then Eq. ( 185) can be written
+˜Cr=CrM (191)
+Taking the transpose (or, equivalently, the Hermitian conjugate) on both sides of
+this equation we find
+˜Cr=MCr (192)
+implying
+[M,Cr] = 0 (193)
+for allr. Since the Lrare a basis for gl( d,C) we deduce
+[M,adA] = 0 (194)
+for allA∈gl(d,C). Eq. (186) is a straightforward consequence of this, the fact
+that gl(d,C) has the direct sum decomposition CI⊕sl(d,C), the fact that sl( d,C)
+is simple, and Schur’s lemma [ 52–55]. However, for the benefit of the reader who is
+not so familiar with the theory of Lie algebras we will give the argument in a little
+more detail.26
+Given arbitrary A=/summationtextd2
+r=1arLr, let/bardblA/an}bracketri}ht/an}bracketri}htdenote the column vector
+/bardblA/an}bracketri}ht/an}bracketri}ht=
+a1
+a2
+...
+ad2
+(195)
+So
+/bardblLr/an}bracketri}ht/an}bracketri}ht=
+1
+0
+...
+0
+/bardblL2/an}bracketri}ht/an}bracketri}ht=
+0
+1
+...
+0
+/bardblLd2/an}bracketri}ht/an}bracketri}ht=
+0
+0
+...
+1
+(196)
+In view of Lemma 8we then have
+/bardblI/an}bracketri}ht/an}bracketri}ht=1
+κd2/summationdisplay
+r=1lr/bardblLr/an}bracketri}ht/an}bracketri}ht (197)
+Since
+Tr(A) =d2/summationdisplay
+r=1arlr=κ/an}bracketle{t/an}bracketle{tI/bardblA/an}bracketri}ht/an}bracketri}ht (198)
+we have that A∈sl(d,C) if and only if /an}bracketle{t/an}bracketle{tI/bardblA/an}bracketri}ht/an}bracketri}ht= 0.
+Now observe that it follows from Lemma 8and the definition of Mthat
+M/bardblI/an}bracketri}ht/an}bracketri}ht=κ/bardblI/an}bracketri}ht/an}bracketri}ht (199)
+IfMis a multiple of the identity we have Mrs=κδrsand the lemma is proved.
+OtherwiseMhas at least one more eigenvalue, βsay. Let Ebe the corresponding
+eigenspace. Since Eis orthogonal to /bardblI/an}bracketri}ht/an}bracketri}htit follows from Eq. ( 198) thatE⊆sl(d,C).
+SinceMcommutes with every adjoint representation matrix we have
+adAE⊆E (200)
+for allA∈sl(d,C). SoEis an ideal of sl( d,C). However sl( d,C) is a simple Lie
+algebra, meaning it has no proper ideals [ 52–55]. So we must have E= sl(d,C). It
+follows that if we define
+˜Lr=Lr−lr
+dI (201)
+then
+M/bardblLr/an}bracketri}ht/an}bracketri}ht=lr
+dM/bardblI/an}bracketri}ht/an}bracketri}ht+M/bardbl˜Lr/an}bracketri}ht/an}bracketri}ht (202)
+=κlr
+d/bardblI/an}bracketri}ht/an}bracketri}ht+β/bardbl˜Lr/an}bracketri}ht/an}bracketri}ht (203)
+=d2/summationdisplay
+s=1(βδrs+γlrls)/bardblLs/an}bracketri}ht/an}bracketri}ht (204)
+whereγ=1
+d/parenleftig
+1−β
+κ/parenrightig
+. Eqs. (186), (187) and (188) are now immediate (in view of
+Lemma8).27
+It remains to establish the bounds on β,γ. LetA=/summationtextd2
+r=1arLrbe any non-zero
+element of sl( d,C). Then/summationtextd2
+r=1arlr= 0, so in view of Eq. ( 186) we have
+0<Tr(A2) =βd2/summationdisplay
+r=1a2
+r (205)
+It follows that β >0. Also, using Lemma 8once more, we find
+lr=1
+κd2/summationdisplay
+s=1lsTr(LrLs)
+=βlr
+κ+γlr
+κd2/summationdisplay
+s=1l2
+s
+=lr/parenleftbiggβ
+κ+dγ/parenrightbigg
+(206)
+Since thelrcannot all be zero this implies
+β
+κ= 1−dγ (207)
+Sinceβ
+κ>0 we deduce that γ <1
+d. /square
+Eq. (186) only depends on the Crbeing Hermitian. If we make the assumption
+that theCrhave theQ-QTproperty we get a stronger statement:
+Corollary 10. LetLr,CrstandCrbe as defined in the statement of Theorem 7.
+Suppose that the Crhave the spectral decomposition
+Cr=Pr−PT
+r (208)
+wherePris a rankd−1projector which is orthogonal to its own transpose. Then
+(1)For allr
+Tr(Lr) =ǫ′
+rl (209)
+(2)For allr,s
+Tr(LrLs) =d
+d+1δrs+ǫ′
+rǫ′
+s
+d/parenleftbigg
+l2−1
+d+1/parenrightbigg
+(210)
+(3)
+d2/summationdisplay
+r=1ǫ′
+rLr=dlI (211)
+for some real constant l>0and signsǫ′
+r=±1.
+Proof.The proof relies on the fact that the Killing form for gl( d,C) is related to
+the Hilbert-Schmidt inner product by [ 55]
+Tr(adAadB) = 2dTr(AB)−2Tr(A)Tr(B) (212)
+Specializing to the case A=B=Lrand making use of the Q-QTproperty we find
+d−1 =dTr(L2
+r)−l2
+r (213)
+Using Lemma 9we deduce
+l2
+r=dβ−d+1
+1−dγ(214)28
+It follows that
+lr=ǫ′
+rl (215)
+for some real constant l≥0 and signs ǫ′
+r=±1. The fact that the Lrare a basis
+for gl(d,C) means the lrcannot all be zero. So we must in fact have l>0. Using
+this result in Eq. ( 188) we find
+β+d2l2γ=dl2(216)
+while Eq. ( 214) implies
+dβ+dl2γ=d−1+l2(217)
+This gives us a pair of simultaneous equations in βandγ. Solving them we obtain
+β=d
+d+1(218)
+γ=1
+dl2/parenleftbigg
+l2−1
+d+1/parenrightbigg
+(219)
+Substituting these expressions into Eqs. ( 186) and (187) we deduce Eqs. ( 210)
+and (211). /square
+The next lemma shows that each Lris a linear combination of a rank-1projector
+and the identity:
+Lemma 11. LetLbe any Hermitian matrix ∈gl(d,C)which is not a multiple of
+the identity. Then
+rank(ad L)≥2(d−1) (220)
+The lower bound is achieved if and only if Lis of the form
+L=ηI+ξP (221)
+wherePis a rank- 1projector and η,ξare any pair of real numbers. The eigenvalues
+ofadLare then ±ξ(each with multiplicity d−1) and0(with multiplicity d2−2d+2).
+Proof.Letλ1≥λ2≥ ··· ≥λdbe the eigenvalues of Larranged in decreasing
+order, and let |b1/an}bracketri}ht,|b2/an}bracketri}ht,...,|bd/an}bracketri}htbe the correspondingeigenvectors. We mayassume,
+without loss of generality, that the |br/an}bracketri}htare orthonormal. We have
+adL/parenleftbig
+|br/an}bracketri}ht/an}bracketle{tbs|/parenrightbig
+=/bracketleftbig
+L,|br/an}bracketri}ht/an}bracketle{tbs|/bracketrightbig
+= (λr−λs)|br/an}bracketri}ht/an}bracketle{tbs| (222)
+So the eigenvalues of ad Lareλr−λs. SinceLis not a multiple of the identity we
+must haveλr/ne}ationslash=λr+1for somerin the range 1 ≤r≤d−1. We then have that
+λs−λt/ne}ationslash= 0 if either s≤r<tort≤r<s. There are 2 r(d−r) such pairs s,t. So
+rank(ad L)≥2r(d−r)≥2(d−1) (223)
+Suppose, now that the lower bound is achieved. Then r(d−r) =d−1, implying
+thatr= 1 ord−1. Also we must have λs=λs+1for alls/ne}ationslash=r. So either
+L=λ2I+(λ1−λ2)|b1/an}bracketri}ht/an}bracketle{tb1| (224)
+or
+L=λd−1I−(λd−1−λd)|bd/an}bracketri}ht/an}bracketle{tbd| (225)
+Either way Land the spectrum of ad Lare as described. /square
+The final ingredient needed to complete the proof is29
+Lemma 12. LetLr,CrstandCrbe as defined in the statement of Theorem 7.
+Suppose that the Crhave the spectral decomposition
+Cr=Pr−PT
+r (226)
+wherePris a rankd−1projector which is orthogonal to its own transpose. Let l,
+ǫ′
+rbe as in the statement of Corollary 10. Then there is a fixed sign ǫ=±1such
+that
+Πr=ǫǫ′
+rLr−ǫl−1
+dI (227)
+is a rank- 1projector for all r.
+Proof.Define
+L′
+r=ǫ′
+rLr−l−1
+dI (228)
+Then it follows from Corollary 10that
+Tr(L′
+r) = 1 (229)
+for allr,
+Tr(L′
+rL′
+s) =dδrs+1
+d+1(230)
+for allr,s, and
+d2/summationdisplay
+r=1L′
+r=dI (231)
+It is also easily seen that if we define C′
+rst=ǫ′
+rǫ′
+sǫ′
+tCrstthen
+[L′
+r,L′
+s] =d2/summationdisplay
+t=1C′
+rstL′
+t (232)
+and
+C′
+r=P′
+r−P′
+rT(233)
+whereP′
+ris a rank-1 projector which is orthogonal to its own transpose (se e the
+first part of the proof of Theorem 7). In particular
+rank/parenleftbig
+adL′r/parenrightbig
+= 2(d−1) (234)
+and the eigenvalues of ad L′rall equal to ±1 or 0. So, taking account of the fact
+that Tr(L′
+r) = 1, we can use Lemma 11to deduce that there is a family of rank-1
+projectors Π′
+rand signsξr=±1 such that
+L′
+r=ξrΠ′
+r+1−ξr
+dI (235)
+Ifξr= +1 (respectively −1) for allrthen Eq. ( 227) holds with Π r= Π′
+randǫ= +1
+(respectively −1). Also, if d= 2 thenL′
+ris a rank-1 projector irrespective of the
+value ofξr, so Eq. ( 227) holds with Π r=L′
+randǫ= +1. The problem therefore
+reduces to showing that if d >2 it cannot happen that ξr= +1 for some values
+ofrand−1 for others. We will do this by assuming the contrary and deducing a
+contradiction.
+Letmbe the number of values of rfor whichξr= +1. We are assuming that
+mis in the range 1 ≤m≤d2−1. We may also assume, without loss of generality,30
+that the labelling is such that ξr= +1 for the first mvalues ofr, and−1 for the
+rest. So
+L′
+r=/braceleftigg
+Π′
+r ifr≤m
+2
+dI−Π′
+rifr>m(236)
+Now define
+˜Trst= Tr/parenleftbig
+L′
+rL′
+sL′
+t/parenrightbig
+(237)
+Eqs. (230) and (231) mean that the same argument which led to Eq. ( 9) can be
+used to deduce
+L′
+rL′
+s=d+1
+d
+d2/summationdisplay
+t=1˜TrstL′
+t
+−K2
+rsI (238)
+SinceL′
+1is a projector it follows that
+L′
+1L′
+s=/parenleftbig
+L′
+1/parenrightbig2L′
+s=d+1
+d
+d2/summationdisplay
+t=1˜T1stL′
+1L′
+t
+−K2
+1sL′
+1 (239)
+By essentially the same argument which led to Eq. ( 118) we can use this to infer
+/parenleftbig˜T′
+1/parenrightbig2=d
+d+1˜T1+2d2
+(d+1)2/bardble1/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{te1/bardbl (240)
+where˜T′
+1is the matrix with matrix elements ˜T′
+1rsand/bardble1/an}bracketri}ht/an}bracketri}htis the vector defined by
+Eq.(116). Asbefore /bardble1/an}bracketri}ht/an}bracketri}htisaneigenvectorof ˜T′
+1witheigenvalue2d
+d+1. Consequently
+the matrix
+˜Q1=d+1
+d˜T′
+1−2/bardble1/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{te1/bardbl (241)
+is a projector. But that means Tr( ˜Q1) must be an integer. We now use this to
+derive a contradiction.
+It follows from Eq. ( 236) that
+(L′
+r)2=/braceleftigg
+L′
+r r≤m
+2(d−2)
+d2I−d−4
+dL′
+rr>m(242)
+Consequently
+˜T1rr=/braceleftigg
+K2
+1r r≤m
+2(d−2)
+d2−d−4
+dK2
+1rr>m(243)
+and so
+Tr(˜Q1) =d+1
+dd2/summationdisplay
+r=1˜T1rr−2
+=d+1−4d2+2m(d−2)
+d3(244)
+So if Tr( ˜Q1) is an integer/parenleftbig
+4d2+2n(d−2)/parenrightbig
+/d3must also be an integer. But the
+fact that 1 ≤m<d2, together with the fact that d>2 means
+4
+d<4d2+2m(d−2)
+d3<2 (245)31
+Ifd= 3 or 4 there are no integers in this interval, which gives us a contrad iction
+straight away. If, on the other hand, d≥5 there is the possibility
+4d2+2m(d−2)
+d3= 1 (246)
+implying
+m=d2(d−4)
+2(d−2)(247)
+This equationhasthe solution d= 6,m= 9(this is in fact the only integersolution,
+ascanbe seenfrom ananalysisofthe possible primefactorizationso fthe numerator
+and denominator on the right hand side). To eliminate this possibility de fine
+L′′
+r=2
+dI−L′
+d2+1−r (248)
+for allr. It is easily verified that
+Tr(L′′
+rL′′
+s) =dδrs+1
+d+1(249)
+d2/summationdisplay
+r=1L′′
+r=dI (250)
+and
+L′′
+r=/braceleftigg
+Πr r≤d2−m
+2
+dI−Πrr>d2−m(251)
+So we can go through the same argument as before to deduce
+d2−m=d2(d−4)
+2(d−2)(252)
+Eqs. (247) and (252) have no joint solutions at all with d/ne}ationslash= 0, integer or otherwise.
+/square
+To complete the proof of Theorem 7observe that Eqs. ( 210) and (227) imply
+Tr(ΠrΠs) =dδrs+1
+d+1(253)
+So the Π rare a SIC-set. Moreover
+Lr=ǫr(Πr+αI) (254)
+whereǫr=ǫǫ′
+randα= (ǫl−1)/d.
+6.The Algebra sl(d,C)
+The motivation for this paper is the hope that a Lie algebraic perspec tive may
+cast some light on the SIC-existence problem, and on the mathemat ics of SIC-
+POVMs generally. We have focused on gl( d,C) as that is the case where the con-
+nection with Lie algebras seems most straightforward. However, it may be worth
+mentioning that a SIC-POVM also gives rise to an interesting geometr ical structure
+in sl(d,C) (the Lie algebra consisting of all trace-zero d×dcomplex matrices).32
+Let Πrbe a SIC-set and define
+Br=/radicaligg
+d+1
+2(d2−1)/parenleftbigg
+Πr−1
+dI/parenrightbigg
+(255)
+SoBr∈sl(d,C). Let
+/an}bracketle{tA,A′/an}bracketri}ht= Tr(ad AadA′) = 2dTr(AA′) (256)
+be the Killing form [ 55] on sl(d,C). Then
+/an}bracketle{tBr,Bs/an}bracketri}ht=/braceleftigg
+1 r=s
+−1
+d2−1r/ne}ationslash=s(257)
+So theBrform a regular simplex in sl( d,C). Since sl( d,C) isd2−1 dimensional
+theBrare an overcomplete set. However, the fact that
+d2/summationdisplay
+r=1Br= 0 (258)
+means that for each A∈sl(d,C) there is a unique set of numbers arsuch that
+A=d2/summationdisplay
+r=1arBr (259)
+and
+d2/summationdisplay
+r=1ar= 0 (260)
+Thearcan be calculated using
+ar=d2−1
+d2/an}bracketle{tA,Br/an}bracketri}ht (261)
+Similarly, given any linear transformation M: sl(d,C)→sl(d,C), there is a unique
+set of numbers Mrssuch that
+MBr=d2/summationdisplay
+s=1MrsBs (262)
+and
+d2/summationdisplay
+s=1Mrs=d2/summationdisplay
+s=1Msr= 0 (263)
+for allr. TheMrscan be calculated using
+Mrs=d2−1
+d2/an}bracketle{tBs,MBr/an}bracketri}ht (264)
+In short, the Brretain many analogous properties of, and can be used in much the
+same way as, a basis. It could be said that they form a simplicial basis.33
+7.Further Identities
+In the preceding pages we have seen that there are five different f amilies of ma-
+trices naturally associated with a SIC-POVM: namely, the projecto rsQrtogether
+with the matrices
+Jr=Qr−QT
+r (265)
+¯Rr=Qr+QT
+r (266)
+Rr=Qr+QT
+r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (267)
+Tr=d
+d+1Qr+2d
+d+1/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (268)
+(see Section 3). As we noted previously, it is possible to define everything in terms
+of the adjoint representation matrices Jrand the rank-1 projectors /bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl:
+Qr=1
+2Jr(Jr+I) (269)
+¯Rr=J2
+r (270)
+Rr=J2
+r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (271)
+Tr=d
+2(d+1)Jr(Jr+I)+2d
+d+1/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (272)
+In that sense the structure constants of the Lie algebra, supple mented with the
+vectors/bardbler/an}bracketri}ht/an}bracketri}ht, determine everything else.
+In the next section we will show that there are some interesting geo metrical
+relationships between the hyperplanes onto which Qr,QT
+rand¯Rrproject. In this
+section, as a preliminary to that investigation, we prove a number of identities
+satisfied by the Q,Jand¯Rmatries. We start by computing their Hilbert-Schmidt
+inner products:
+Theorem 13. For allr,s
+Tr/parenleftbig
+QrQs/parenrightbig
+=d3δrs+d2−d−1
+(d+1)2(273)
+Tr/parenleftbig
+QrQT
+s/parenrightbig
+=d2(1−δrs)
+(d+1)2(274)
+Tr/parenleftbig
+JrJs/parenrightbig
+=2(d2δrs−1)
+d+1(275)
+Tr/parenleftbig¯Rr¯Rs/parenrightbig
+=2(d−1)(d2δrs+2d+1)
+(d+1)2(276)
+Tr/parenleftbig
+Jr¯Rs/parenrightbig
+= 0 (277)
+Proof.Let us first calculate some auxiliary quantities. It follows from the de finition
+ofTr, andthe factthat the matrix P=1
+dGdefined byEq.( 63) isarankdprojector,
+that
+Tr(TrTs) =d2/summationdisplay
+u,v=1TruvTsvu34
+=d2/summationdisplay
+u,v=1K2
+uvGruGusGsvGvr
+=d
+d+1d2/summationdisplay
+u=1K2
+ruK2
+su+d4
+d+1d2/summationdisplay
+u,v=1PruPusPsvPvr
+=d2(dδrs+d+2)
+(d+1)3+d4
+d+1/vextendsingle/vextendsinglePrs/vextendsingle/vextendsingle2
+=d2(dδrs+d+2)
+(d+1)3+d2
+d+1K2
+rs
+=d2/parenleftbig
+d(d+2)δrs+2d+3/parenrightbig
+(d+1)3(278)
+Also
+Tr/parenleftbig
+TrTT
+s/parenrightbig
+=d2/summationdisplay
+u,v=1TruvTsuv
+=d2/summationdisplay
+u=1GruGsu
+d2/summationdisplay
+v=1GuvGuvGvrGvs
+
+=2d
+d+1d2/summationdisplay
+u=1GruGsuGurGus
+=2d2
+(d+1)2/parenleftbig
+1+K2
+rs/parenrightbig
+=2d2(dδrs+d+2)
+(d+1)3(279)
+where we made two applications of Eq. ( 23) (i.e.the fact that every SIC-POVM is
+a 2-design). Finally, it is a straightforward consequence of the defi nitions ofTr,TT
+r
+and/bardbler/an}bracketri}ht/an}bracketri}htthat
+/an}bracketle{t/an}bracketle{ter/bardblTs/bardbler/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{ter/bardblTT
+s/bardbler/an}bracketri}ht/an}bracketri}ht
+=d+1
+2dd2/summationdisplay
+u,v=1TsuvK2
+ruK2
+rv
+=1
+2d(d+1)
+d2Tsrr+dd2/summationdisplay
+v=1Tsrv+dd2/summationdisplay
+u=1Tsur+d2/summationdisplay
+u,v=1Tsuv
+
+=d
+2(d+1)/parenleftbig
+3K2
+rs+1/parenrightbig
+=d(3dδrs+d+4)
+2(d+1)2(280)35
+and
+/an}bracketle{t/an}bracketle{ter/bardbles/an}bracketri}ht/an}bracketri}ht=d+1
+2dd2/summationdisplay
+u=1K2
+ruK2
+su
+=dδrs+d+2
+2(d+1)(281)
+Using these results in the expressions
+Tr/parenleftbig
+QrQs/parenrightbig
+= Tr/parenleftigg/parenleftbiggd+1
+dTr−2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightbigg/parenleftbiggd+1
+dTs−2/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardbl/parenrightbigg/parenrightigg
+(282)
+and
+Tr/parenleftbig
+QrQT
+s/parenrightbig
+= Tr/parenleftigg/parenleftbiggd+1
+dTr−2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightbigg/parenleftbiggd+1
+dTT
+s−2/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardbl/parenrightbigg/parenrightigg
+(283)
+the first two statements follow. The remaining statements are imme diate conse-
+quences of these and the fact that
+Jr=Qr−QT
+r (284)
+¯Rr=Qr+QT
+r (285)
+/square
+Now define
+/bardblv0/an}bracketri}ht/an}bracketri}ht=1
+dd2/summationdisplay
+r=1/bardblr/an}bracketri}ht/an}bracketri}ht (286)
+where/bardblr/an}bracketri}ht/an}bracketri}htis the basis defined in Eq. ( 117). The following result shows (among
+other things) that the subspaces onto which the Qr(respectively QT
+r,Rr) project
+span the orthogonal complement of /bardblv0/an}bracketri}ht/an}bracketri}ht.
+Theorem 14. For allr
+Qr/bardblv0/an}bracketri}ht/an}bracketri}ht=QT
+r/bardblv0/an}bracketri}ht/an}bracketri}ht=Jr/bardblv0/an}bracketri}ht/an}bracketri}ht=Rr/bardblv0/an}bracketri}ht/an}bracketri}ht= 0 (287)
+Moreover
+d2/summationdisplay
+r=1Qr=d2/summationdisplay
+r=1QT
+r=d2
+d+1/parenleftbig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightbig
+(288)
+d2/summationdisplay
+r=1Jr= 0 (289)
+d2/summationdisplay
+r=1¯Rr=2d2
+d+1/parenleftbig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightbig
+(290)
+Proof.Some of this is a straightforward consequence of the fact that Jris the
+adjoint representative of Π r. Since
+d2/summationdisplay
+s=1Πs=dI (291)36
+we must have
+d2/summationdisplay
+s,t=1JrstΠt=d2/summationdisplay
+s=1adΠrΠs= 0 (292)
+In view of the antisymmetry of the Jrstit follows that
+d2/summationdisplay
+r=1Jr= 0 (293)
+and
+Jr/bardblv0/an}bracketri}ht/an}bracketri}ht= 0 (294)
+Using the relations
+Qr=1
+2Jr(Jr+I) (295)
+QT
+r=1
+2Jr(Jr−I) (296)
+¯Rr=J2
+r (297)
+we deduce
+Qr/bardblv0/an}bracketri}ht/an}bracketri}ht=QT
+r/bardblv0/an}bracketri}ht/an}bracketri}ht=¯Rr/bardblv0/an}bracketri}ht/an}bracketri}ht= 0 (298)
+It remains to prove Eqs. ( 288) and (290). It follows from Eq. ( 120) that
+d2/summationdisplay
+r=1Qrst=d+1
+dd2/summationdisplay
+r=1Trst−2d2/summationdisplay
+r=1/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardblt/an}bracketri}ht/an}bracketri}ht
+= (d+1)K2
+st−d+1
+dd2/summationdisplay
+r=1K2
+rsK2
+rt
+=d2δst−1
+d+1(299)
+from which it follows
+d2/summationdisplay
+r=1Qr=d2/summationdisplay
+r=1QT
+r=d2
+d+1/parenleftbig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightbig
+(300)
+Eq. (290) follows from this and the fact that Rr=Qr+QT
+r.
+/square
+8.Geometrical Considerations
+In this section we show that there are some interesting geometrica l relationships
+between the subspaces onto which the operators Qr,QT
+rand¯Rrproject. The
+original motivation for this work was an observation concerning the subspaces onto
+which the ¯Rrproject. ¯Rris a real matrix, and so it defines a 2( d−2) subspace
+inRd2, which we will denote Rr. We noticed that for each pair of distinct indices
+randsthe intersection Rr∩Rsis a 1-dimensional line. This led us to speculate
+that a set of hyperplanes parallel to the Rrmight be the edges of an interesting
+polytope. We continue to think that this could be the case. Unfortu nately we have
+not been able to prove it. However, it appears to us that the result s we obtained37
+while trying to prove it have an interest which is independent of the tr uth of the
+motivating speculation.
+We will begin with some terminology. Let Pbe any projector (on either RN
+orCN), letPbe the subspace onto which Pprojects, and let |ψ/an}bracketri}htbe any non-zero
+vector. Then we define the angle between |ψ/an}bracketri}htandPin the usual way, to be
+θ= cos−1/parenleftigg/vextenddouble/vextenddoubleP|ψ/an}bracketri}ht/vextenddouble/vextenddouble
+/vextenddouble/vextenddouble|ψ/an}bracketri}ht/vextenddouble/vextenddouble/parenrightigg
+(301)
+(soθis the smallest angle between |ψ/an}bracketri}htand any of the vectors in P).
+Suppose, now, that P′is another projector, and let P′be the subspace onto
+whichP′projects. We will say that P′is uniformly inclined to Pif every vector in
+P′makes the same angle θwithP. Ifθ= 0 this means that P′⊆P, while ifθ=π
+2
+it means P′⊥P. Suppose, on the other hand, that 0 < θ <π
+2. Let|u′
+1/an}bracketri}ht,...,|u′
+n/an}bracketri}ht
+be any orthonormal basis for P′, and define |ur/an}bracketri}ht= secθP|u′
+r/an}bracketri}ht. Then/an}bracketle{tur|ur/an}bracketri}ht= 1
+for allr. Moreover, if P,P′are complex projectors,
+/an}bracketle{tu′
+r+eiφu′
+s|P|u′
+r+eiφu′
+s/an}bracketri}ht= 2cos2θ/parenleftig
+1+Re/parenleftbig
+eiφ/an}bracketle{tur|us/an}bracketri}ht/parenrightbig/parenrightig
+(302)
+for allφandr/ne}ationslash=s. On the other hand it follows from the assumption that P′is
+uniformly inclined to Pthat
+/an}bracketle{tu′
+r+eiφu′
+s|P|u′
+r+eiφu′
+s/an}bracketri}ht= 2cos2θ (303)
+for allφandr/ne}ationslash=s. Consequently
+/an}bracketle{tur|us/an}bracketri}ht=δrs (304)
+for allr,s. It is easily seen that the same is true if P,P′are real projectors.
+Suppose we now make the further assumption that dim( P′) = dim( P) =n. Then
+|u1/an}bracketri}ht,...,|un/an}bracketri}htis an orthonormal basis for P, and we can write
+P=n/summationdisplay
+r=1|ur/an}bracketri}ht/an}bracketle{tur| (305)
+P′=n/summationdisplay
+r=1|u′
+r/an}bracketri}ht/an}bracketle{tu′
+r| (306)
+Observe that
+/an}bracketle{tu′
+r|us/an}bracketri}ht=/an}bracketle{tu′
+r|P|us/an}bracketri}ht= cosθ/an}bracketle{tur|us/an}bracketri}ht= cosθδrs (307)
+for allr,s. Consequently
+P′|ur/an}bracketri}ht= cosθ|ur/an}bracketri}ht (308)
+for allr. It follows that
+/vextenddouble/vextenddoubleP′|ψ/an}bracketri}ht/vextenddouble/vextenddouble=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay
+r=1cosθ/an}bracketle{tur|ψ/an}bracketri}ht|u′
+r/an}bracketri}ht/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble= cosθ/vextenddouble/vextenddouble|ψ/an}bracketri}ht/vextenddouble/vextenddouble (309)
+for all|ψ/an}bracketri}ht ∈P. SoPis uniformly inclined to P′at the same angle θ.
+It follows from Eqs. ( 305) and (306) that
+PP′P= cos2θP (310)
+P′PP′= cos2θP′(311)
+Eq. (310), or equivalently Eq. ( 311), is not only necessary but also sufficient for
+the subspaces to be uniformly inclined. In fact, let P,P′be any two subspaces38
+which have the same dimension n, but which are not assumed at the outset to be
+uniformly inclined, and let P,P′be the corresponding projectors. Suppose
+PP′P= cos2θP (312)
+for someθin the range 0 ≤θ≤π
+2. It is immediate that P=P′ifθ= 0, and
+P⊥P′ifθ=π
+2. Either way, the subspaces are uniformly inclined. Suppose, on
+the other hand, that 0 <θ<π
+2. Let|u′
+1/an}bracketri}ht,...,|u′
+n/an}bracketri}htbe any orthonormal basis for P′,
+and define |ur/an}bracketri}ht= secθP|u′
+r/an}bracketri}ht. Eq. (305) then implies
+P= sec2θn/summationdisplay
+r=1P|u′
+r/an}bracketri}ht/an}bracketle{tu′
+r|P=n/summationdisplay
+r=1|ur/an}bracketri}ht/an}bracketle{tur| (313)
+Given any |ψ/an}bracketri}ht ∈Pwe have
+|ψ/an}bracketri}ht=P|ψ/an}bracketri}ht=n/summationdisplay
+r=1/an}bracketle{tur|ψ/an}bracketri}ht|ur/an}bracketri}ht (314)
+Since dim( P) =nit follows that the |ur/an}bracketri}htare linearly independent. In particular
+|ur/an}bracketri}ht=P|ur/an}bracketri}ht=n/summationdisplay
+s=1/an}bracketle{tus|ur/an}bracketri}ht|us/an}bracketri}ht (315)
+Since the |ur/an}bracketri}htare linearly independent this means
+/an}bracketle{tus|ur/an}bracketri}ht=δrs (316)
+So the|ur/an}bracketri}htare an orthonormal basis for P. It follows, that if |ψ′/an}bracketri}htis any vector in
+P′, then
+/vextenddouble/vextenddoubleP|ψ′/an}bracketri}ht/vextenddouble/vextenddouble=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay
+r=1/an}bracketle{tu′
+r|ψ′/an}bracketri}htP|u′
+r/an}bracketri}ht/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble= cosθ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay
+r=1/an}bracketle{tu′
+r|ψ′/an}bracketri}ht|ur/an}bracketri}ht/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble= cosθ/vextenddouble/vextenddouble|ψ′/an}bracketri}ht/vextenddouble/vextenddouble(317)
+implying that P′is uniformly inclined to Pat angleθ.
+It will be convenient to summarise all this in the form of a lemma:
+Lemma 15. LetP,P′be any two subspaces, real or complex, having the same
+dimensionn. LetP,P′be the corresponding projectors. Then the following state-
+ments are equivalent:
+(a)P′is uniformly inclined to Pat angleθ.
+(b)Pis uniformly inclined to P′at angleθ.
+(c)
+PP′P= cos2θP (318)
+(d)
+P′PP′= cos2θP′(319)
+Suppose these conditions are satisfied for some θin the range 0< θ <π
+2, and
+let|u1/an}bracketri}ht,...|un/an}bracketri}htbe any orthonormal basis for P. Then there exists an orthonormal
+basis|u′
+1/an}bracketri}ht,...,|u′
+n/an}bracketri}htforP′such that
+P′|ur/an}bracketri}ht= cosθ|u′
+r/an}bracketri}ht (320)
+P|u′
+r/an}bracketri}ht= cosθ|ur/an}bracketri}ht (321)
+We are now in a position to state the main results of this section. Let Qr
+(respectively ¯Qr) be the subspace onto which Qr(respectively QT
+r) projects. We
+then have39
+Theorem 16. For each pair of distinct indices r,sthe subspaces Qr,¯Qrhave the
+orthogonal decomposition
+Qr=Q0
+rs⊕Qrs (322)
+¯Qr=¯Q0
+rs⊕¯Qrs (323)
+where
+Q0
+rs⊥Qrs dim(Q0
+rs) = 1 dim( Qrs) =d−2
+¯Q0
+rs⊥¯Qrs dim(¯Q0
+rs) = 1 dim( ¯Qrs) =d−2
+We have
+(a)Relation of the subspaces QrandQs:
+(1)Q0
+rs⊥QsrandQrs⊥Q0
+sr.
+(2)Q0
+rsandQ0
+srare inclined at angle cos−1/parenleftbig1
+d+1/parenrightbig
+.
+(3)QrsandQsrare uniformly inclined at angle cos−1/parenleftig
+1√d+1/parenrightig
+.
+(b)Relation of the subspaces ¯Qrand¯Qs:
+(1)¯Q0
+rs⊥¯Qsrand¯Qrs⊥¯Q0
+sr.
+(2)¯Q0
+rsand¯Q0
+srare inclined at angle cos−1/parenleftbig1
+d+1/parenrightbig
+.
+(3)¯Qrsand¯Qsrare uniformly inclined at angle cos−1/parenleftig
+1√d+1/parenrightig
+.
+(c)Relation of the subspaces Qrand¯Qs:
+(1)Q0
+rs⊥¯Qsr,Qrs⊥¯Q0
+srandQrs⊥¯Qsr.
+(2)Q0
+rsand¯Q0
+srare inclined at angle cos−1/parenleftbigd
+d+1/parenrightbig
+.
+The relations between these subspaces are, perhaps, easier to a ssimilate if pre-
+sented pictorially. In the following diagrams the line joining each pair of subspaces
+is labelled with the cosine of the angle between them. In particular a 0 o n the line
+joining two subspaces indicates that they are orthogonal.
+Q0
+rs Qrs
+Q0
+sr Qsr0
+01
+d+11√d+1
+
+0❅
+❅
+❅
+❅
+❅
+❅❅0¯Q0
+rs¯Qrs
+¯Q0
+sr¯Qsr0
+01
+d+11√d+1
+
+0❅
+❅
+❅
+❅
+❅
+❅❅0
+Q0
+rs Qrs
+¯Q0
+sr¯Qsr0
+0d
+d+10
+
+0❅
+❅
+❅
+❅
+❅
+❅❅040
+Wewillprovethistheorembelow. Beforedoingso,however,letusst atetheother
+mainresult ofthis section. Let Rrbe the subspace ontowhichthe ¯Rrproject. Since
+¯Rris a real matrix we regard Rras a subspace of Rd2. We have
+Theorem 17. For each pair of distinct indices r,sthe subspace Rrhas the decom-
+position
+Rr=R0
+rs⊕R1
+rs⊕Rrs (324)
+whereR0
+rs,R1
+rs,Rrsare pairwise orthogonal and
+dim(R0
+rs) = 1 dim( R1
+rs) = 1 dim( Rrs) = 2d−4 (325)
+We have
+(1)R0
+rs=R0
+sr.
+(2)R1
+rs⊥RsrandRrs⊥R1
+sr.
+(3)R1
+rsandR1
+srare inclined at angle cos−1/parenleftbigd−1
+d+1/parenrightbig
+.
+(4)RrsandRsrare uniformly inclined at angle cos−1/parenleftig/radicalig
+1
+d+1/parenrightig
+In particular, the subspaces ¯Rrand¯Rsintersect in a line.
+In diagrammatic form the relations between these subspaces are
+R0
+rs=R0
+srR1
+rs Rrs
+R1
+sr Rsr0
+0d−1
+d+1/radicalig
+1
+d+1
+
+0❅
+❅
+❅
+❅
+❅
+❅❅0✟✟✟✟✟0
+❍❍❍❍❍00
+0
+where, as before, each line is labelled with the cosine of the angle betw een the two
+subspaces it connects.
+Proof of Theorem 16.Let/bardbl1/an}bracketri}ht/an}bracketri}ht,...,/bardbld2/an}bracketri}ht/an}bracketri}htbethestandardbasisfor Hd2, asdefined
+by Eq. (117). For each pair of distinct indices r,sdefine
+/bardblfrs/an}bracketri}ht/an}bracketri}ht=i√
+d+1Qr/bardbls/an}bracketri}ht/an}bracketri}ht (326)
+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=−i√
+d+1QT
+r/bardbls/an}bracketri}ht/an}bracketri}ht (327)
+The significance of these vectors is that /bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl(respectively /bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl) will
+turn out to be the projectoronto the 1-dimensionalsubspace Q0
+rs(respectively ¯Q0
+rs).41
+Note that the fact that Qris Hermitian means
+QT
+r=Q∗
+r (328)
+(whereQ∗
+ris the matrix whose elements are the complex conjugates of the cor re-
+sponding elements of Qr). Consequently
+/an}bracketle{t/an}bracketle{tt/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=/parenleftig
+/an}bracketle{t/an}bracketle{tt/bardblfrs/an}bracketri}ht/an}bracketri}ht/parenrightig∗
+(329)
+for allr,s,t.
+It is easily seen that /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
+rs/an}bracketri}ht/an}bracketri}htare normalized. In fact, it follows from
+Eqs. (116) and (120) that
+/an}bracketle{t/an}bracketle{tfrs/bardblfrs/an}bracketri}ht/an}bracketri}ht= (d+1)/an}bracketle{t/an}bracketle{ts/bardblQr/bardbls/an}bracketri}ht/an}bracketri}ht
+=(d+1)2
+dTrss−2(d+1)/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbls/an}bracketri}ht/an}bracketri}ht
+=(d+1)2
+d/parenleftbig
+K2
+rs−K4
+rs/parenrightbig
+= 1 (330)
+for allr/ne}ationslash=s. In view of Eq. ( 329) we then have
+/an}bracketle{t/an}bracketle{tf∗
+rs/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=/parenleftig
+/an}bracketle{t/an}bracketle{tfrs/bardblfrs/an}bracketri}ht/an}bracketri}ht/parenrightig∗
+= 1 (331)
+for allr/ne}ationslash=s. The fact that QrQT
+r= 0 means we also have
+/an}bracketle{t/an}bracketle{tfrs/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht= 0 (332)
+for allr/ne}ationslash=s.
+Note that, although we required that r/ne}ationslash=sin the definitions of /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht,
+the definitions continue to make sense when r=s. However, the vectors are then
+zero (as can be seen by setting r=sin Eq. (121)).
+The vectors /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
+rs/an}bracketri}ht/an}bracketri}htsatisfy a number of identities, which it will be conve-
+nient to collect in a lemma:
+Lemma 18. For allr/ne}ationslash=s
+/bardblfrs/an}bracketri}ht/an}bracketri}ht=−/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht+i/radicalbigg
+2
+d/parenleftig
+/bardbles/an}bracketri}ht/an}bracketri}ht−/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
+(333)
+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=−/bardblfsr/an}bracketri}ht/an}bracketri}ht−i/radicalbigg
+2
+d/parenleftig
+/bardbles/an}bracketri}ht/an}bracketri}ht−/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
+(334)
+(where/bardbler/an}bracketri}ht/an}bracketri}htis the vector defined by Eq. ( 116))
+Qr/bardblfrs/an}bracketri}ht/an}bracketri}ht=/bardblfrs/an}bracketri}ht/an}bracketri}ht QT
+r/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht (335)
+QT
+r/bardblfrs/an}bracketri}ht/an}bracketri}ht= 0 Qr/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht= 0 (336)
+Qs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−1
+d+1/bardblfsr/an}bracketri}ht/an}bracketri}ht QT
+s/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=−1
+d+1/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht(337)
+QT
+s/bardblfrs/an}bracketri}ht/an}bracketri}ht=−d
+d+1/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht Qs/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht=−d
+d+1/bardblfsr/an}bracketri}ht/an}bracketri}ht(338)
+/an}bracketle{t/an}bracketle{tfrs/bardblfsr/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tf∗
+rs/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht=−1
+d+1(339)42
+/an}bracketle{t/an}bracketle{tfrs/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tf∗
+rs/bardblfsr/an}bracketri}ht/an}bracketri}ht=−d
+d+1(340)
+Proof.It follows from Eqs. ( 116) and (120) that
+/an}bracketle{t/an}bracketle{tt/bardblfrs/an}bracketri}ht/an}bracketri}ht+/an}bracketle{t/an}bracketle{tt/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht=i√
+d+1/parenleftbig
+Qrts−Qsrt/parenrightbig
+=i√
+d+1/parenleftbiggd+1
+d/parenleftbig
+Trts−Tsrt/parenrightbig
+−2/parenleftbig
+/an}bracketle{t/an}bracketle{tt/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbls/an}bracketri}ht/an}bracketri}ht−/an}bracketle{t/an}bracketle{tr/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardblt/an}bracketri}ht/an}bracketri}ht/parenrightbigg
+=i/radicalbigg
+2
+d/parenleftbig
+/an}bracketle{t/an}bracketle{tt/bardbles/an}bracketri}ht/an}bracketri}ht−/an}bracketle{t/an}bracketle{tt/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightbig
+(341)
+where we used the fact that Trts=Tsrtin the third step, and the fact that /an}bracketle{t/an}bracketle{tt/bardbles/an}bracketri}ht/an}bracketri}htis
+real in the last. This establishes Eq. ( 333). Eq. (334) is obtained by taking complex
+conjugates on both sides, and using the fact that the vectors /bardbles/an}bracketri}ht/an}bracketri}htare real.
+Eqs. (335) and (336) are immediate consequences of the definitions, and the fact
+thatQrQT
+r= 0. Turning to the proof of Eqs. ( 337) and (338), it follows from
+Eqs. (119) and (120) that
+Qs/bardbles/an}bracketri}ht/an}bracketri}ht= 0 (342)
+Using this and the fact that Qs/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht= 0 in Eq. ( 333) we find
+Qs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−i/radicalbigg
+2
+dQs/bardbler/an}bracketri}ht/an}bracketri}ht (343)
+Since
+/bardbler/an}bracketri}ht/an}bracketri}ht=/radicaligg
+d
+2(d+1)/parenleftig
+/bardblr/an}bracketri}ht/an}bracketri}ht+/bardblv0/an}bracketri}ht/an}bracketri}ht/parenrightig
+(344)
+and taking account of the fact that Qs/bardblv0/an}bracketri}ht/an}bracketri}ht= 0 (see Eq. ( 287)) we deduce
+Qs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−i/radicalbigg
+1
+d+1Qs/bardblr/an}bracketri}ht/an}bracketri}ht=−1
+d+1/bardblfsr/an}bracketri}ht/an}bracketri}ht (345)
+Taking complex conjugates on both sides of this equation we deduce the second
+identity in Eq. ( 337).
+In the same way, acting on both sides of Eq. ( 333) withQT
+swe find
+QT
+s/bardblfrs/an}bracketri}ht/an}bracketri}ht=−/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht−i/radicalbigg
+2
+dQT
+s/bardbler/an}bracketri}ht/an}bracketri}ht
+=−/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht−i/radicalbigg
+1
+d+1QT
+s/bardblr/an}bracketri}ht/an}bracketri}ht
+=−d
+d+1/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht (346)
+Taking complex conjugates on both sides of this equation we deduce the second
+identity in Eq. ( 338).
+Turning to the last group of identities we have
+/an}bracketle{t/an}bracketle{tfrs/bardblfsr/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tfrs/bardblQr/bardblfsr/an}bracketri}ht/an}bracketri}ht=−1
+d+1/an}bracketle{t/an}bracketle{tfrs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−1
+d+1(347)
+and
+/an}bracketle{t/an}bracketle{tfrs/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tfrs/bardblQr/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht=−d
+d+1/an}bracketle{t/an}bracketle{tfrs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−d
+d+1(348)43
+The other two identities are obtained by taking complex conjugates on both sides
+of the two just derived. /square
+This lemma provides a substantial part of what we need to prove the theorem.
+The remaining part is provided by
+Lemma 19. For allr/ne}ationslash=s
+QrQsQr=1
+d+1Qr−d
+(d+1)2/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (349)
+QrQT
+sQr=d2
+(d+1)2/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (350)
+Proof.It follows from Eq. ( 120) that
+QrQsQr=d+1
+dQrTsQr−2Qr/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardblQr (351)
+QrQT
+sQr=d+1
+dQrTT
+sQr−2Qr/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardblQr (352)
+In view of Eqs. ( 344), (287) and the definition of /bardblfrs/an}bracketri}ht/an}bracketri}htwe have
+Qr/bardbles/an}bracketri}ht/an}bracketri}ht=/radicaligg
+d
+2(d+1)Qr/bardbls/an}bracketri}ht/an}bracketri}ht=−i√
+d√
+2(d+1)/bardblfrs/an}bracketri}ht/an}bracketri}ht (353)
+Substituting this expression into Eqs. ( 351) and (352) we obtain
+QrQsQr=d+1
+dQrTsQr−d
+(d+1)2/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (354)
+QrQT
+sQr=d+1
+dQrTT
+sQr−d
+(d+1)2/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (355)
+The problem therefore reduces to showing
+QrTsQr=d
+(d+1)2Qr (356)
+QrTT
+sQr=d2
+(d+1)2/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (357)
+Using Eq. ( 120) we find
+/an}bracketle{t/an}bracketle{ta/bardblQrTsQr/bardblb/an}bracketri}ht/an}bracketri}ht=(d+1)2
+d2/an}bracketle{t/an}bracketle{ta/bardblTrTsTr/bardblb/an}bracketri}ht/an}bracketri}ht
+−1
+2/parenleftbigg2(d+1)
+d/parenrightbigg3
+2/parenleftig
+K2
+ra/an}bracketle{t/an}bracketle{ter/bardblTsTr/bardblb/an}bracketri}ht/an}bracketri}ht+K2
+rb/an}bracketle{t/an}bracketle{ta/bardblTrTs/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
++2(d+1)
+dK2
+raK2
+rb/an}bracketle{t/an}bracketle{ter/bardblTs/bardbler/an}bracketri}ht/an}bracketri}ht (358)
+/an}bracketle{t/an}bracketle{ta/bardblQrTT
+sQr/bardblb/an}bracketri}ht/an}bracketri}ht=(d+1)2
+d2/an}bracketle{t/an}bracketle{ta/bardblTrTT
+sTr/bardblb/an}bracketri}ht/an}bracketri}ht
+−1
+2/parenleftbigg2(d+1)
+d/parenrightbigg3
+2/parenleftig
+K2
+ra/an}bracketle{t/an}bracketle{ter/bardblTT
+sTr/bardblb/an}bracketri}ht/an}bracketri}ht+K2
+rb/an}bracketle{t/an}bracketle{ta/bardblTrTT
+s/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
++2(d+1)
+dK2
+raK2
+rb/an}bracketle{t/an}bracketle{ter/bardblTT
+s/bardbler/an}bracketri}ht/an}bracketri}ht (359)44
+Using the definitions of Tr,/bardbler/an}bracketri}ht/an}bracketri}htand Eq. ( 23) (the 2-design property) we find, after
+some algebra,
+/an}bracketle{t/an}bracketle{ta/bardblTrTsTr/bardblb/an}bracketri}ht/an}bracketri}ht=d2
+(d+1)2/parenleftig
+K2
+raTrsb+K2
+rbTras+K2
+rsTrab+K2
+raK2
+rb/parenrightig
+(360)
+/an}bracketle{t/an}bracketle{ter/bardblTsTr/bardblb/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
+2(d+1)/parenrightbigg3
+2/parenleftig
+2K2
+rsK2
+rb+K2
+rb+Trsb/parenrightig
+(361)
+/an}bracketle{t/an}bracketle{ta/bardblTrTs/bardbler/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
+2(d+1)/parenrightbigg3
+2/parenleftig
+2K2
+rsK2
+ra+K2
+ra+Tras/parenrightig
+(362)
+/an}bracketle{t/an}bracketle{ter/bardblTs/bardbler/an}bracketri}ht/an}bracketri}ht=d
+2(d+1)/parenleftbig
+3K2
+rs+1/parenrightbig
+(363)
+and
+/an}bracketle{t/an}bracketle{ta/bardblTrTT
+sTr/bardblb/an}bracketri}ht/an}bracketri}ht=d2
+(d+1)2/parenleftig
+GraGasGsbGbr
++K2
+raTrsb+K2
+rbTras+K2
+raK2
+rb/parenrightig
+=d2
+(d+1)2/parenleftig
+(d+1)TrasTrsb
++K2
+raTrsb+K2
+rbTras+K2
+raK2
+rb/parenrightig
+(364)
+/an}bracketle{t/an}bracketle{ter/bardblTT
+sTr/bardblb/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
+2(d+1)/parenrightbigg3
+2/parenleftig
+K2
+rsK2
+rb+K2
+rb+2Trsb/parenrightig
+(365)
+/an}bracketle{t/an}bracketle{ta/bardblTrTT
+s/bardbler/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
+2(d+1)/parenrightbigg3
+2/parenleftig
+K2
+rsK2
+ra+K2
+ra+2Tras/parenrightig
+(366)
+/an}bracketle{t/an}bracketle{ter/bardblTT
+s/bardbler/an}bracketri}ht/an}bracketri}ht=d
+2(d+1)/parenleftbig
+3K2
+rs+1/parenrightbig
+(367)
+where in deriving Eq. ( 364) we used the fact that GraGasGsbGbr= (d+1)TrasTrsb
+(in view of the fact that r/ne}ationslash=s). Substituting these expressions into Eqs. ( 358)
+and (359) we deduce Eqs. ( 356) and (357). /square
+Now define the rank d−1 projectors
+Qrs=Qr−/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl (368)
+QT
+rs=QT
+r−/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl (369)
+andletQ0
+rs,Qrs,¯Q0
+rsand¯Qrsbe, respectively, the subspacesontowhich /bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl,
+Qrs,/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardblandQ∗
+rsproject. It is immediate that we have the orthogonal
+decompositions
+Qr=Q0
+rs⊕Qrs (370)
+¯Qr=¯Q0
+rs⊕¯Qrs (371)
+Using Lemma 18we find
+Qsr/bardblfrs/an}bracketri}ht/an}bracketri}ht=Qrs/bardblfsr/an}bracketri}ht/an}bracketri}ht= 0 (372)45
+implying that Q0
+rs⊥QsrandQrs⊥Q0
+sr, and
+/vextendsingle/vextendsingle/an}bracketle{t/an}bracketle{tfrs/bardblfsr/an}bracketri}ht/an}bracketri}ht/vextendsingle/vextendsingle=1
+d+1(373)
+implying that Q0
+rsandQ0
+srare inclined at angle cos−1/parenleftbig1
+d+1/parenrightbig
+. Using Lemma 18
+together with Lemma 19we find
+QrsQsrQrs=QrsQsQrs
+=QrQsQr−/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardblQsQr−QrQs/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl
++/an}bracketle{t/an}bracketle{tfrs/bardblQs/bardblfrs/an}bracketri}ht/an}bracketri}ht/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl
+=1
+d+1Qr−1
+d+1/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl
+=1
+d+1Qrs (374)
+which in view of Lemma 15implies that QrsandQsrare uniformly inclined at angle
+cos−1/parenleftbig1√d+1/parenrightbig
+. This proves part (a) of the theorem. Parts (b) and (c) are prov ed
+similarly.
+Proof of Theorem 17.Define
+/bardblgrs/an}bracketri}ht/an}bracketri}ht=1√
+2/parenleftbig
+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht+/bardblfrs/an}bracketri}ht/an}bracketri}ht/parenrightbig
+(375)
+/bardbl¯grs/an}bracketri}ht/an}bracketri}ht=i√
+2/parenleftbig
+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht−/bardblfrs/an}bracketri}ht/an}bracketri}ht/parenrightbig
+(376)
+By construction the components of /bardblgrs/an}bracketri}ht/an}bracketri}ht,/bardbl¯grs/an}bracketri}ht/an}bracketri}htin the standard basis are real, so
+we can regard them as ∈Rd2. They are orthonormal:
+/an}bracketle{t/an}bracketle{tgrs/bardblgrs/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{t¯grs/bardbl¯grs/an}bracketri}ht/an}bracketri}ht= 1 and /an}bracketle{t/an}bracketle{tgrs/bardbl¯grs/an}bracketri}ht/an}bracketri}ht= 0 (377)
+It is also readily verified, using Lemma 18, that
+¯Rr/bardblgrs/an}bracketri}ht/an}bracketri}ht=/bardblgrs/an}bracketri}ht/an}bracketri}ht (378)
+¯Rr/bardbl¯grs/an}bracketri}ht/an}bracketri}ht=/bardbl¯grs/an}bracketri}ht/an}bracketri}ht (379)
+So
+Rrs=¯Rr−/bardblgrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgrs/bardbl−/bardbl¯grs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯grs/bardbl (380)
+is a rank 2 d−4 projector. If we define R0
+rs,R1
+rsandRrsto be, respectively,
+the subspaces onto which /bardblgrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgrs/bardbl,/bardbl¯grs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯grs/bardblandRrsproject we have the
+orthogonal decomposition
+Rr=R0
+rs⊕R1
+rs⊕Rrs (381)
+It follows from Eqs. ( 333) and (334) that
+/bardblgrs/an}bracketri}ht/an}bracketri}ht=−/bardblgsr/an}bracketri}ht/an}bracketri}ht (382)
+implying that R0
+rs=R0
+srfor allr/ne}ationslash=s. It is also easily verified, using Lemma 18,
+that/vextendsingle/vextendsingle/an}bracketle{t/an}bracketle{t¯grs/bardbl¯gsr/an}bracketri}ht/an}bracketri}ht/vextendsingle/vextendsingle=d−1
+d+1(383)
+from which it follows that R1
+rsandR1
+srare inclined at angle cos−1/parenleftbigd−1
+d+1/parenrightbig
+. We next
+observe that
+Rrs=Qrs+QT
+rs (384)46
+Using Lemma 18once again we deduce
+Rrs/bardbl¯gsr/an}bracketri}ht/an}bracketri}ht=Rsr/bardbl¯grs/an}bracketri}ht/an}bracketri}ht= 0 (385)
+from which it follows that R1
+rs⊥RsrandRrs⊥R1
+sr. Finally, we know from
+Theorem 16thatQT
+rsQsr=QrsQT
+sr= 0. Consequently
+RrsRsrRrs=QrsQsrQrs+QT
+rsQT
+srQT
+rs
+=d
+d+1Qrs+d
+d+1QT
+rs
+=1
+d+1Rrs (386)
+In view of Lemma 15it follows that RrsandRsrare uniformly inclined at angle
+cos−1/parenleftbig1√d+1/parenrightbig
+.
+Further Identities. We conclude this section with another set of identities in-
+volving the vectors /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht,/bardblgrs/an}bracketri}ht/an}bracketri}htand/bardbl¯grs/an}bracketri}ht/an}bracketri}ht.
+Define
+/bardbl¯er/an}bracketri}ht/an}bracketri}ht=/radicalbigg
+2d
+d−1/bardbler/an}bracketri}ht/an}bracketri}ht−/radicalbigg
+d+1
+d−1/bardblv0/an}bracketri}ht/an}bracketri}ht (387)
+where/bardblv0/an}bracketri}ht/an}bracketri}htis the vector defined by Eq. ( 286). It is readily verified that
+/an}bracketle{t/an}bracketle{t¯er/bardbl¯er/an}bracketri}ht/an}bracketri}ht= 0 and /an}bracketle{t/an}bracketle{t¯er/bardblv0/an}bracketri}ht/an}bracketri}ht= 0 (388)
+So/bardbl¯er/an}bracketri}ht/an}bracketri}ht,/bardblv0/an}bracketri}ht/an}bracketri}htis an orthonormal basis for the 2-dimensional subspace spanned b y
+/bardbler/an}bracketri}ht/an}bracketri}ht,/bardblv0/an}bracketri}ht/an}bracketri}ht. Note that
+Qr/bardbl¯er/an}bracketri}ht/an}bracketri}ht=QT
+r/bardbl¯er/an}bracketri}ht/an}bracketri}ht=¯Rr/bardbl¯er/an}bracketri}ht/an}bracketri}ht= 0 (389)
+We then have
+Theorem 20. For allr
+1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl=Qr (390)
+1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl=QT
+r (391)
+2
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblgrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgrs/bardbl=¯Rr (392)
+2
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardbl¯grs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯grs/bardbl=¯Rr (393)
+and
+1
+d−1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl=QT
+r+/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl+1
+d2−1/parenleftig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig
+(394)47
+1
+d−1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl=Qr+/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl+1
+d2−1/parenleftig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig
+(395)
+2
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblgsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgsr/bardbl=¯Rr (396)
+2
+d−3d2/summationdisplay
+s=1
+(s/negationslash=r)/bardbl¯gsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯gsr/bardbl=¯Rr+4(d−1)
+d−3/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl+4
+(d+1)(d−3)/parenleftig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig
+(397)
+Proof.It follows from the definition of /bardblfrs/an}bracketri}ht/an}bracketri}htthat
+1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl=d2/summationdisplay
+s=1
+(s/negationslash=r)Qr/bardbls/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ts/bardblQr
+=Qr
+d2/summationdisplay
+s=1/bardbls/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ts/bardbl
+Qr
+=Qr (398)
+where in the second step we used the fact that Qr/bardblr/an}bracketri}ht/an}bracketri}ht= 0 (as can be seen by setting
+r=sin Eq. (121)). Eq. (391) is obtained by taking the complex conjugate on both
+sides.
+We also have
+1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl=−d2/summationdisplay
+s=1
+(s/negationslash=r)Qr/bardbls/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ts/bardblQT
+r
+=−Qr
+d2/summationdisplay
+s=1/bardbls/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ts/bardbl
+QT
+r
+=−QrQT
+r
+= 0 (399)
+Taking the complex conjugate on both sides we find
+1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl= 0 (400)
+Consequently
+2
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/bardblgrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgrs/bardbl=1
+d+1d2/summationdisplay
+s=1
+(s/negationslash=r)/parenleftig
+/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl
++/bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+rs/bardbl+/bardblf∗
+rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl/parenrightig
+=¯Rr (401)48
+Eq. (393) is proved similarly.
+To prove the second group of identities we have to work a little harde r. Using
+Eqs. (116) and (120) we find
+1
+d−1d2/summationdisplay
+s=1
+(s/negationslash=r)/an}bracketle{t/an}bracketle{ta/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardblb/an}bracketri}ht/an}bracketri}ht=d+1
+d−1d2/summationdisplay
+s=1/an}bracketle{t/an}bracketle{ta/bardblQs/bardblr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tr/bardblQs/bardblb/an}bracketri}ht/an}bracketri}ht
+=(d+1)3
+d2(d−1)d2/summationdisplay
+s=1/parenleftig
+TsarTsrb−K2
+saK2
+srTsrb
+−K2
+srK2
+sbTsar+K2
+saK4
+srK2
+sb/parenrightig
+(402)
+(where we used the fact that Qs/bardbls/an}bracketri}ht/an}bracketri}ht= 0 in the first step). After some algebra we
+find
+d2/summationdisplay
+s=1TsarTsrb=d
+d+1/parenleftigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg
++Trba/parenrightigg
+(403)
+d2/summationdisplay
+s=1K2
+saK2
+srTsrb=d
+d+1/parenleftigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+2d+1
+d(d+1)/parenrightigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg
++1
+d+1Trba/parenrightigg
+(404)
+d2/summationdisplay
+s=1K2
+srK2
+sbTsar=d
+d+1/parenleftigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+2d+1
+d(d+1)/parenrightigg
++1
+d+1Trba/parenrightigg
+(405)
+d2/summationdisplay
+s=1K2
+saK4
+srK2
+sb=d
+(d+1)/parenleftigg
+d+2
+d+1/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg
++d
+(d+1)3δab+d+2
+(d+1)3/parenrightigg
+(406)
+wherewe usedEq. ( 23) to derivethe firstexpression. Substituting these expressions
+into Eq. ( 402) and using
+/an}bracketle{t/an}bracketle{ta/bardblQT
+r/bardblb/an}bracketri}ht/an}bracketri}ht=d+1
+d/parenleftigg
+Trba−/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg/parenleftigg/radicalbigg
+d−1
+d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
+d/parenrightigg/parenrightigg
+(407)
+we deduce Eq. ( 394). Taking complex conjugates on both sides we obtain Eq. ( 395).
+Eq. (396) is an immediate consequence of Eq. ( 392) and the fact that /bardblgsr/an}bracketri}ht/an}bracketri}ht=
+−/bardblgrs/an}bracketri}ht/an}bracketri}htfor allr,s.49
+To prove Eq. ( 397) observe that it follows from Eqs. ( 394)–(396) that
+d2/summationdisplay
+s=1
+(s/negationslash=r)/parenleftig
+/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl+/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl/parenrightig
+=d2/summationdisplay
+s=1
+(s/negationslash=r)/parenleftig
+2/bardblgsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgsr/bardbl−/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl−/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl/parenrightig
+= 2/parenleftig
+¯Rr−(d−1)/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl
+−1
+d+1/parenleftig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig/parenrightbigg
+(408)
+Hence
+2
+d−3d2/summationdisplay
+s=1
+(s/negationslash=r)/bardbl¯gsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯gsr/bardbl=1
+d−3d2/summationdisplay
+s=1
+(s/negationslash=r)/parenleftig
+/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl+/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl
+−/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl−/bardblf∗
+sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
+sr/bardbl/parenrightig
+=¯Rr+4(d−1)
+d−3/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl+4
+(d+1)(d−3)/parenleftig
+I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig
+(409)
+/square
+9.TheP-PTProperty
+In the preceding sections the Q-QTproperty has played a prominent role. In
+this section we show that in the particular case ofa Weyl-Heisenberg covariantSIC-
+POVM, and with the appropriate choice of gauge, the Gram project or (defined in
+Eq. (63)) has an analogous property, which we call the P-PTproperty. Specifically
+one has
+PPT=PTP=/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardbl (410)
+where/bardblh/an}bracketri}ht/an}bracketri}htis a normalized vector whose components in the standard basis are a ll
+real. In odd dimensions the components of /bardblh/an}bracketri}ht/an}bracketri}htin the standard basis can be simply
+expressed in terms of the Wigner function of the fiducial vector. I t could be said
+thattheprojectors PandPTarealmostorthogonal(bycontrastwiththeprojectors
+QrandQT
+rwhich are completely orthogonal). More precisely Phas the spectral
+decomposition
+P=¯P+/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardbl (411)
+where¯Pis a rank (d−1) projector with the property
+¯P¯PT= 0 (412)
+This means that the matrix
+JP=P−PT(413)
+is a pure imaginary Hermitian matrix with the property that J2
+Pis a real rank
+2d−2 projector ( c.f.the discussion in Section 4).
+Although we are mainly interested in the P-PTproperty as it applies to SIC-
+POVMs, itshould benoted that itactuallyholdsforanyWeyl-Heisenbe rgcovariant
+POVM (with the appropriate choice of gauge). So we will prove the ab ove propo-
+sitions for this more general case.50
+Let us begin by fixing notation. Let |0/an}bracketri}ht,...,|d−1/an}bracketri}htbe an orthonormal basis for
+d-dimensional Hilbert space and let XandZbe the operators whose action on the
+|r/an}bracketri}htis
+X|a/an}bracketri}ht=|a+1/an}bracketri}ht (414)
+Z|a/an}bracketri}ht=ωa|a/an}bracketri}ht (415)
+whereω=e2πi
+dand the addition of indices in the first equation is modd. We then
+define the Weyl-Heisenberg displacement operators by (adopting t he convention
+used in, for example, ref. [ 16])
+Dp=τp1p2Xp1Zp2(416)
+wherepis the vector ( p1,p2) (p1,p2being integers) and τ=e(d+1)πi
+d. Generally
+speaking the decision to insert the phase τp1p2is a matter of convention, and many
+authors define it differently, or else omit altogether. However, for the purposes of
+this section it is essential, as a different choice of phase at this stage would lead to a
+different gauge in the class of POVMs to be defined below, and the Gra m projector
+would then typically not have the P-PTproperty.
+Note thatτ2=τd2=ωin every dimension. If the dimension is odd we can write
+τ=ωd+1
+2. Soτis adthroot of unity. However, if the dimension is even τd=−1.
+This has the consequence that
+Dp+du= (−1)u1p2+u2p1Dp (417)
+Soin even dimension p=q(modd) does notnecessarilyimply Dp=Dq(although
+the operators are, of course, equal if p=q(mod 2d))
+In every dimension (even or odd) we have
+D†
+p=D−p (418)
+for allp
+(Dp)n=Dnp (419)
+for allp,nand
+DpDq=τ/angbracketleftp,q/angbracketrightDp+q (420)
+for allp,q. In the last expression /an}bracketle{tp,q/an}bracketri}htis the symplectic form
+/an}bracketle{tp,q/an}bracketri}ht=p2q1−p1q2 (421)
+Now let|ψ/an}bracketri}htbe any normalized vector (not necessarily a SIC-fiducial vector), and
+define
+|ψp/an}bracketri}ht=Dp|ψ/an}bracketri}ht (422)
+Let
+L=/summationdisplay
+p∈Z2
+d|ψp/an}bracketri}ht/an}bracketle{tψp| (423)
+It is easily seen that/bracketleftbig
+Dp,L/bracketrightbig
+= 0 (424)
+for allp.51
+We now appeal to the fact that there is no non-trivial subspace of Hdwhich
+the displacement operators leave invariant. To see this assume the contrary. Then
+there would exist non-zero vectors |φ/an}bracketri}ht,|χ/an}bracketri}htsuch that
+/an}bracketle{tφ|Dp|χ/an}bracketri}ht= 0 (425)
+for allp. Writing the left-hand side out in full this gives
+d−1/summationdisplay
+a=0ωp2a/an}bracketle{tφ|a+p1/an}bracketri}ht/an}bracketle{ta|χ/an}bracketri}ht= 0 (426)
+for allp1,p2. Taking the discrete Fourier transform with respect to p2, we have
+/an}bracketle{tφ|a+p1/an}bracketri}ht/an}bracketle{ta|χ/an}bracketri}ht= 0 (427)
+for alla,p1, implying that either |φ/an}bracketri}ht= 0 or|χ/an}bracketri}ht= 0—contrary to assumption. We
+can therefore use Schur’s lemma [ 55] to deduce that
+L=kI (428)
+for some constant k. Taking the trace on both sides of this equation we infer
+thatk=d. We conclude that1
+d|ψp/an}bracketri}ht/an}bracketle{tψp|is a POVM. We refer to POVMs of this
+general class as Weyl-Heisenberg covariant POVMs. We refer to th e vector |ψ/an}bracketri}ht
+which generates the POVM as the fiducial vector (with no implication t hat it is
+necessarily a SIC-fiducial).
+Now consider the Gram projector
+P=/summationdisplay
+p,q∈Z2
+dPp,q/bardblp/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tq/bardbl (429)
+where
+Pp,q=1
+d/an}bracketle{tψp|ψq/an}bracketri}ht (430)
+and where we label the matrix elements of Pand the standard basis kets with the
+vectorsp,qrather than with the single integer indices r,sas in the rest of this
+paper. We know from Theorem 1thatPis a rankdprojector.
+In view of Eqs. ( 418) and (420) we have
+/an}bracketle{t/an}bracketle{tp/bardblP/bardblq/an}bracketri}ht/an}bracketri}ht=Pp,q
+=1
+dτ−/angbracketleftp,q/angbracketright/an}bracketle{tψ|Dq−p|ψ/an}bracketri}ht
+=1
+dd−1/summationdisplay
+a=0τp1p2+q1q2ωaq2−(q1+a)p2/an}bracketle{tψ|a+q1−p1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht(431)
+Hence
+/an}bracketle{t/an}bracketle{tp/bardblPPT/bardblq/an}bracketri}ht/an}bracketri}ht=/summationdisplay
+u∈Zd/an}bracketle{t/an}bracketle{tp/bardblP/bardblu/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tq/bardblP/bardblu/an}bracketri}ht/an}bracketri}ht
+=1
+d2d−1/summationdisplay
+a,b,u1,u2=0τp1p2+q1q2ωu2(u1+a+b)−(u1+a)p2−(u1+b)q2
+×/an}bracketle{tψ|a+u1−p1/an}bracketri}ht/an}bracketle{tψ|b+u1−q1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht/an}bracketle{tb|ψ/an}bracketri}ht52
+=1
+dd−1/summationdisplay
+a,b=0τp1p2+q1q2ωp2b+q2a/an}bracketle{tψ|−b−p1/an}bracketri}ht/an}bracketle{tb|ψ/an}bracketri}ht/an}bracketle{tψ|−a−q1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht
+=/an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardblq/an}bracketri}ht/an}bracketri}ht (432)
+where/bardblh/an}bracketri}ht/an}bracketri}htis the vector with components
+/an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}ht=1√
+dd−1/summationdisplay
+a=0τp1p2ωp2a/an}bracketle{tψ|−a−p1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht (433)
+It is easily verified that /bardblh/an}bracketri}ht/an}bracketri}htis normalized, and that /an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}htis real.
+Finally, suppose that the dimension is odd. Then the Wigner function o f the
+state|ψ/an}bracketri}htis [56,57]
+W(p) =1
+d/an}bracketle{tψ|DpUPD†
+p|ψ/an}bracketri}ht=1
+d/an}bracketle{tψ|D2pUP|ψ/an}bracketri}ht (434)
+whereUPistheparityoperator,whoseactiononthestandardbasisis UP|a/an}bracketri}ht=|−a/an}bracketri}ht.
+It is straightforward to show
+/an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}ht=√
+dW(−2−1p) (435)
+where 2−1= (d+1)/2 is the multiplicative inverse of 2 considered as an element of
+Zd:i.e.the unique integer 0 ≤m<dsuch that 2 m= 1 (modd).
+10.Conclusion
+A curious fact about SIC-POVMs is that, although they are charac terized by
+their being highly symmetric structures, they do not wear this prop erty on their
+sleeve (so to speak). If one casually inspects the components of a SIC-fiducial,
+without knowing in advance that that is what they are, there does n ot seem to be
+anything special about them at all. Indeed, so far from there being any obvious
+pattern to the components, they seem, to a casual inspection, lik e a completely
+random collection of numbers. Moreover, this is just as true of an e xact fiducial as
+it is of a numerical one (see, for instance, the tabulations in Scott a nd Grassl [ 46]).
+It is only when one looks at them through the right pair of spectacles , and takes the
+trouble to calculate the overlaps Tr(Π rΠs), that the symmetry becomes apparent.
+The situation is a little reminiscent of a hologram, which only takes on th e aspect of
+a meaningful image when it is viewed in the right way. If one wanted to s ummarize
+the content of this paper in a nutshell it could be said that we have ex hibited some
+other pairs of spectacles—other ways of looking at a SIC—which cau se its inner
+secrets (or at any rate some of its inner secrets) to become manif est.
+Rather than focusing on the SIC-vectors |ψr/an}bracketri}ht, as is usually done, we havefocused
+on the angle tensors θrsandθrst, and on the T,JandRmatrices defined in
+terms of them. This is an important change of emphasis because, ra ther than
+being tied to any particular SIC, these quantities characterize ent ire families of
+unitarily equivalent SICs. Like the components of a SIC-fiducial, the angle tensors
+appear, to a casual inspection, like a random collection of numbers. However, if
+one examines the spectra of the T,JandRmatrices one realizes that, underlying
+the appearance of randomness, there is a high degree of order. I f one then goes on
+to examine the geometrical relationships between the subspaces o nto which the Q,
+QTand¯Rmatrices project, as we did in Section 8, one finds yet more instances
+of structure and order. To our minds what is particularly interestin g about all of53
+this is that none of it is obviously suggested by the defining property of a SIC, that
+Tr(ΠrΠs) = 1/(d+1) forr/ne}ationslash=s.
+In the course of this paper we have several times expressed the h ope that the
+Lie algebraic perspective on a SIC will lead to a solution to the existenc e problem.
+Of course, that is only a hope, and it may not be fulfilled. However, we feel on
+rather safer ground when we suggest that the solution is likely to co me, if not from
+this investigation, then from one which is like it to the extent that it fo cuses on a
+feature of a SIC which is not immediately apparent to the eye.
+Specializing to the case of a Weyl-Heisenberg covariant SIC, a fiducia l vector|ψ/an}bracketri}ht
+is a solution to the equations
+/vextendsingle/vextendsingle/an}bracketle{tψ|Dp|ψ/an}bracketri}ht/vextendsingle/vextendsingle2=dδp,0+1
+d+1(436)
+Allowing for the arbitrariness of the overall phase of |ψ/an}bracketri}ht, and taking |ψ/an}bracketri}htto be nor-
+malized, this gives us d2−1 conditions on only 2 d−2 independent real parameters.
+The equations are thus over-determined, and very highly over-de termined when d
+is large. Nevertheless, they have turned out to be soluble in every c ase which has
+been investigated to date. It seems likely that progress will depend on finding the
+structural feature which is responsible for this remarkable fact. The motivation for
+this paper is the belief that it may be structural features of the Lie algebra gl(d,C)
+which are responsible. That suggestion may or may not be correct. But if it turns
+out to be incorrect, the amount of effort which has been expended on this problem
+over a period of more than ten years, so far without fruit, sugges ts to us that the
+solution will depend on finding some other structural feature of a S IC, which is not
+obvious, and which has hitherto escaped attention.
+11.Acknowledgements
+The authors thank I. Bengtsson for discussions. Two authors, D MA and CAF,
+were supported in part by the U. S. Office of Naval Research (Gran t No. N00014-
+09-1-0247). Research at Perimeter Institute is supported by th e Government of
+Canada through Industry Canada and by the Province of Ontario t hrough the
+Ministry of Research & Innovation.
+References
+[1] S.G. Hoggar, Geom. Dedic. 69, 287 (1998).
+[2] G. Zauner, “Quantum designs—foundations of a non-commutat ive theory of
+designs” (in German), Ph.D. thesis, University of Vienna, 1999. Ava ilable on-
+line athttp://www.mat.univie.ac.at/˜neum/papers/physpapers.html .
+[3] C.M. Caves, “Symmetric Informationally Complete POVMs,” UNM
+Information Physics Group internal report. Available online at
+http://info.phys.unm.edu/caves/reports/reports.html (1999).
+[4] C.A. Fuchs and M. Sasaki, Quant. Inf. Comp. 3, 277 (2003). Also available as
+quant-ph/030292.
+[5] J.M. Renes, R. Blume-Kohout, A.J. Scott and C.M. Caves, J. Math. Phys. 45,
+2171 (2004). Also available as quant-ph/0310075.
+[6] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum and Semicl. Optics
+6, L19 (2005). Also available as math-ph/0403057.54
+[7] C.A. Fuchs, Quantum Information and Computation 4, 467 (2004). Also avail-
+able as quant-ph/0404122.
+[8] J.ˇReh´ aˇ cek, B.-G. Englert and D. Kaszlikowski, Phys. Rev. A 70, 052321
+(2004). Also available as quant-ph/0405084.
+[9] W.K. Wootters, Found. Phys. 36, 112 (2006). Also available as quant-
+ph/0406032.
+[10] I. Bengtsson, quant-ph/0406174.
+[11] M. Grassl, in Proceedings ERATO Conference on Quantum Information Sci-
+ence 2004 (Tokyo, 2004). Also available as quant-ph/0406175.
+[12] J.M. Renes, Quant. Inf. Comp. 5, 80 (2005). Also available as quant-
+ph/0409043.
+[13] I. Bengtsson and ˚A. Ericsson, Open Sys. and Information Dyn. 12, 187 (2005).
+Also available as quant-ph/0410120.
+[14] R. K¨ onig and R. Renner, J. Math. Phys. 46, 122108 (2005) Also available as
+quant-ph/0410229.
+[15] M. Ziman and V. Buˇ zek, Phys. Rev. A 72, 022343 (2005). Also available as
+quant-ph/0411135.
+[16] D.M. Appleby, J. Math. Phys. 46, 052107 (2005). Also available as quant-
+ph/0412001.
+[17] A. Klappenecker and M. R¨ otteler, Proc. 2005 IEEE International Sympo-
+sium on Information Theory, Adelaide , 1740 (2005). Also available as quant-
+ph/0502031.
+[18] A. Klappenecker, M. R¨ otteler, I. Shparlinski and A. Winterho f,J. Math. Phys.
+46, 082104 (2005). Also available as quant-ph/0503239.
+[19] M. Grassl, Electronic Notes in Discrete Mathematics 20, 151 (2005).
+[20] M.A. Ballester, quant-ph/0507073.
+[21] D. Gross, “Finite Phase Space Methods in Quantum Information ”, Diploma
+Thesis, Potsdam (2005). Available online at http://gross.qipc.org/diplom.pdf .
+[22] S. Colin, J. Corbett, T. Durt and D. Gross, J. Opt. B: Quantum and Semicl.
+Optics7, S778 (2005).
+[23] C. Godsil and A. Roy, European J. Combin. ,30, 246 (2009). Also available as
+quant-ph/0511004.
+[24] S.D. Howard, A.R. Calderbank and W. Moran, EURASIP J. Appl. Sig. Pro-
+cess.2006, 85865 (2006).
+[25] A.J. Scott, J. Phys. A 39, 13507 (2006). Also available as quant-ph/0604049.
+[26] T. Durt, quant-ph/0604117 (2006).
+[27] S.T. Flammia, J. Phys. A 39, 13483 (2006). Also available as quant-
+ph/0605050.
+[28] M. Grassl, conference presentation at MAGMA 2006 Confer-
+ence(Tehnische Universit¨ at Berlin, 2006). Available online at
+http://magma.maths.usyd.edu.au/Magma/2006 .
+[29] I.H. Kim, Quant. Inf. Comp. 8, 730 (2007). Also available as quant-
+ph/0608024.
+[30] D.M. Appleby, Opt. Spect. 103, 416 (2007). Also available as quant-
+ph/0611260.
+[31] L. Bos and S. Waldron, New Zealand J. Math. 36, 113 (2007).
+[32] A. Roy and A.J. Scott, J. Math. Phys. 48, 072110 (2007). Also available as
+quant-ph/0703025.55
+[33] O. Albouy and M.R. Kibbler, Journal of Russian Laser Research 28, 429
+(2007). Also available as arXiv:0704.0511.
+[34] D.M. Appleby, H.B. Dang and C.A. Fuchs, arXiv:0707.2071.
+[35] M. Khatirinejad, Journal of Algebraic Combinatorics 28, 333 (2008).
+[36] B.G. Bodmann, P.G. Casazza, D. Edidin and R. Balan, Information Sci-
+ences and Systems, CISS 2008, Proceedings of 42nd Annual Con ference, p. 721
+(2008).
+[37] M.R. Kibler, J. Phys. A 41, 375302 (2008). Also available as arXiv:0807.2837.
+[38] I. Bengtsson and H. Granstr¨ om, Open Sys. Information Dyn. 16, 145 (2009).
+Also available as arXiv:0808.2974.
+[39] M. Grassl, conference presentation at Seeking SICs: A Workshop on Quantum
+Frames and Designs (Perimeter Institute, Waterloo, 2008). Available online at
+http://pirsa.org/08100069 .
+[40] M. Grassl, Lecture Notes in Computer Science 5393, 89 (2008).
+[41] M. Fickus, J. Fourier Anal. Appl. 15, 413 (2009).
+[42] D.M. Appleby, Foundations of Probability and Physics-5, V¨ axj¨ o 2008, AI P
+Conf. Proc. 1101, 223 (2009). Also available as arXiv:0905.1428.
+[43] C.A. Fuchs and R. Schack, arXiv:0906.2187 (2009).
+[44] D.M. Appleby, arXiv:0909.5233 (2009).
+[45] D.M. Appleby, ˚A. Ericsson and C.A. Fuchs, arXiv:0910.2750 (2009).
+[46] A.J. Scott and M. Grassl, arXiv:0910.5784 (2009).
+[47] I. Bengtsson and K. ˙Zyczkowski, Geometry of Quantum States (Cambridge
+University Press, Cambridge, 2006).
+[48] C.A. Fuchs, quant-ph/0205039.
+[49] P.O. Boykin, M. Sitharam, P.H. Tiep and P. Wocjan, Quantum Inf. Comput.
+7, 371 (2007). Also available as quant-ph/0506089.
+[50] N. Mukunda, Arvind, S. Chaturvedi and R. Simon, Phys. Rev. A 65, 012102
+(2001). Also available as quant-ph/0107006.
+[51] G.R. Goodson and R.A. Horn, Lin. Alg. App. 430, 1025 (2009).
+[52] N. Jacobson, Lie Algebras (Dover Publications, New York, 1979).
+[53] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory
+(Graduate Texts in Mathematics, Springer-Verlag, New York, 197 0).
+[54] W. Fulton and J. Harris, Representation Theory (Graduate Texts in Mathe-
+matics, Springer-Verlag, New York, 1991).
+[55] A.O. Barut and R R¸ aczka, Theory of Group Representations and Applications
+(World Scientific, Singapore, 1986).
+[56] W.K. Wootters, Ann. Phys. (N.Y.) 176, 1 (1987).
+[57] A. Vourdas, Rep. Prog. Phys. 67, 267 (2004).
\ No newline at end of file