SIC-POVMs: A new computer study

Section I Introduction

Interest in sets of equiangular lines began at least 60 years ago Haantjes48 ; VanLint66 ; Lemmens73 ; Delsarte75 ; Delsarte77 and continues to this day Bannai09 . In this article we report on a new computer study of what has now become one of the most urgent unanswered questions: Do there exist $d^{2}$ equiangular lines, the maximum possible, in all finite complex dimensions $d$? This question is attracting increasing attention from the quantum physics community Renes04 ; Grassl04 ; Appleby05 ; Grassl05 ; Grassl06 ; Grassl08a ; Grassl08b ; Klappenecker05 ; Colin05 ; Wootters06 ; Flammia06 ; Scott06 ; Appleby07 ; Belovs08 ; Appleby09 ; Fuchs09 ; Bengtsson09 ; Appleby09b ; Appleby09c and, more recently, from the communities of design theory Khatirinejad08 ; Godsil09 and frame theory Howard05 ; Waldron07 ; Fickus09 . The believed affirmative answer, originally conjectured 10 years ago by Gerhard Zauner Zauner99 , has now been confirmed exactly in dimensions $d=2,3$ Delsarte75 , $4,5$ Zauner99 , 6 Grassl04 , 7 Appleby05 , 8 Hoggar98 ; Grassl05 $9,\dots,13,15$ Grassl05 ; Grassl06 ; Grassl08a ; Grassl08b and 19 Appleby05 , and to high numerical precision in all dimensions $d\leq 45$ Renes04 . The fundamental question remains unresolved, however, inviting speculation as to whether the true answer is negative, with Zauner’s conjecture failing in some large untested dimension, or simply that the affirmative answer is truly difficult to prove, with Zauner’s conjecture remaining open for many years to come. In either case, a new computer study is timely.

Section II SIC-POVMs

In quantum theory, a set of $d^{2}$ equiangular lines in $d$ complex dimensions is the underlying mathematical object defining a symmetric informationally complete positive-operator-valued measure (SIC-POVM) Renes04 . These measures describe the measurement-outcome statistics of a particularly attractive choice of a ‘standard’ informationally complete quantum measurement, both from a foundational perspective Fuchs09 and for the purpose of quantum state tomography Scott06 . In precise terms, a SIC-POVM $P$ is a positive-operator-valued measure (POVM) (see e.g. ref. Busch96 ) that maps each of its $d^{2}$ possible measurement outcomes, denoted $x_{1},\dots,x_{d^{2}}$ say, to one of $d^{2}$ subnormalised rank-one projectors on the Hilbert space of $d$-dimensional pure quantum states d,

with the defining property that equiangularity is enjoyed under the Hilbert-Schmidt inner product:

Considering the rays in d upon which each ${|x_{k}\rangle\langle x_{k}|}$ projects, a SIC-POVM is of course equivalent to a set of $d^{2}$ equiangular lines through the origin of d. It is therefore natural to identify the outcome set as a subset of complex projective space, $\mathscr{X}\colonequals\{x_{1},\dots,x_{d^{2}}\}\subset{\C P^{d-1}}$. A set of equiangular lines $\mathscr{C}\subset{\C P^{d-1}}$, $|{\langle x|y\rangle}|^{2}=\alpha<1$ for all $x\neq y\in\mathscr{C}$, is then a type of complex projective code, called a ${\C P^{d-1}}$ 1-distance set (see e.g. refs. Rankin55 ; Conway88 ; Levenshtein98 ). It is known that any such $\mathscr{C}$ obeys the so-called absolute bound on its size: $|\mathscr{C}|\leq d^{2}$. The maximum requires the same common angle enjoyed by SIC-POVMs. Alternatively, in terms of line packings, for any set $\mathscr{S}\subset{\C P^{d-1}}$ of size $|\mathscr{S}|=d^{2}$, it is known that $\max_{x\neq y\in\mathscr{S}}|{\langle x|y\rangle}|^{2}\geq 1/(d+1)$ with equality only if $\mathscr{S}$ is equiangular Rankin55 .

A dual characterisation of SIC-POVMs comes from design theory (see e.g. refs. Levenshtein98 ; Harpe05 ; Hoggar82 or sec. 2 of ref. Roy07 for a concise introduction). A finite set $\mathscr{D}\subset{\C P^{d-1}}$ is called a complex projective $t$-design if

where $\mu$ is the Haar measure. In these terms, SIC-POVMs are precisely equivalent to tight complex projective 2-designs. These are 2-designs that meet the absolute bound on their size: $|\mathscr{D}|\geq d^{2}$. All 2-designs with $|\mathscr{D}|=d^{2}$ are necessarily sets of equiangular lines and these are the only 2-designs with this structure. This characterisation has a straightforward generalisation in terms of weighted $t$-designs Levenshtein98 or cubature formulas Harpe05 .

A third characterisation of SIC-POVMs is in terms of frame theory (see e.g. refs. Kovacevic07 ; Christensen03 ). In this context, a set of unit vectors that specifies a SIC-POVM, $\{{|x_{k}\rangle}\}_{k=1}^{d^{2}}\subset{}^{d}$, is called a maximally equiangular tight frame Fickus09 . More importantly, under the projection $P\mapsto P-{\operatorname{tr}}(P)I/d$ into the real vector space of traceless Hermitian operators (the natural generalisation of the Bloch-sphere representation to higher dimensions Scott06 ; Appleby09 ) a SIC-POVM again maps to a tight frame (in this case a simplex), which means the representation

is afforded by any quantum state $\rho$ (see refs. Scott06 ; Scott08 for descriptions of such tight informationally complete POVMs). This state-inversion formula for $\rho$ in terms of its measurement statistics {\operatorname{tr}}\bigl{(}P(x_{k})\rho\bigr{)} immediately proves informational completeness Busch91 for SIC-POVMs. Moreover, amongst all minimally informationally complete POVMs (i.e. those having $d^{2}$ outcomes), this representation is unique to SIC-POVMs. These considerations have lead some to argue Scott06 ; Appleby07 that SIC-POVMs should be promoted to the unique status of standard informationally complete POVMs, being as close as possible to orthonormal bases for the space of quantum states. Indeed, SIC-POVMs would be the best choice in any bid to standardise experimental reporting in quantum state tomography, being the most robust minimally informationally complete POVMs against statistical error Scott06 .

Section III Weyl-Heisenberg SIC-POVMs and the Clifford group

The most promising route towards a general construction of SIC-POVMs involves translating a fiducial vector under the Weyl displacement operators Weyl50 ; Schwinger60 :

where $p=(p_{1},p_{2})\in{}^{2}$, $\tau=e^{\pi i(d+1)/d}$, $\omega=\tau^{2}=e^{2\pi i/d}$ (meaning $\tau^{d^{2}}=\tau^{2d}=\omega^{d}=1$), and we have fixed an orthonormal basis for d: ${|0\rangle},\dots,{|d-1\rangle}$. Defining the symplectic form

and together generate a variant of the Heisenberg group:

Modulo its center, ${\operatorname{I}}(d)\colonequals\{e^{i\xi}I:\xi\in\R\}$, the Heisenberg group is simply a direct product of cyclic groups, ${\operatorname{H}}(d)/{\operatorname{I}}(d)\cong{{}_{d}}^{2}$, where ${}_{d}=\Z/d\Z=\{0,\dots,d-1\}$.

It was conjectured in ref. Renes04 that, in every finite dimension, a SIC-POVM can be constructed as the orbit of a suitable fiducial vector ${|\phi\rangle}\in{}^{d}$ under the action of the displacement operators:

The condition for equiangularity (2) then becomes

To bolster this conjecture, such Weyl-Heisenberg covariant SIC-POVMs were found with high numerical precision in all dimensions $d\leq 45$. Unbeknownst to the authors of ref. Renes04 , however, a stronger conjecture had already been put forward by Gerhard Zauner in his doctoral dissertation Zauner99 . Zauner claimed that, in every finite dimension, a fiducial vector for a Weyl-Heisenberg covariant SIC-POVM can be found in an eigenspace of the matrix1

In all finite dimensions there exists a fiducial vector for a Weyl-Heisenberg covariant SIC-POVM that is an eigenvector of ${Z}$.

Setting $\xi=\pi(d-1)/12$, it can be shown Zauner99 that ${Z}$ has order 3: ${Z}^{3}=I$. The eigenspace with eigenvalue $e^{2\pi ik/3}$ will be labeled $\mathcal{Z}_{k}$ ($k=0,1,2$). Then

Under the action of conjugation, ${Z}$ defines an automorphism of the Heisenberg group, ${Z}^{-1}{\operatorname{H}}(d){Z}={\operatorname{H}}(d)$, and therefore belongs to the normaliser of ${\operatorname{H}}(d)$ in ${\operatorname{U}}(d)$,

which is called the Clifford group in quantum information theory, but more widely recognised as a variant of the Jacobi group Berndt98 . The significance of ${\operatorname{C}}(d)$ to SIC-POVMs follows from eq. (12): if ${|\phi\rangle}$ is a fiducial vector for a Weyl-Heisenberg covariant SIC-POVM, then so is $U{|\phi\rangle}$ for any $U\in{\operatorname{C}}(d)$.

An explicit description of the Clifford group can be easily deduced in odd dimensions, which we now summarise. In general dimensions, for each symplectic matrix $F\in{\operatorname{SL}_{2}}({}_{d})$ and $p\in{{}_{d}}^{2}$, let

Now assume $d$ is odd. Then for each ${({F}\mspace{1.0mu}|\mspace{1.0mu}{p})}\in{\operatorname{c}}(d)$ there is a unique unitary $C_{{({F}\mspace{1.0mu}|\mspace{1.0mu}{p})}}\in{\operatorname{C}}(d)$, up to the multiplication of a phase $e^{i\xi}$, for which

for all $q\in{}^{2}$. All Clifford operators take this action and, in fact, $C_{g}$ is a faithful projective unitary representation of ${\operatorname{c}}(d)$ (i.e. $C_{g}C_{h}=e^{i\xi(g,h)}C_{gh}$ for some $\xi:{\operatorname{c}}(d)\times{\operatorname{c}}(d)\rightarrow\R$) that defines the group isomorphism

The Clifford group describes all automorphisms of ${\operatorname{H}}(d)$ that leave its center pointwise fixed: the inner automorphisms are just displacement operators, $D_{p}=C_{{({I}\mspace{1.0mu}|\mspace{1.0mu}{p})}}$, while the outer automorphisms are specified by the operators $C_{{({F}\mspace{1.0mu}|\mspace{1.0mu}{0})}}$, which are more widely recognised as metaplectic operators Weil63 ; Feichtinger08 . Since eq. (21) defines each Clifford operator uniquely, up to a phase $e^{i\xi}$, it can be used to derive formulas, e.g., by multiplying on the right by ${D_{q}}^{\dagger}$ and summing over $q$ to obtain

where $\eta(F)\colonequals|\{q\in{{}_{d}}^{2}|Fq=q\}|=|{\operatorname{tr}}\,C_{{({F}\mspace{1.0mu}|\mspace{1.0mu}{0})}}|^{2}$. To check eq. (21), simply replace $r$ by $r+q$ in the sum, and use ${\langle Fr,Fq\rangle}=(\det F){\langle r,q\rangle}$ and eq. (7) repeatedly to obtain:

since $\det F=1{\>(\operatorname{mod}\,d)}$ and $\tau^{d}=1$ for odd $d$. In even dimensions we would need $\det F=1{\>(\operatorname{mod}\,2d)}$, however, or we would need to generalise eq. (21) so that the factor $\tau^{(\det F-1)q_{1}q_{2}}$ appears on the right. Both possibilities indicate how the above description of ${\operatorname{C}}(d)$ might be amended to work in general. The former choice was taken by Appleby Appleby05 and the latter by Feichtinger et al. Feichtinger08 . We will follow Appleby’s approach, which we now summarise.

In even dimensions, the metaplectic operators can introduce sign changes to eq. (21) unless we instead take $F\in{\operatorname{SL}_{2}}({}_{2d})$, as done by Appleby Appleby05 . We also take $p\in{{}_{2d}}^{2}$ for simplicity. In general dimensions, let

Now, and hereafter, for each ${({F}\mspace{1.0mu}|\mspace{1.0mu}{p})}\in{\operatorname{c}}({\bar{d}})$ define

if there exists an element $\beta^{-1}\in{}_{{\bar{d}}}$ with $\beta^{-1}\beta=1{\>(\operatorname{mod}\,{\bar{d}})}$; otherwise, we find an integer $x\in{}_{{\bar{d}}}$ with the property that $(\delta+x\beta)^{-1}\in{}_{{\bar{d}}}$ (whose existence is guaranteed (Appleby05, , Lemma 3)) and take $V_{F}=V_{F_{1}}V_{F_{2}}$, using eq. (29) now for $V_{F_{1}}$ and $V_{F_{2}}$, where

fulfill the decomposition $F=F_{1}F_{2}$. In odd dimensions, eq. (28) is equivalent to eq. (23). In all dimensions, now choosing $\xi=0$ in eq. (13), Zauner’s matrix is ${Z}=C_{{({{F_{z}}}\mspace{1.0mu}|\mspace{1.0mu}{0})}}$, where

With these definitions, Appleby (Appleby05, , Theorem 1) showed that the map $g\mapsto C_{g}$ defines a group isomorphism

Combining eqs. (19) and (32) we can deduce the size of the Clifford group. It is known that

where the product is over all primes $p$ dividing $d$, which means $|{\operatorname{c}}({\bar{d}})|=32|{\operatorname{c}}(d)|$ for even $d$. But since $|{\ker(C)}|=32$ in even dimensions, we can conclude that

Finally, note that there is a complex-conjugation symmetry apparent in eq. (12). Let $\hat{J}=\hat{J}^{\dagger}$ be the anti-unitary operator with action $\hat{J}\sum_{k}c_{k}{|k\rangle}=\sum_{k}{c_{k}}^{*}{|k\rangle}$ for a vector rewritten in our standard basis. Then,

It thus follows from eq. (12) that $\hat{J}{|\phi\rangle}$ is a fiducial vector for a Weyl-Heisenberg covariant SIC-POVM whenever ${|\phi\rangle}$ is. To analyze this additional symmetry, define the matrix group

where {\operatorname{ESL}_{2}}({}_{d})\colonequals\bigl{\{}F\in{\operatorname{Mat}_{2,2}}({}_{d}):\det F=\pm 1{\>(\operatorname{mod}\,d)}\bigr{\}}={\operatorname{SL}_{2}}({}_{d})\cup J\,{\operatorname{SL}_{2}}({}_{d}) is the union of all symplectic and anti-symplectic matrices. The extended Clifford group is then defined as the group of all unitary and anti-unitary operators that normalise ${\operatorname{H}}(d)$, i.e., the disjoint union

Appleby (Appleby05, , Theorem 2) showed that

through the map $E:{\operatorname{ec}}({\bar{d}})\rightarrow{\operatorname{EC}}(d)$, where

From eq. (39), we know that $\ker(E)=\ker(C)$ and thus have $|{\operatorname{PEC}}(d)|=2|{\operatorname{PC}}(d)|$.

Lastly, in eqs. (32) and (40) we have defined projective versions of the Clifford and extended Clifford groups, respectively. The notation $[U]\colonequals\{e^{i\xi}U\}_{\xi\in\R}$ will be used in the following section to denote members of these groups, which are equivalence classes of unitary and anti-unitary matrices that differ only by a factor of unit modulus. We will also use the shorthand notation

Section IV Numerical computer solutions

The task of finding SIC-POVMs is facilitated by the following recharacterisation of $t$-designs (see e.g. ref. Roy07 ): for any finite $\mathscr{S}\subset{\C P^{d-1}}$, and any positive integer $t$,

with equality if and only if $\mathscr{S}$ is a $t$-design. This allows us to check whether a proposed set $\mathscr{S}\subset{\C P^{d-1}}$ forms a $t$-design by considering only the angles between members. It also shows that $t$-designs can be found numerically by parameterising $\mathscr{S}$ and minimising the LHS of eq. (45). The lower bound was derived by Welch Welch74 . Setting ${|x_{p_{1}+p_{2}d+1}\rangle}=D_{p}{|\phi\rangle}$, $|\mathscr{S}|=d^{2}$ and $t=2$, we can translate eq. (45) to our case: for any ${|\phi\rangle}\in{}^{d}$,

with equality if and only if ${|\phi\rangle}$ is a fiducial vector for a Weyl-Heisenberg covariant SIC-POVM. The second form of this sum was used to obtain the numerical results reported next.2\c@chapter

Our extensive computer searches discovered fiducial vectors in all dimensions $d\leq 67$, improving considerably on the solutions reported previously Renes04 . These are listed in appendix \thechapter.A and may be assumed exact to the 38 digits quoted. Note that each solution $\phi$ will generate an entire orbit of related fiducial vectors under the action of the extended Clifford group:

Those listed in the appendix generate unique orbits. Moreover, we are confident that this list is complete for $d\leq 50$, where the computer search was exhaustive.3\c@chapter

In table LABEL:tbl:symmetries we list the total number of unique Weyl-Heisenberg SIC-POVMs in each dimension in terms of orbits of fiducial vectors. Note that the length of each orbit will be $|{\operatorname{PEC}}(d)|$, unless there is a symmetry present in $\phi$, described by its stabiliser

where $[U]\colonequals\{e^{i\xi}U\}_{\xi\in\R}$. We then have $|{\operatorname{orb}}(\phi)|=|{\operatorname{PEC}}(d)|/|{\operatorname{S}}(\phi)|$, giving

which is $5760\times(3/3+1/6)=6720$ unique SIC-POVMs in dimension 15, for example. The stabiliser of each orbit is also given in table LABEL:tbl:symmetries. The numerical initial fiducial vector ${|\phi\rangle}$ and its stabiliser ${\operatorname{S}}(\phi)$ were always chosen in a way that the stabiliser elements take the form ${[{F}\mspace{1.0mu}|\mspace{1.0mu}{0}]}$ (recall that ${[{F}\mspace{1.0mu}|\mspace{1.0mu}{p}]}\colonequals[E_{{({F}\mspace{1.0mu}|\mspace{1.0mu}{p})}}]$) and, therefore, only the matrices $F$ are quoted.4\c@chapter The number of stabilised fiducial vectors in each orbit is also provided, which is further divided into the number of stabilised SIC-POVMs times the number of stabilised vectors per SIC-POVM. For example, ${\operatorname{orb}}(\phi_{15a})$ contains 24 SIC-POVMs stabilised by {[{{F_{z}}}\mspace{1.0mu}|\mspace{1.0mu}{0}]}=\bigl{[}\begin{smallmatrix}{0}&{14}\\ {1}&{14}\end{smallmatrix}\big{|}\begin{smallmatrix}{0}\\ {0}\end{smallmatrix}\bigr{]}, each containing 3 stabilised vectors; ${\operatorname{orb}}(\phi_{15d})$ contains 12 SIC-POVMs stabilised by \bigl{[}\begin{smallmatrix}{4}&{11}\\ {4}&{0}\end{smallmatrix}\big{|}\begin{smallmatrix}{0}\\ {0}\end{smallmatrix}\bigr{]}, each containing a single stabilised vector.

In agreement with Appleby’s analysis Appleby05 of the previous solutions Renes04 , in dimensions $d>3$ each stabiliser of the new solutions is a cyclic group of order a multiple of 3, and the vast majority, up to group conjugacy, have the symmetry described by Zauner’s order-3 unitary ${[{{F_{z}}}\mspace{1.0mu}|\mspace{1.0mu}{0}]}$, where (eq. (31))

Exceptions occur in dimensions $d=9k+3=12(b)$, $21(e)$, $30(d)$, $39(g,h,i,j)$, $48(e,g)$, $57,\dots$, in which case solutions stabilised by the order-3 unitary ${[{{F_{a}}}\mspace{1.0mu}|\mspace{1.0mu}{0}]}$ exist, as indicated in parentheses, where

The solution $\phi_{12b}$ was known previously Grassl05 , but the remaining are new. Following Appleby Appleby05 , call an order-3 (unitary) element ${[{F}\mspace{1.0mu}|\mspace{1.0mu}{p}]}\in{\operatorname{PEC}}(d)$ canonical if ${\operatorname{tr}}(F)=-1{\>(\operatorname{mod}\,d)}$ and $F\neq I$. Note that Zauner’s unitary ${[{{F_{z}}}\mspace{1.0mu}|\mspace{1.0mu}{0}]}$ is canonical. In fact, it can be shown that all other canonical elements are conjugate to it when $9\nmid(d-3)$; otherwise, there are exactly two conjugacy classes of canonical elements in ${\operatorname{PEC}}(d)$, containing ${[{{F_{z}}}\mspace{1.0mu}|\mspace{1.0mu}{0}]}$ and ${[{{F_{a}}}\mspace{1.0mu}|\mspace{1.0mu}{0}]}$, respectively.5\c@chapter Order-3 canonical unitaries are therefore special, in that, in the dimensions tested, every such unitary stabilises a fiducial vector.

We can deduce two more general symmetries present in the new solutions: in dimensions $d=k^{2}-1=8(b)$, $15(d)$, $24(c)$, $35(i,j)$, $48(f)$, $63,\dots$, the order-2 permutation ${[{{F_{b}}}\mspace{1.0mu}|\mspace{1.0mu}{0}]}$, where

stabilises the solutions indicated; in dimensions $d=(3k\pm 1)^{2}+3=4(a)$, $7(b)$, $19(d,e)$, $28(c)$, $52,\dots$, the order-2 anti-unitary ${[{{F_{c}}}\mspace{1.0mu}|\mspace{1.0mu}{0}]}$, where

stabilises the solutions indicated. As with Zauner’s symmetry, the reason for these extra symmetries is unknown, but their existence assists the discovery of new analytical solutions.

In table LABEL:tbl:Zsymmetries we list the total number of fiducial vectors stabilised by Zauner’s unitary matrix ${Z}\in{[{{F_{z}}}\mspace{1.0mu}|\mspace{1.0mu}{0}]}$ in terms of fiducial-vector orbits. The number of fiducial vectors contained in each ${Z}$-eigenspace is also given ($\mathcal{Z}_{k}$ is as defined above eq. (14)), which is further divided into the number of ${Z}$-stabilised SIC-POVMs times the number of ${Z}$-stabilised vectors per SIC-POVM. The latter is 3 when $3\mid d$, and 1 otherwise. The vectors tend to populate the largest eigenspace(s) only. This is $\mathcal{Z}_{0}$, except in dimensions $d=3k-1$, where $\mathcal{Z}_{0}$ and $\mathcal{Z}_{1}$ have the same largest dimension. The ${Z}$-stabilised vectors then divide evenly between these eigenspaces. Exceptions to this rule occur in dimensions $d=9k-1=8(b)$, $17(c)$, $26(d)$, $35(h,i,j)$, $44(e,f)$, $53,\dots$, where orbits exist containing only ${Z}$-stabilised vectors from $\mathcal{Z}_{2}$, as indicated in parentheses.

It is of some interest whether real fiducial vectors exist Khatirinejad08 . A search through our orbits shows that, for $d\leq 50$, these exist only in dimensions $d=3(c)$, $7(b)$, $19(e,d)$ and $39(i,j)$, of which, those in dimension 39 are new. Since \bigl{(}\begin{smallmatrix}{5}&{29}\\ {29}&{28}\end{smallmatrix}\bigr{)}\bigl{(}\begin{smallmatrix}{0}&{7}\\ {28}&{6}\end{smallmatrix}\bigr{)}{}^{3}\bigl{(}\begin{smallmatrix}{5}&{29}\\ {29}&{28}\end{smallmatrix}\bigr{)}{}^{-1}=J{\>(\operatorname{mod}\,39)}, examples from each of the two orbits can be found by multiplying the vectors given in appendix \thechapter.A by the unitary \bigl{[}\begin{smallmatrix}{5}&{29}\\ {29}&{28}\end{smallmatrix}\big{|}\begin{smallmatrix}{0}\\ {0}\end{smallmatrix}\bigr{]}.

Also of interest are Schmidt decompositions of the fiducial vectors for different bipartitions ${}^{d}\cong{}^{d_{1}}\otimes{}^{d_{2}}$, where $d_{1}<d_{2}$ are coprime, under the identification ${|k\rangle}\rightarrow{|k{\>(\operatorname{mod}\,d_{1})}\rangle}\otimes{|k{\>(\operatorname{mod}\,d_{2})}\rangle}$. This choice preserves group covariance of SIC-POVMs by realising the group isomorphism ${\operatorname{H}}(d)\cong{\operatorname{H}}(d_{1})\times{\operatorname{H}}(d_{2})$ Gross08 . Given the singular value decomposition of the $d_{1}\times d_{2}$ matrix $M(\phi)\colonequals\sum_{k_{1},k_{2}}({\langle k_{1}|}\otimes{\langle k_{2}|}){|\phi\rangle}{|k_{1}\rangle}{\langle k_{2}|}=\sum_{l}\sqrt{\lambda_{l}}{|u_{l}\rangle}{\langle{v_{l}}^{*}|}$ (complex conjugation in the standard basis), the Schmidt decomposition of ${|\phi\rangle}$ is defined as

where the $\lambda_{l}$ are called Schmidt coefficients. When $d_{1}=2$ (and, therefore, $d_{2}$ must be odd), these coefficients can be calculated independently of the fiducial state, \lambda=\big{(}1\pm\sqrt{3/(d+1)}\big{)}/2, leading to some interest in the approach Gross08 . This is because, along with normalisation $\sum_{l}\lambda_{l}=1$, the Schmidt coefficients always satisfy

using group covariance (where the displacement $D_{q}^{(k)}$ acts on ${}^{d_{k}}$), the property of 2-designs [eq. (3)], and ref. Lubkin78 to evaluate the integral. Some other special Schmidt coefficients are apparent from our solutions (to the numerical precision given): when $d_{1}=3$, we find that \lambda=0,\big{(}1\pm 1/\sqrt{13}\big{)}/2 in dimension $d=12(b)$, $\lambda=0,1/2,1/2$ in dimension $d=15(d)$, and $\lambda=1/7,3/7,3/7$ in dimension $d=48(f,g)$; when $d_{1}=5$, we find $\lambda=0,0,1/3,1/3,1/3$ in dimension $d=35(i,j)$. Nice Schmidt coefficients are seen to follow from the ${F_{b}}$-symmetry [eq. (52)] except for $d=24(c)$, where we deduce that \lambda=\big{(}\sqrt{21}-1\big{)}/10, \big{(}11-\sqrt{21}\pm\sqrt{26\sqrt{21}-98}\big{)}/20, when $d_{1}=3$.

Section V Symbolical computer solutions

Algebraic solutions for Weyl-Heisenberg SIC-POVMs have been reported by Zauner Zauner99 for dimensions $d=2,3,4,5$, together with a solution for $d=8$ due to Hoggar Hoggar98 which is covariant with respect to the three-qubit Pauli group. Appleby Appleby05 added Weyl-Heisenberg solutions for $d=7$ and $d=19$. The second author has reported on algebraic solutions for dimension $d=6,\ldots,13$ and $d=15$ in a series of conference talks Grassl04 ; Grassl05 ; Grassl06 ; Grassl08a ; Grassl08b . Here we close the gap at $d=14$ and add new solutions for $d=24,35,48$.

The general approach used for finding algebraic solutions is to solve the system of polynomial equations for a fiducial vector that lies in an eigenspace of a prescribed symmetry, for example, the Zauner matrix (see e.g. Grassl08b for more details). The smaller the dimension of the eigenspace, the fewer the variables that are needed, and the higher the chances are of being able to compute a solution. The new solutions for $d=24,35,48$ have been obtained with the help of the comparatively large symmetry groups identified in these dimensions, upon inspection of the numerical solutions.

In order to be able to express the Hermitian inner product and the squared modulus via polynomials, we have to replace each complex variable $c_{j}={\langle j|\phi\rangle}$ by two real variables, i.e., $c_{j}=a_{j}+ib_{j}$ where $i^{2}=-1$. In general, the resulting system of polynomial equations will have both real and complex solutions for the variables $a_{j}$ and $b_{j}$, but we are only interested in those solutions for which all components are real.

Note that in an algebraic extension of the rationals , there is no a priori notion of an element being positive or real. An element $\alpha$, for example, which is defined via $\alpha^{2}-2=0$ can either be mapped to $\sqrt{2}$ or $-\sqrt{2}$. Furthermore, the roots of $f(z)=z^{4}-2z^{2}-1$ are $\beta\colonequals\pm\sqrt{1\pm\alpha}$. Depending on the choice of the sign of $\alpha=\pm\sqrt{2}$, $\beta$ is mapped to a real or a complex number. Now assume that we are given a number field $\K=\Q(\theta)$ where $\theta$ is a root of an irreducible polynomial $f(z)\in\Q[z]$ of degree $2m$ with rational coefficients. Without loss of generality, we additionally assume that contains a square root of $-1$. We can then identify a subfield $\LL\leq\K$ such that $\K=\LL(\tau)$ with $\tau^{2}=-1$ and $\LL=\Q(\theta_{1})$ with $f(\theta_{1})=0$ for an irreducible polynomial $f\in\Q[z]$ of degree $m$ that has at least one real root. A field with this property is said to have a real embedding. Fixing such a representation of , we can identify the elements of with the real numbers, and complex conjugation corresponds to an automorphism $\gamma_{c}$ of that maps $\tau$ to $-\tau$ and fixes all elements of . As we will illustrate below, a field automorphism of that commutes with the “complex conjugation” $\gamma_{c}$ and stabilises the Weyl-Heisenberg group as a set, maps a Weyl-Heisenberg SIC-POVM to a possibly different Weyl-Heisenberg SIC-POVM.

Using the computer algebra system Magma Magma , we have computed at least one fiducial vector for dimensions $d=4,\dots,15,19,24,35,48$. The symbolical fiducial vectors can be found in appendix \thechapter.B. While the labeling of the orbits is the same as that introduced for the numerical fiducial vectors, the vectors themselves are not necessarily the very same as those given in appendix \thechapter.A. Their lengths are also unnormalised. Note that in addition to basic arithmetic operations we need only to compute square roots, and third or fifth roots. This is related to the fact that in all cases for which we have been able to compute a solution, the solution can be found in a number field with solvable Galois group, and hence the field extension can be expressed in terms of radicals.6\c@chapter

As an intermediate step, we compute a so-called Gröbner basis $\mathcal{G}$ of the ideal generated by the system of polynomial equations. Unfortunately, the polynomials in $\mathcal{G}$ tend to have large coefficients, and in some cases even a couple of hundred digits, e.g. for $d=11$. Likewise, the defining polynomials of the number field containing a solution quite often have large degrees and huge coefficients. In most of the cases, we succeeded in computing a more compact representation of the solution based on the non-unique decomposition of the field. In Table 1 we give the degree of the number field over which a fiducial vector has been found and which contains the square root $\tau$ of $-1$ as canonical generator for the extension $\C=\R(\tau)$. Note that this field does not necessarily contain a primitive $d$-th root of unity. One such example is $d=19$, where the extension degree is $8$, but a $19$-th root of unity is defined by an irreducible polynomial of degree $18$. Hence the overall extension degree is at least $\mathop{\rm lcm}(8,18)=72$. In some other cases, the field is not closed under all possible automorphisms induced by mapping one root of the defining polynomial to another. In those cases, the field has to be extended to obtain a normal field, and the Galois group is larger than the degree of the original field. In any case, the Galois group is independent of the chosen representation of the field.

The representation of the fiducial vector, and thereby the SIC-POVM, by algebraic numbers allowed us to identify a relation between the two orbits under the extended Clifford group for $d=9$. Fiducial vectors of both orbits labeled $9a$ and $9b$ can be found in an eigenspace of the Zauner matrix, and they are defined over the same number field

where $\sigma_{1}$ and $\sigma_{2}$ are roots of the polynomials $f_{1}(z)=z^{3}-3z-1$ and $f_{2}(z)=z^{3}-18z+12$, respectively. It turns out that the following automorphism of 9 connects the two orbits:

and fixes $\sigma_{1}$, $\sigma_{2}$ and $\tau$. Note that the change of signs of the square roots of $3$ and $5$ implies the mapping for $\sigma_{3}$, since the defining polynomial of $\sigma_{3}$ then changes as well. The set of triple products $C_{jkl}={\langle x_{j}|x_{k}\rangle}{\langle x_{k}|x_{l}\rangle}{\langle x_{l}|x_{j}\rangle}$ computed for all triples of states in a single Weyl-Heisenberg orbit, however, is not invariant under $\gamma$. This implies that the two orbits are not related by a unitary or anti-unitary transformation (see ref. Appleby08 ). Similarly, the two orbits $13a$ and $13b$ are related by a field automorphism of order $8$, implied by simultaneously changing the signs of $\sqrt{5}$ and $\sqrt{7}$. For dimension $11$, the orbits $11a$ and $11b$ are related by a field automorphism of order $8$, implied by simultaneously changing the signs of $\sqrt{2}$ and $\sqrt{3}$, while the orbit $11c$ is unrelated to them. The orbits $14a$ and $14b$ are related by simultaneously changing the signs of $\sqrt{3}$ and $\sqrt{5}$.

It is not yet clear to us when such a relation exists. Unlike complex conjugation, which is intrinsic to the problem and captured by the extended Clifford group, we cannot predict which other field automorphisms relate the different orbits since we know the relevant number field only after having found a solution.

Section VI Conclusion

Ten years have now passed since it was first conjectured, by Gerhard Zauner Zauner99 , that maximal sets of $d^{2}$ equiangular lines exist in all finite complex dimensions $d$. Despite considerable attention now attracted to this question from the quantum physics community, where sets of $d^{2}$ equiangular lines define SIC-POVMs Renes04 , Zauner’s conjecture remains open to this day.

The primary purpose of this article is to report the results of the authors’ new computer study on SIC-POVMs: numerical solutions for fiducial vectors that generate Weyl-Heisenberg covariant SIC-POVMs are provided in appendix \thechapter.A for all $d\leq 67$. These confirm Zauner’s conjecture in all such dimensions, except $d=66$, which remains inconclusive. Moreover, a putatively complete list of solutions is given for $d\leq 50$. It is our hope that this list will act as an important resource for the growing community of researchers interested in the SIC-POVM problem. A preliminary symmetry analysis has lead to new algebraic solutions in dimensions $d=24$, $35$ and $48$. These are collected with other known Weyl-Heisenberg covariant algebraic solutions in appendix \thechapter.B (see also ref. Appleby05 ). Additional symmetries are likely to exist in arbitrarily large dimensions, and might help in proving the existence of an infinite series of corresponding SIC-POVMs.

Although our confidence in its truth has grown considerably, we seem no closer to a proof of Zauner’s conjecture than Gerhard Zauner was at the time of his doctoral dissertation. The observation by Zauner, and subsequently by others, of the apparent deep connection between the Heisenberg group and maximal sets of equiangular lines, appears to be born more out of luck than discernment. Our incomplete understanding of the automorphism group of Weyl-Heisenberg covariant SIC-POVMs is most impoverishing: simplifying to prime dimensions, although the outer automorphism group of ${\operatorname{H}}(d)$ is ${\operatorname{GL}}_{2}({}_{d})$, only members of the subgroup ${\operatorname{ESL}_{2}}({}_{d})$ are currently known to naturally define automorphisms of Weyl-Heisenberg SIC-POVMs (as described in sec. III). The relation found in dimension $d=9$ between the two different extended Clifford orbits of solutions (labeled $9a$ and $9b$) hints of a bigger picture that could paint currently unrelated orbits in the same light (see sec. V). Without a deeper analysis of the known SIC-POVM solutions, however, we can only speculate on what this bigger picture might be.

Section VII Notes

1. In his doctoral dissertation (Zauner99, , p. 65), following the derivation of SIC-POVMs for dimensions $2,\ldots,7$, Gerhard Zauner makes the following observation:

Die Beispiele für $b=2,3,4,5,6,7$ legen die folgende Vermutung nahe. Für alle $b\geq 2$ gibt es im ($\left[\frac{b}{3}\right]+1$)-dimensionalen Eigenraum zum Eigenwert $1$ der $b\times b$ Matrix $\mathbf{Z}$ Vektoren, welche mit dem Ansatz (3.14) maximale Quantendesigns mit Grad $1$ erzeugen. Für $b=3m+2$ gibt es im gleichdimensionalen Eigenraum zum Eigenwert $\alpha$ ebensolche Vektoren.

In the present context, this quotation can be translated as follows ($\mathbf{Z}$ is the transpose of ${Z}$, in fact, but we will follow Appleby Appleby05 ):

The examples for dimension $d=2,3,4,5,6,7$ give rise to the following conjecture. For all dimensions $d\geq 2$ the eigenspace of dimension $\lfloor\frac{d}{3}\rfloor+1$ with eigenvalue $1$ of the $d\times d$ matrix $Z$ (see eq. (13)) contains fiducial vectors of a Weyl-Heisenberg covariant SIC-POVM. For $d=3m+2$ the eigenspace of the same dimension with eigenvalue $e^{2\pi i/3}$ contains fiducial vectors as well.

2. Solutions for fiducial vectors were found by minimising the LHS of eq. (46), the cost function, until the bound on the RHS was met. This was performed using the MATLAB® Optimization Toolbox™ MATLAB with the cost function and its derivatives implemented in C and linked in. The solutions were then further refined to 38 digits with the multiprecision capabilities of PARI/GP PARIGP . A small number of AMD Opteron™ 252 dual-processor machines were used, though for a significant amount of time.

3. In each dimension $d$, the current list of known ${\operatorname{PEC}}(d)$ orbits of fiducial vectors is considered complete when, upon initialising each trial to a random vector under the Haar measure, the search consecutively encounters $30(n+1)$ solutions that generate one of the $n$ known orbits. Assuming each orbit is found with equal probability, the probability of missing an orbit is then no more than $e^{-30}$. Under this criteria, the list is complete for $d\leq 47$. In dimensions $d=48,\dots,50$, we have encountered enough of the known orbits to be confident that our list is complete here also, but the computations are ongoing.

4. Note that when $d$ is even, however, the subgroup of ${\operatorname{ESL}_{2}}({}_{2d})$ that defines ${\operatorname{S}}(\phi)\leq{\operatorname{PEC}}(d)$ can have an order that is a multiple of $|{\operatorname{S}}(\phi)|$. We have chosen all stabiliser orders to double when treated as subgroups of ${\operatorname{ESL}_{2}}({}_{2d})$, e.g. $|\langle{F_{z}}\rangle|=2|\langle{[{{F_{z}}}\mspace{1.0mu}|\mspace{1.0mu}{0}]}\rangle|$.

5. See Flammia (Flammia06, , Conjecture 4) and note that ${F_{z}}$ and $J{F_{z}}J$ belong to different conjugacy classes in ${\operatorname{SL}_{2}}({}_{{\bar{d}}})$ when $3\mid d$.

6. When expressing the elements of the original number field in terms of radicals, we generally have to extend the field to one of larger degree. The information given in table 1 is with respect to the original field.

References

Appendix \thechapter.A Numerical solutions

Appendix \thechapter.B Symbolical solutions