Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension six

I Introduction

The existence of maximal sets of mutually unbiased (MU) bases in every dimension is a very important open problem in foundations of quantum mechanics. Two orthogonal bases are MU if they are as different as possible in Hilbert space, in the sense that the projection of every element of the first base onto every element of the second one has the same absolute value. This kind of bases has several applications in quantum information theory: quantum key distribution protocols Bennett ; Brub ; Cerf , entanglement detection Spengler , dense coding, teleportation, entanglement swapping, covariant cloning and state tomography (see Durt and references therein). They are also interesting in mathematics since their connection with affine planes Gibbons and finite geometries Bengtsson3 . Additionally, they are useful to solve the Mean King Problem Aharonov . In a Hilbert space of dimension $d$ we can construct maximal sets of $d+1$ MU bases when $d$ is prime or prime power. Otherwise, analytical Zauner ; Archer ; Jaming ; Grassl and numerical Butterley ; Bengtsson1 ; Brierley ; Brierley3 ; Jaming2 efforts to construct $d+1$ MU bases fail and it is suspected that they do not exist. The lowest dimension where this problem remains open is six, where most of the previously mentioned works have tried to find a solution. This paper presents a new method that numerically solves the problem to find the maximal set of MU bases that can be obtained from a given pair of MU bases. The most important advantage of our method is that the computational cost is independent of the pair of MU bases considered. Our method is not an algorithm because it does not stop with a definite answer but it converges very quickly even in higher dimensions.

This work is organized as follows: In Section II, we briefly introduce complex Hadamard matrices and mutually unbiased bases. In Section III, we present the method to find MU vectors and we discuss its convergence. We successfully test our approach in Section IV by searching the known 48 vectors MU to the identity and the Fourier matrix in dimension six, and we obtain known triplets containing the identity and the Diţă matrix. Section V contains our main results: we could not find triplets of MU bases from considering complex Hadamard matrix belonging to the non-affine families $K_{6}^{(2)}$ and $K_{6}^{(3)}$ existing in dimension six as well as other families contained in them. This study have allowed us to find new symmetries for the family $K_{6}^{(2)}$.

II Complex Hadamard matrices and mutually unbiased bases

This section contains the minimal information about complex Hadamard matrices and mutually unbiased bases required to make the paper self-contained; more details can be found in Bengtsson1 and Brierley4 , for example. Two orthonormal bases $\{|\varphi_{k}\rangle\}$ and $\{|\phi_{l}\rangle\}$ defined on a $d$-dimensional Hilbert space are mutually unbiased (MU) if they satisfy the property

for every $k,l=0,\dots,d-1$. Maximal sets of $d+1$ MU bases have been found in every prime Ivanovic and prime power Wootters dimension. A lower bound on their members can be established in the general case of $d=p_{1}^{r_{1}}\dots p_{n}^{r_{n}}$, where $p_{1}^{r_{1}}<\dots<p_{n}^{r_{n}}$: it is known how to construct at least $p_{1}^{r_{1}}+1$ MU bases Bengtsson2 . Here, $d=p_{1}^{r_{1}}\dots p_{n}^{r_{n}}$ is the prime power decomposition of the number $d$. In the particular case of $d=6$ the lower bound is three, and this is the maximal number of MU bases attained so far.

A square matrix is a complex Hadamard matrix if it has unimodular entries and orthonormal columns. Such matrices exist in every dimension and the Fourier matrices represent the simplest proof of their existence. For example, in dimension four the Fourier matrix is given by

where $\omega=e^{2\pi i/4}$. Also, the tensor product of complex Hadamard matrices is a complex Hadamard matrix. The simplest example is given by the tensor product of two Fourier matrices defined in dimension two:

which gives us a real Hadamard matrix. Complex Hadamard matrices have been extensively studied in recent years and they are very hard to find when $d>5$ Brierley . We say that two complex Hadamard matrices $H_{1}$ and $H_{2}$ are equivalent ($H_{1}\sim H_{2}$) if there exist unitary diagonal operators $D_{1},D_{2}$ and permutation operators $P_{1},P_{2}$, such that

A complex Hadamard matrix may belong to a continuous set of inequivalent complex Hadamard matrices called a family. A family of complex Hadamard matrices $H(x)$ is affine if it can be cast in the form

and the symbol $\circ$ denotes the Hadamard product $(A\circ B)_{lm}=A_{lm}B_{lm}$. The number $s$ of independent parameters corresponds to the dimension of the family. If a continuous family of inequivalent complex Hadamard matrices is not affine we say it is non-affine. For example, the families stemming from the Fourier matrix in dimension six $(F^{(2)}_{6})$ and the Diţă family $(D^{(1)}_{6})$ are affine families, whereas the Karlsson families $K^{(2)}_{6}$ and $K^{(3)}_{6}$ are non-affine. The notation here considered is consistent with the catalog of complex Hadamard matrices presented by Bruzda-Tadej-Życzkowski Bruzda . In this notation, the upper index denotes the dimension of the family and the lower index the dimension of the space where it is defined. If a complex Hadamard matrix $H$ belongs neither to an affine nor to a non-affine family we call it isolated. In other words, it is impossible to obtain a complex Hadamard matrix inequivalent to $H$ from infinitesimal perturbations to its entries. A set of MU bases is inextensible if no further orthonormal basis MU to every base of the set exists. A well-known fact is that any set of $d+1$ MU bases is inextensible, and it is conjectured that every triplet of MU bases in dimension six is inextensible. We mention that the complete set of inextensible MU bases in dimensions $d\leq 5$ has been found in Brierley2 by considering Buchberger’s algorithm Buchberger . This algorithm is a generalization of gaussian elimination to non-linear multivariate polynomial equations. Also, a characterization of triplets of MU bases in $d=6$ have been given if the second MU basis belongs to an affine family of Hadamard matrices Brierley3 . In a recent paper, it has been analytically proven that given any triplet of MU product bases in dimension six it is not possible to find even a single vector MU to the triplet McNulty .

III MU vectors as fixed points

where $k,l=0,\dots,d-1$. In order to find a solution we perform the following steps:

Choose a quantum state $|\Psi_{0}\rangle$ at random, which will be called the seed.

Decompose the state $|\Psi_{0}\rangle$ in the basis $\{|\varphi_{k}\rangle\}$,

Modify the amplitudes of the expansion coefficients $c_{k}$ in order to impose the information about $A$,

In the last step, we have replaced the amplitudes of the coefficients $\{c_{k}\}$ compatible with those of the observable $A$; note that we did not modify the phase factors $c_{k}/|c_{k}|$ because we can not draw any conditions about them from the data $\{p^{(A)}_{k}\}$. The physical imposition operator implements the transformations just described in one operation,

when $|\Psi_{0}\rangle$ happens to be orthogonal to the state $|\varphi_{k}\rangle$, for any $k=0,\dots,d-1$, we define

This operator is non-linear and its action on every quantum state is well-defined. The action of this operator on a randomly chosen state $|\Psi_{0}\rangle$ can be interpreted as incorporating what we learn about the unknown state when the observable $A$ is measured. In other words, the initial state $|\Psi_{0}\rangle$ has no information about the quantum system considered while the state $T_{A,p^{(A)}}|\Psi_{0}\rangle$ contains all the information we have acquired by measuring $A$ in the unknown state. Note that $T_{A,p^{(A)}}$ is idempotent because applying it once exhausts the information available about $A$.

Next, we proceed in a similar way with the second observable $B$ in order to try to find a solution of the set of Eqs.(7,8), defining the physical imposition operator associated with the observable B,

generally does not contain the complete information about both $A$ and $B$: some of the information about $A$ is destroyed when $T_{B,p^{(B)}}$ is imposed, which is a consequence of the commutation rule $[A,B]\neq 0$. If $A$ and $B$ commute, it is trivial to find a solution to Eqs. (7,8), namely

In general, the state $|\Psi_{1}\rangle$ has the complete information about $B$ and only partial information about $A$, so the composite operator $T_{B,p^{(B)}}T_{A,p^{(A)}}$ is not idempotent. Therefore, we can iterate the procedure just described and analyze the convergence of the sequence

It has been proven Goyeneche2 that every solution of the system of coupled equations (7,8) is an attractive fixed point of $T_{B,p^{(B)}}T_{A,p^{(A)}}$. Moreover, this property also holds for a general set of observables $A,B,C,\dots$ and probability distributions $p^{(A)},p^{(B)},p^{(C)},\dots$ The iterations are robust under adding redundant information, and the sequences converge if and only if the probability distributions are compatible, in the sense that the Heisenberg uncertainty principle is not violated.

The problem of constructing MU vectors is a particular case of the quantum state reconstruction problem just described. Let $A$ and $B$ be two observables with a pair of MU eigenbases $\mathcal{B}_{A}$ and $\mathcal{B}_{B}$. A vector is MU to the pair $\{\mathcal{B}_{A},\mathcal{B}_{B}\}$ if it has equally weighted probability distributions with respect to both observables, that is,

for every $k,l=0,\dots,d-1$. Interestingly, when the eigenvectors bases are MU, every basin of attraction is found to be of the same size, verified numerically in every prime dimension $2\leq d\leq 37$ Goyeneche2 , as well as in every simulation reported below for $d=6$. This property indicates that the efficiency of the algorithm is maximal when the eigenvector bases of the observables are MU, because the number of randomly chosen seed states needed to find all solutions is minimized. This observation conforms with the idea that the redundancy of information is minimal when the observables have MU eigenvector bases.

III.2 Convergence

In order to analyze the convergence of the sequence $|\Psi_{n}\rangle$ defined in Eq.(17) we need to define a metric for quantum states. We want to determine when a solution given by our method is a solution of the coupled system of equations given by Eqs. (7,8). Let $A$ be an observable having the eigenvectors base $\{|\varphi_{k}\rangle\}_{k=0,\dots,d-1}$ and let $|\phi\rangle$ and $|\psi\rangle$ be two arbitrary quantum states. The distance between the probability distributions associated with the observable $A$ in the states $|\phi\rangle$ and $|\psi\rangle$ can be defined by means of Hellinger’s metric Hellinger ,

This metric compares two probability distributions of the eigenvalues of a single observable and it is important to realize that this is a metric for probability distributions, not for states. In the present context, we need to consider more than one observable and the corresponding probability distributions. Therefore, we introduce the Hellinger metric for $m$ observables, the so-called distributional metric Goyeneche2 ,

where $D_{A^{j}}(|\phi\rangle,|\psi\rangle)$ is the Hellinger distance of the observable $A^{j}$, defined in Eq.(19).

tells us how close the state $|\Psi_{n}\rangle$ is to being MU to $\mathcal{B}_{A}$ and $\mathcal{B}_{B}$. We will say that a sequence has converged when

which means that the absolute error of the amplitudes is less than $8\times 10^{-4}$ on average. Numerical simulations suggest that the absolute error of every amplitude of a solution is very close to the averaged error just mentioned. Given that the desired solutions are (stable) attractive fixed points, our approximations must be close to the exact solutions of the problem. In the next section, we test our method by constructing known sets of states MU to a number of pairs consisting of the identity and a complex Hadamard matrix of order six.

IV Testing the method: Tao, Fourier and Diţă matrices

It turns out that the 48 vectors can be grouped into three sets, each corresponding to one orbit under the Weyl-Heisenberg group. To see this, let us first define the displacement operators $D_{p}$ by

Three vectors generating the mentioned orbits under Weyl-Heisenberg group are given by

Finally, in condensed form, the 48 MU vectors can be written as

Both the first and second set define a circulant matrix each while the last set give rise to six circulant matrices. These observations generalize to other dimensions $d$ as follows.

and $\omega=e^{2\pi i/24}$. The MU vectors are given by the columns of the matrices $H_{1}$ and $H_{2}$. Interestingly, both of them are equivalent to a member of the Fourier family. We have verified that these analytical expressions are indeed solutions of the problem.

V Karlson’s non-affine families

Karlsson has found a two-parameter non-affine family of complex Hadamard matrices $K_{6}^{(2)}$ in dimension six Karlsson , which contains the families $D_{6}^{(1)},M_{6}^{(1)}$ and two subfamilies of the Fourier family. The Diţă family $D_{6}^{(1)}(t)$ is equivalent to the four corners $K_{6}^{(2)}(\pm\pi/2,\pm\pi/2)$ whereas $K_{6}^{(2)}(x,0)\sim F_{6}^{(2)}(x,x)$ and $K_{6}^{(2)}(0,x)\sim(F_{6}^{(2)}(x,x))^{t}$. Also, Matolcsi family determines one of the diagonals, that is, $K_{6}^{(2)}(x,x)\sim M_{6}^{(1)}(x)$. All these subfamilies are explicitly obtained from $K_{6}^{(2)}$ in Karlsson’s paper Karlsson . Note that a subset of the Fourier family and its transpose define the horizontal and vertical axes, respectively, of the parameter space of $K_{6}^{(2)}$. Also, the Fourier matrix $F_{6}^{(2)}(0,0)$ is equivalent to the center $K_{6}^{(2)}(0,0)$.

where $z_{1}=e^{ix_{1}}$ and $z_{2}=e^{ix_{2}}$, $-\pi/2\leq x_{1},x_{2}\leq\pi/2$ and the four functions

are defined in terms of a single function, namely

where $P_{34}$ and $P_{56}$ are permutations matrices. Consequently, one may restrict both parameters $x_{1}$ and $x_{2}$ to the interval $[-\pi/2,\pi/2]$.

Inspired by the symmetries of the graph shown in Fig. 1 and taking into account Eqs.(35) and Eq.(36), we notice that

using the symmetry $f(x_{1},x_{2})=f(x_{2},x_{1})$, we also obtain

Eqs.(41) to (43) reveal the symmetry apparent in Fig. 1, and we consider it unlikely that any further symmetries exist. The family $K_{6}^{(2)}(x_{1},x_{2})$ with $-\pi/2\leq x_{1},x_{2}\leq\pi/2$ is divided into eight triangles of the same area, each of them containing one copy of the complete family. Therefore, it is sufficient to consider values in the triangle $x_{1}\in[0,\pi/2]$, $x_{2}\leq x_{1}$, i.e. the shaded area in Fig. 2. In this figure, we show that the Matolcsi’s family $M_{6}^{(1)}$ is located on the both diagonals of the square. As we will show later, we can construct triplets of MU bases from $M_{6}^{(1)}$, which means that it should appear in Fig. 1. However, the set $\{M_{6}^{(1)}{x}\}$ is of measure zero within the family $K_{6}^{(2)}(x_{1},x_{2})$; since the parameters $(x_{1},x_{2})$ are chosen at random, the probability of observe it vanishes.

V.2 Karlsson’s tri-parametric family

A tri-parametric non-affine family of complex Hadamard matrices has been recently found by Karlsson Karlsson3 , reading explicitly:

Here, $\mathcal{M}$ denotes the Möbius transformation, defined by

with $\alpha_{A}=A_{12}^{2}$, $\beta_{A}=A_{11}^{2}$, and $\alpha_{B}=B_{12}^{2}$, $\beta_{B}=B_{11}^{2}$, and $\theta,\phi,\psi\in[0,\pi)$.

This family contains the non-affine family $K_{6}^{(2)}$ and it also contains the complete set of the so-called $H_{2}$-reducible matrices. In dimension six, a complex Hadamard matrix is $H_{2}$-reducible if it contains nine $2\times 2$ submatrices that are Hadamard matrices. Let us analyze an interesting particular case. It follows from Eq.(47) and (48) that the subfamily $K_{6}^{(3)}(0,\phi,\psi)$ do not depend on the parameter $\phi$. In this case, the Möbius transformations in Eqs.(50–52) turn into the identity irrespective of the value of $z$. Therefore, we obtain an affine one-parameter family

which is contained in the Fourier family $F_{6}^{(2)}$. Here, the matrix $R(\psi)$ is defined by

where $\bullet$ means a null entry. The permutation matrix $P_{46}$ interchanges rows 4 and 6. This subfamily allows a triplet for any $\psi\in[0,\pi)$, because it is contained in the Fourier family $F_{6}^{(2)}(a,b)$, which admit a triplet for any value of $a$ and $b$.

Numerical simulations from considering the family $K_{6}^{(3)}$ are shown in Fig. 3(a) to Fig. 3(d). As we can see, these figures strongly suggest the existence of reflection symmetries in the three variables $\theta,\phi$ and $\psi$. However, we have not been able to find them analytically even in the simplest case $\psi=0$. There are highly non-trivial diagonal matrices and row permutations which do not allow us to reveal the hidden symmetries. On the other hand, if a random value of $\psi$ is considered we always obtain the same kind of two dimensional objects. No fractal structures have been detected for any value of the parameters. Consequently, we have a representative description of the general case, in the sense that the evolution of the parameter $\psi$ gives us a smooth connection between these four figures (Fig. 3(a) to Fig. 3(d)).

VI Summary and conclusions

We have presented an efficient numerical method to construct sets of mutually unbiased bases in finite dimension. The main advantage of our method appears when non-affine families of complex Hadamard matrices are considered, where the standard method to solve coupled polynomial equations (Buchberger’s algorithm) often stalls due to excessive memory requirements. Our method numerically solves the problem for non-affine families with the same computational cost as for affine families.

Table 1 summarizes the results obtained in this paper, indicating the maximal set of MU bases that can be constructed from pairs of MU bases associated with various families in dimension six. In the cases of affine families and the non-affine family $B_{6}^{(1)}$ we found triplets in the entire range of the parameters, whereas the isolated matrix $S_{6}^{(0)}$ does not allow a triplet. This property of $S_{6}^{(0)}$ has been previously found by Brierley and Weigert Brierley6 .

In the simulation realized for the non-affine family $B_{6}^{(1)}$ we have considered 10.000 random choices of its parameter, and we obtained a triplet in every case. This is the only non-affine family where we found a triplet for every value. The non-affine family $M_{6}^{(1)}$ defined in the range $t\in(\pi/2,\pi]\cup(3\pi/2,2\pi]$ does not allow a triplet in its full range. We have realized three simulations considering 1,000; 10,000 and 100,000 random choices of the parameter $t$ and we obtain the same results. That is, a part of the family $M_{6}^{(1)}$ does not allow us to construct a triplet of MU bases (see Table 1).

In the cases of Karlsson’s families $K_{6}^{(2)}$ and $K_{6}^{(3)}$ we have considered 2 millon and 8 millon random choices, respectively, sampling the entire parameter ranges of both families. We have shown that these two families extend to triplets at most. The property that a triplet of MU bases can be found only for a reduced set of parameters of a family is a new result presented here by the first time. In addition, we identified new symmetries that reduce the range of the parameters of the family $K_{6}^{(2)}$.

Finally, in our investigation of more than ten million complex Hadamard matrices belonging to non-affine families we could not find a single vector being MU to a triplet. This evidence supports the conjecture that no more than three MU bases can be constructed in dimension six.

VII Acknowledgments

I specially thank to Stefan Weigert for his invaluable help in order to make possible this article. This work is supported by Grants FONDECyT N${}^{\text{\text@underline{o}}}$ 3120066 and MSI P010-30F.

References