Constructing mutually unbiased bases in dimension six

The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex $\left(6\times 6\right)$ Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.

Figure 1

The set of known Hadamard matrices in dimension six consists of special Hadamard matrices $F\left(0,0\right)\equiv F_6,D\left(0\right)\equiv D_0$, $C$, and $S$, located on the vertical symmetry axis; of the affine families $D\left(x\right)$, $F\left(\mathbf{x}\right)$, and $F^T\left(\mathbf{x}\right)$; and of the nonaffine families $M\left(t\right)$, $B\left(\theta \right)$, $X\left(a,b\right)$, and $X^T\left(a,b\right)$ (see Appendix for definitions). Note that the sets $X\left(a,b\right)$ and $X^T\left(a,b\right)$ cover the interior of the upper circle twice.

Figure 2

The number $N_v$ of vectors $|v⟩$ which are MU with respect to the columns of the identity $I$ and Diţă matrices $D\left(x\right)$; the parameter $x$ assumes 72 parameter values $x∊{\Gamma }_D$ defined in Eq. (21) and 500 randomly chosen ones in the fundamental interval $\left[-1/8,1/8\right]$ of the parameter $x$. The inset illustrates the impact on $N_v$ near the discontinuity $x\simeq 0.0177$ if an approximate set of equations is used (cf. Sec. 4C).

Figure 3

The number $N_v$ of vectors $|v⟩$ which are MU with respect to the columns of the identity $I$ and (a) symmetric Hadamard matrices $M\left(t\right)$; the parameter $t$ assumes 60 parameter values $t∊{\Gamma }_M$ defined in Eq. (23) and 300 randomly chosen ones in the fundamental interval [0,1/2] and of (b) Hermitean matrices $B\left(\theta \right)$; the parameter $\theta$ assumes 34 parameter values $\theta ∊{\Gamma }_B$ defined in Eq. (26) and 300 randomly chosen ones in the fundamental interval $\left[{\theta }_0,1-{\theta }_0\right]$. The phase ${\theta }_0$ has been defined in Eq. (A8).

Figure 4

The number $N_v$ of vectors $|v⟩$ which are MU with respect to the columns of the identity $I$ and Szöllősi Hadamard matrices $X\left(a,b\right)$ for 50 randomly chosen parameter values (a) on the line $\Lambda$ connecting $F\left(1/6,0\right)$ to $C$ and (b) on the line ${\Lambda }^{\prime }$ connecting $F\left(1/6,0\right)$ to $B\left({\theta }^{\prime }\right)$; the maximum modulus ${\left|a+ib\right|}_{\text{max}}$ is defined by Eq. (A14).

Figure 5

The set of all Hadamard matrices $H$ which have been considered (cf. Fig. and Tables ): for each $H$, a second MU Hadamard matrix can always be found except for the isolated spectral matrix $S$; consequently, triples of MU bases are the norm while quartets of MU bases do not exist.