Characterization of high-dimensional entangled systems via mutually unbiased measurements

Appendix A Supplemental material

After recording the coincidence count rates $C_{k}$ for each choice of $n,i,m$ and $j$, and the single-channel count rates $A_{k}$ and $B_{k}$, we convert the count rates to detection probabilities through

where $\Delta t$ is the gate time of our coincidence-counting electronics and $\Upsilon$ an appropriate normalization factor. The term $A_{k}B_{k}\Delta t$ corresponds to the uncorrelated accidental count rate $U_{k}$.

depends on the type of tomographic reconstruction performed. The factor $Q$ indicates the number of $d\times d$ quadrants in the correlations matrix for the set of measurements of choice. The product $d^{2}Q$ corresponds to the total number of independent measurements. For an overcomplete tomography, where we set $\sum_{k}^{d^{2}}p_{k}=1$ for any given choice of $m$ and $n$, $Q=(d+1)^{2}$; see Fig. 5(a). For a tomographically complete reconstruction that uses the presented subset of MUBs measurements, $Q=\left[1+(d-1)\right]^{2}=d^{2}$; see Fig. 5(b).

A.2 Completeness of tomographic reconstruction

One can express the density matrix $\rho$ as a linear combination of a complete basis of $d^{2}\times d^{2}$ matrices $\Gamma_{\mu}$ with complex coefficients $\gamma_{\mu}$ Thew:2002a:

where $D=d^{2}$ is the dimension of our bipartite system. The basis matrices $\Gamma_{\mu}$ have the following properties:

where $\kappa$ is any $d^{2}\times d^{2}$ matrix. A suitable set of Hermitian matrices $\Gamma_{\mu}$ for the decomposition of $\rho$ is given by the generalized Gell-Mann matrices for dimension $D$.

A necessary and sufficient condition for the completeness of the set of tomographic states $\left\{\left|\psi_{\mu}\right\rangle\right\}$ (associated with the two-qudit observables $\Pi_{\mu}$) is given by the invertibility of the matrix

which allows us to express the complex coefficients $\gamma_{\mu}$ in terms of probabilities $p_{\mu}=\left\langle\psi_{\mu}\right|\rho\left|\psi_{\mu}\right\rangle$ Altepeter:2005:

Let us define the orthonormal set of basis vectors $u_{i}$ in dimension $d$, whose elements are given by $(u_{i})_{j}=\delta_{ij}$. For a choice of two single-particle MUBs vectors for the qu$d$it subsystems $A$ and $B$

we can express the elements $j=1,\,\dots\,D$ of the corresponding vector for the $D$-dimensional state space of the bipartite system as

where $(\alpha,\,\beta)$ are all pairwise permutations of indices $\left\{1,\,\dots\,d\right\}$. From the $D$-dimensional vectors $v_{AB}$ we then define the states $\left|\psi_{\mu}\right\rangle$ that describe the measurements on the composite system.

After calculating the states $\left|\psi_{\mu}\right\rangle$ for the subset of MUBs measurements for complete tomography defined previously, we find the invertible matrix $B$ through Eq. (10).

A.3 Quantum state reconstruction via numerical optimization

The number of joint measurements required to perform our reconstruction procedure is given by:

which corresponds to the minimum number of parameters required to perform a complete quantum state tomography. The number of measurements required by the overcomplete quantum state reconstruction strategy outlined in Ref. Agnew:2011 requires instead the following total number of measurements:

The numerical optimization to find the density matrix $\rho$ that provides the best fit to the experimental probabilities from Eq. (6) is carried out by performing a random search over the parameter space of a complex left-triangular matrix $T$ Altepeter:2005, from which a physical guessed density matrix is derived:

We reconstructed the states from $d=2$ to $5$ using both methods. A quantitative comparison of the results is shown in Tab. 1.

A.4 Mutually unbiased vectors

The complete sets of mutually unbiased vectors used in the quantum state tomography, for dimensions from $d=2$ to $5$, are reported in this section. For each vector $v_{mi}$, $m$ indicates the basis among the $d+1$ available in dimension $d$ and $i$ the vector within the basis. Each vector provides the corresponding set of complex coefficients for the superposition of the basis modes of choice; see Fig. 6 for $d=2$ and Fig. 7 for $d=5$.

The experimental procedure and the reconstruction technique can be readily extended to higher dimensions. The existence of full sets of $d+1$ MUBs has however only been proven for dimensions $d$ that are prime numbers or powers of a prime. Finding MUBs in higher prime power dimensions, especially sets that may be suitable for practical implementations, remains challenging. It should also be noted that, despite MUBs being particularly advantageous to efficiently reconstruct the density matrix of an unknown state encoded in the spatial modes of a single photon, as $d$ increases the complicated structures of the modes involved may negatively affect the detection efficiency.

A.4.2 Coefficients for d=3fragmentsd3d=3

A.4.3 Coefficients for d=4fragmentsd4d=4

A.4.4 Coefficients for d=5fragmentsd5d=5