Are SIC-POVMs optimal for quantum state reconstruction?

Question

Mutually unbiased bases (MUBs) are pairs of orthonormal bases $\{u_j\}_j,\{v_j\}_j\in\mathbb C^N$ such that $$|\langle u_j,v_k\rangle|= \frac{1}{\sqrt N},$$ for all $j,k=1,...,N$. These are useful for a variety of reasons, e.g. because they provide "optimally independent information", in the sense that if we want to reconstruct a state from a set of projective measurements, choosing MUBs maximises the amount of gained information. A standard reference here is Wooters and Fields (1998).

On the other hand, SIC-POVMs are informationally complete POVMs $\{\mu_b\}_{b=1}^{N^2}$ such that $\mu_b=\Pi_b/N$, with $\Pi_b$ rank-1 projections such that $$\langle \Pi_a,\Pi_b\rangle = \frac{N\delta_{ab}+1}{N+1}.$$ Two relevant references are here (Rastegin 2014) and (Appleby et al. 2017).

It seems intuitive that such SIC-POVMs should be, similarly to MUBs, optimal for state reconstruction, in the sense of them being the optimal - with respect to the number of required samples - choice of POVM to reconstruct an arbitrary given quantum state. However, I haven't seen this stated explicitly.

Are SIC-POVMs optimal in this sense? More precisely, given a state $\rho$ that we want to reconstruct via a POVM $\mu$, and some target degree of reconstruction accuracy, is it true that the expected number of required samples is minimal if $\mu$ is a SIC-POVM?

Answer 1

First of all, here's a short disclaimer: I'm not an in-depth expert in this field, I'm just currently getting in contact with tomography more and more often :) So take the following with a grain of salt. It might be incomplete in the sense that better results have been shown somewhere.

We consider the problem of reconstructing a $d$-dimensional quantum state $\rho$ from measurements with respect to a POVM $M_1,\dots,M_m$. Recall that a POVM is defined by $d\times d$ operators $M_i$ that fulfill $M_i \geq 0$ and $\sum_{i=1}^m M_i = I$. The measurements define a linear map $\mathcal{M}: \mathbb H_d\rightarrow \mathbb R^m$ on the space of Hermitian $d\times d$ matrices $\mathbb H_d$, given by $$ \mathcal M(\rho) := \sum_{i=1}^m \mathrm{tr} ( M_i \rho) \, e_i, $$ where $e_i\in\mathbb R^m$ is the standard basis.

Note that the measurement map is linear in the state $\rho$. This approach assumes that we can perform iid measurements on sequential, single copies of $\rho$. If we can measure multiple copies at once, we can define more general measurement maps such as $$ \tilde{\mathcal M}(\rho) := \sum_{i=1}^{\tilde m} \mathrm{tr} ( \tilde M_i \rho^{\otimes k}) \, e_i. $$ Note that $\tilde{\mathcal{M}}$ is a polynomial of order $k$ in the state $\rho$. Such non-linear measurement maps can outperform the linear ones in terms of sampling rate (see e.g. Ref. 1-2).

In any case, the POVM has to be informationally complete, meaning that it should be possible to uniquely reconstruct the state from the measurement statistics. However, that doesn't mean that the measurement map is invertible. We only require it to have a left inverse ("reconstruction formula"). This is equivalent to saying that the $M_i$ span the space of Hermitian matrices $\mathbb H_d$. We can then use frame theory to define a left inverse using a dual frame, see e.g. Ref. 3. This is what is usually meant by linear reconstruction or linear inversion.

At this point, let me note that any POVM coming from a complex projective 2-design is informationally complete (in fact these are exactly the rank-one tight informationally complete POVMs, see again Ref. 3). A complex projective 2-design is a set of states $(\psi_i)_{i=1,\dots,N}$ such that $$ \frac{1}{N}\sum_{i=1}^N \left( |\psi_i\rangle\langle\psi_i| \right)^{\otimes 2} = \int_{\mathbb CP^{d-1}} \left( |\psi\rangle\langle\psi| \right)^{\otimes 2}\,\mathrm{d}\psi = \frac{2}{d+1} \Pi_\mathrm{sym}, $$ where $ \Pi_\mathrm{sym}$ is the projector onto the symmetric subspace of $\mathbb C^d \otimes \mathbb C^d$. Here are some examples of 2-designs:

• Stabiliser states
• SIC-POVMs (equiangular 2-design)
• MUBs

Hence, from this point of view, MUBs and SIC-POVMs are equally suited for tomography.

Great, but you asked for the number of samples. In general, the needed number of samples will not only depend on the POVM but also on the precise reconstruction method and the required precision (in some distance measure). The ideas I have sketched above lead to linear inversion and least squares methods, but there are also alternative methods such as maximum likelihood estimation. Moreover, tomography under rank constraints is practically very important. If $\rho$ is guaranteed to have rank $\leq r$, this can be used to reduce the sampling complexity significantly.

To be precise, if we want to reconstruct a rank $r$ state up to error $\varepsilon$ in trace distance, then we need at least $\tilde\Omega(r^2 d/\varepsilon^2)$ many samples (with independent measurements as above, see Ref. 1). As far as I know, this bound can be achieved with compressed sensing techniques (see Ref. 4-6 and Ref. 1, Sec. IIA for discussion) and least squares (Ref. 7), but not with 2-designs. Using 2-designs, it is possible to achieve $O(r^2 d \log d /\varepsilon^2)$ which is almost optimal (Ref. 7).

In particular, I am not aware of any result connecting SIC-POVMs (or 2-designs in general) to optimal sample complexities. But as mentioned in the disclaimer, there might be one I have overlooked.

Further helpful discussions of previous literature can be found in Ref. 1, 7, and 8.

References

1. Haah et al., "Sample-Optimal Tomography of Quantum States", IEEE TIT 2017. arXiv
2. O'Donnell, Wright: "Efficient quantum tomography", ACM 2016. arXiv
3. Scott: "Tight informationally complete quantum measurements", J. Phys. A 2006 arXiv
4. Kueng et al.: "Low rank matrix recovery from rank one measurements" Appl. Comput. Harmon. Anal 2017 arXiv
5. Flammia et al.: "Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators", NJP 2012 IOP open access
6. Gross et al.: "Quantum state tomography via compressed sensing", PRL 2010 arXiv
7. Guţă et al.: "Fast state tomography with optimal error bounds", J. Phys. A 2020 IOP open access
8. Stilck França, Brandão, and Kueng: "Fast and robust quantum state tomography from few basis measurements" arXiv