Adaptive compressive tomography with no a priori information

TL;DR

This paper introduces an adaptive compressive tomography method that efficiently reconstructs quantum states with minimal measurement configurations, requiring no prior assumptions beyond the system dimension.

Contribution

It presents a novel adaptive tomography scheme inspired by compressed sensing, eliminating the need for prior information about the quantum state.

Findings

Significantly reduces measurement configurations needed for quantum state reconstruction

Successfully implemented and tested experimentally

Achieves accurate reconstruction without prior state assumptions

Abstract

Quantum state tomography is both a crucial component in the field of quantum information and computation, and a formidable task that requires an incogitably large number of measurement configurations as the system dimension grows. We propose and experimentally carry out an intuitive adaptive compressive tomography scheme, inspired by the traditional compressed-sensing protocol in signal recovery, that tremendously reduces the number of configurations needed to uniquely reconstruct any given quantum state without any additional a priori assumption whatsoever (such as rank information, purity, etc) about the state, apart from its dimension.

Equations

$$\rho ={\rho }_r=\sum _{j=0}^{r-1}|{\lambda }_j⟩p_j⟨{\lambda }_j|$$

$$\hat{\rho }={\rho }_r+\sum _{j\ne k=0}^{r-1}|{\lambda }_j⟩c_{jk}⟨{\lambda }_k|+W,$$

Full text

*Introduction.—*The characterization of an unknown (true) quantum state $\rho_{\text{t}}\geq 0$ of Hilbert-space dimension $d$ is a subject of immense study in quantum information Chuang and Nielsen (2000); Paris and Řeháček (2004); Teo (2015). To fully reconstruct an arbitrary $\rho_{\text{t}}$, one may perform a set of measurements that is enough to characterize all $d^{2}-1$ independent parameters that define $\rho_{\text{t}}$. Unfortunately, the number of such measurements generally grows polynomially with $d$, or exponentially with the number of subsystems that determine the quantum-source complexity. This poses a technical limitation on how far conventional quantum tomography can go in practical experiments Häffner et al. (2005); Titchener et al. (2018).

If we know a priori that $\mathrm{rank}\{\rho_{\text{t}}=\rho_{r}\}\leq r$ is extremely small, $r\ll d$, then the concept of compressed sensing (CS), whose foundation was first mathematically laid in the context of imaging Donoho (2006); Candès and Tao (2006); Candès and Recht (2009), facilitates the search for a unique estimator by measuring much fewer configurations Gross et al. (2010); Kalev et al. (2015); Steffens et al. (2017); Riofrío et al. (2017). We say that the corresponding data are informationally complete (IC) for $\rho_{r}$. The state-of-the-art CS measurements to be performed given such a prior information have been constructed in Baldwin et al. (2016).

The standard CS procedure, nevertheless, has two important issues that need to be addressed. Firstly, an a priori knowledge about $r$ is necessary to establish a preliminary order-of-magnitude estimate for the number of configurations needed to fully characterize $\rho_{r}$ of rank no larger than $r$. Accuracy of the final estimator is hence highly dependent on the validity of this a priori guess. Secondly, one has no means of verifying whether the measurement data at hand are truly IC for $\rho_{r}$ in the standard scheme. Typically, accuracy surveys with target states are employed Kalev et al. (2015); Steffens et al. (2017); Riofrío et al. (2017) and the value of such a survey relies on the precision of these target states. Therefore, the decision of a priori rank information and presumed choices of target states are ultimately debatable in the presence of experimental errors, rendering the reliability of any related tomography scheme questionable.

In this Letter, we establish a new adaptive tomography paradigm that completely removes the need for any sort of a priori information about $\rho_{r}$ (except for its dimension $d$). Our proposed adaptive compressive tomography (ACT) also includes an efficient recipe to determine informational completeness of the collected data. No target states are ever required to validate the resulting state estimator. The convex boundary of the quantum state space and the positivity constraint plays the principal role in checking whether the accumulated data are IC and adaptively choosing measurements efficiently to uniquely reconstruct $\rho_{r}$, the two of which completely define the purpose of ACT.

To demonstrate ACT, we perform an experiment with the orbital angular momentum (OAM) of single photons and apply ACT to states of various ranks engineered in these degrees of freedom. Both experimental and simulated results show that ACT requires a smaller number of measurements to reconstruct rank-deficient quantum states as compared to conventional CS tomography with known types of CS measurements.

*The quantum state space and ACT.—*In the absence of statistical fluctuations, we measure a randomly chosen computational basis $\{\left|{0}\right>,\left|{1}\right>,\ldots,\left|{d-1}\right>\}$ on $\rho_{r}$ of Hilbert-space dimension $d$. The corresponding Born probabilities $p_{j}=\left<{j}\vphantom{j}\vphantom{j}\right|{\rho_{r}}\left|{j}\vphantom{j}\vphantom{j}\right>$ $(0\leq j\leq d-1)$ specify only the diagonal elements of $\rho_{r}$, and there is in principle a data convex set $\mathcal{C}=\{\rho|\rho\leftrightarrow p_{j}\,\,\forall j\}$ comprising infinitely many estimators $\widehat{\rho}$ that are consistent with $p_{j}$. Evidently, $\rho_{r}\in\mathcal{C}$, and so the only fundamental objective of ACT is to shrink $\mathcal{C}$ to a single point with only $k_{\textsc{ic}}\ll d+1$ IC measurement bases for $r\ll d$. For noiseless situations, this point must be $\rho_{r}$.

To gain insights into how quantum positivity constraint plays a major role in shrinking $\mathcal{C}$, we argue in Appendix A that if one methodically measures $k_{0}=\lceil(r^{2}-r)/(d-1)\rceil+1$ orthonormal bases, one of which being the eigenbasis $\mathcal{B}_{\rho_{r}}$ of $\rho_{r}$, then $\widehat{\rho}=\rho_{r}$ is the unique positive estimator consistent with all measured probabilities. For $r\ll d$, the regime of our interest, $k_{0}=2$. It is however clear that $k_{\textsc{ic}}>2$ in real-world settings where $\rho_{r}$ is completely unknown (apart from its dimension $d$), so the famous no-go answer to Pauli’s phase-retrieval problem Pauli (1933); Carmeli et al. (2015) still stands. Regardless the positivity constraint can still ensure an efficient compression of the IC tomography procedure solely by data analysis.

The goal of ACT is to uniquely reconstruct any given unknown $\rho_{r}$ through adaptively measuring one orthonormal basis at a time according to collected data, as sketched in Fig. 1. In the $k$th step of the adaptive scheme, ACT performs two main procedures. (I) First, ACT checks whether the probabilities $p_{j^{\prime}k^{\prime}}=\mathrm{tr}\!\left\{\rho_{r}\Pi_{j^{\prime}k^{\prime}}\right\}$ obtained from outcomes $\Pi_{j^{\prime}k^{\prime}}>0$ $\left(\Pi_{j^{\prime}k^{\prime}}\Pi_{j^{\prime\prime}k^{\prime}}=\delta_{j^{\prime},j^{\prime\prime}}\right)$ of all the measured orthonormal bases $\mathcal{B}=\{\mathcal{B}_{1},\mathcal{B}_{2},\ldots,\mathcal{B}_{k}\}=\{{\Pi_{01},\ldots,\Pi_{d-1\,\,1}},\ldots,{\Pi_{0k},\ldots,\Pi_{d-1\,\,k}}\}$ so far are IC. Since the accumulated data define a data convex set $\mathcal{C}_{k}$ of size $s_{k}$ that contains all quantum states $\rho$ consistent with $p_{j^{\prime}k^{\prime}}$, this procedure is tantamount to finding out whether $s_{k}$ is zero or not. If $s_{k=k_{\textsc{ic}}}=0$, then the estimator $\widehat{\rho}_{k=k_{\textsc{ic}}}\geq 0$ consistent with the IC data is unique by definition, and equal to $\rho_{r}$ when statistical fluctuation is absent. (II) If $s_{k}\neq 0$, the accumulated data collected are not IC and ACT shall choose the next basis by analyzing $\mathcal{C}_{k}$. Beginning with $k=1$, a “good” adaptive bases sequence should lead to a quick convergence of $\mathcal{C}\rightarrow\rho_{r}$ as ACT progresses.

*(I) Informational completeness certification (ICC).—*To certify whether all collected data are IC or not in the $k$th adaptive step, it suffices to note that since $\mathcal{C}_{k}$ is convex, maximizing and minimizing the linear function $f_{Z}(\rho)=\mathrm{tr}\!\left\{\rho Z\right\}$ for some operator $Z$ over $\rho\in\mathcal{C}_{k}$ respectively give unique solutions $\rho_{\text{max}}$ and $\rho_{\text{min}}$ to the corresponding maximum $f_{\text{max},k}$ and minimum $f_{\text{min},k}$. Without loss of generality, $Z$ is taken to be a random full-rank state. We may define the quantity $s_{\textsc{cvx},k}=(f_{\text{max},k}-f_{\text{min},k})/(f_{\text{max},1}-f_{\text{min},1})$ that is a size monotone (see Appendix B) for $\mathcal{C}_{k}$ in the sense that $s_{k}<s_{k-1}$ if $s_{\textsc{cvx},k}<s_{\textsc{cvx},k-1}$—it is a witness for the shrinkage of $\mathcal{C}_{k}$. As more linearly independent bases are measured, $s_{\textsc{cvx},k}\geq s_{\textsc{cvx},k+1}$, and $s_{\textsc{cvx},k_{\textsc{ic}}}=0$ implies that $s_{k_{\textsc{ic}}}=0$ and that all data collected are IC for a unique reconstruction of $\rho_{r}$. Therefore, at every adaptive step in ACT, we run:

ICC in the $k$th step

Maximize and minimize $f_{Z}(\rho)=\mathrm{tr}\!\left\{\rho Z\right\}$ for a fixed, randomly-chosen full-rank state $Z\neq 1/d$ to obtain $f_{\text{max},k}$ and $f_{\text{min},k}$ subject to

• •

$\rho\geq 0$ , $\mathrm{tr}\!\left\{\rho\right\}=1$ ,

• •

$\mathrm{tr}\!\left\{\rho\Pi_{j^{\prime}k^{\prime}}\right\}=p^{(\textsc{ml})}_{j^{\prime}k^{\prime}}$ for $0\leq j^{\prime}\leq d-1$ and $1\leq k^{\prime}\leq k$ . 2. 2.

Compute $0\leq s_{\textsc{cvx},k}\leq 1$ and check if it is smaller than some threshold $\epsilon$. 3. 3.

If $s_{\textsc{cvx},k}<\epsilon$, terminate ACT. Continue otherwise.

The aforementioned strategy is, as a matter of fact, a semidefinite program (SDP) Vandenberghe and Boyd (1996) that can be efficiently solved by a variety of numerical methods. We should clarify here that while determining whether a set of measurement bases $\mathcal{B}$ possesses the conventional CS property for the entire class of rank-$r$ states is an NP-hard problem Bandeira et al. (2013), ascertaining whether $\mathcal{B}$ gives a unique estimator for one unknown $\rho_{r}$ with the measurement data is, on the other hand, only as computationally difficult as carrying out the semidefinite program in ICC with a worst-case polynomial complexity Vandenberghe and Boyd (1996).

For experimental data $\sum_{j^{\prime}}\nu_{j^{\prime}k^{\prime}}=1$ ($1\leq k^{\prime}\leq k$) with statistical noise, $\mathcal{C}_{k}$ is defined as the maximum-likelihood (ML) convex set in which all $\rho\in\mathcal{C}_{k}$ satisfy the physical constraints $p^{(\textsc{ml})}_{j^{\prime}k^{\prime}}=\mathrm{tr}\!\left\{\rho\Pi_{j^{\prime}k^{\prime}}\right\}$ imposed by the ML principle for quantum states $\left[p^{(\textsc{ml})}_{j^{\prime}k^{\prime}}\rightarrow p_{j^{\prime}k^{\prime}}\,\text{ for }\,N\rightarrow\infty\right]$ Paris and Řeháček (2004); Teo (2015); Řeháček et al. (2007); Shang et al. (2017). All arguments for noiseless data hold exactly for the ML probabilities, so that the working principle of ICC is perfectly robust against arbitrary noise in the sense that $s_{\textsc{cvx},k_{\textsc{ic}}}=0$ still implies $s_{k_{\textsc{ic}}}=0$ for noisy data owing to the preserved convexity of the newly defined $\mathcal{C}_{k}$. Noise only affects the reconstruction accuracy of the final unique estimator relative to $\rho_{r}$, which is a different subject matter for discussion.

*(II) Adaptive selection of measurement bases.—*The optimal orthonormal basis to pick in the $k$th step and measure in the $(k+1)$th step is the one that minimizes $s_{\textsc{cvx},k+1}$. Since $\rho_{r}$ is unknown, we can treat some a posteriori estimator $\widehat{\rho}_{k}$ from $\mathcal{C}_{k}$ as a guess for $\rho_{r}$ to generate simulated data during the minimization of $s_{\textsc{cvx},k+1}$ over all future basis choices. The complicated dependence of $s_{\textsc{cvx},k+1}$ on the future basis however makes its brute-force optimization computationally exhaustive for large $d$.

For a more tractable approach to adaptively measure good bases, we first note that $\mathcal{C}_{k<k_{\textsc{ic}}}$ essentially contains states with eigenbases that are distinct from $\{\mathcal{B}_{1}\ldots\mathcal{B}_{k}\}$ (see Appendix C). So even if we know nothing about $\rho_{r}$, if it is rank-deficient, then taking $\mathcal{B}_{k+1}$ to be the diagonal basis of a rank-deficient $\widehat{\rho}_{k}\in\mathcal{C}_{k}$ ensures a distinct measurement basis in each step that generates a reasonably fast converging sequence $\mathcal{B}_{k}\rightarrow\mathcal{B}_{\rho_{r}}$ as $k$ increases since $\mathcal{C}_{k}\rightarrow\rho_{r}$ at the same time. There is more than one approach to pick eigenbases of rank-deficient states from $\mathcal{C}_{k}$, and as an example we shall consider the minimization of von Neumann entropy function $S(\rho)=-\mathrm{tr}\!\left\{\rho\log\rho\right\}$. A superfast algorithm suitable for minimizing $S$ over $\mathcal{C}_{k}$ exists Shang et al. (2017); Teo et al. (2011). Incidentally, it was reported in Huang et al. (2016); Tran et al. (2016) that entropy minimization also offers high compressive efficiencies in both sparse-signal and low-rank matrix recovery.

*Complete ACT protocol.—*All aforementioned arguments can accommodate real experimental situations, where the relative frequency data do not typically correspond to physical quantum states for $k>1$. The data convex sets contain states that are now consistent with the corresponding physical ML probabilities derived from data, which are statistically consistent with the true probabilities. The final unique estimator $\widehat{\rho}_{k_{\textsc{ic}}}$ would then incur a statistical bias from $\rho_{r}$ that drops as $N$ increases. For many-body quantum sources, the bases generated by ACT are entangled. In practice, product bases are typically much more practical to implement for such sources. While verifying if a rank-deficient $\widehat{\rho}_{k}\in\mathcal{C}_{k}$ can possess a product eigenbasis is computationally difficult, ACT can still be adjusted to feasibly generate near-optimal product bases (pACT) by defining $\mathcal{B}_{k+1}$ to be the product basis that is nearest to the eigenbasis of $\widehat{\rho}_{k}$ with respect to some given norm using a nonlinear optimization routine. Both ACT and pACT for any experimental setting are summarized:

**ACT/pACT

**Beginning with $k=1$ and a random computational basis $\mathcal{B}_{1}$:

Measure $\mathcal{B}_{k}$ and collect the relative frequency data $\sum^{d-1}_{j^{\prime}=0}\nu_{j^{\prime}k}=1$. 2. 2.

From $\left\{\nu_{0k^{\prime}},\ldots,\nu_{d-1\,\,k^{\prime}}\right\}^{k}_{k^{\prime}=1}$, obtain $kd$ physical ML probabilities. 3. 3.

Perform ICC with the ML probabilities and compute $s_{\textsc{cvx},k}$:

• •

If $s_{\textsc{cvx},k}<\epsilon$, terminate ACT and take $\rho_{\text{max}}\approx\rho_{\text{min}}$ as the estimator and report $s_{\textsc{cvx},k}$.

• •

Else Proceed. 4. 4.

Choose a rank deficient $\widehat{\rho}_{k}\in\mathcal{C}_{k}$ [for instance by minimizing the von Neumann entropy $S(\rho)$ in $\mathcal{C}_{k}$]. 5. 5.

Define $\mathcal{B}_{k+1}$ to be the eigenbasis of $\widehat{\rho}_{k}$ for ACT, or a basis close to it for pACT via some prechosen distance minimization technique. 6. 6.

Set $k=k+1$ and repeat.

*Analysis and experiments.—*We put both ACT and pACT tomography schemes to the experimental test by comparing their results with those from measuring random Pauli (RP) bases considered in Kalev et al. (2015); Steffens et al. (2017); Riofrío et al. (2017), the Baldwin–Goyeneche (BG) bases in Baldwin et al. (2016) that generalizes a known five-bases construction for $r=1$ to an IC set of $k_{\textsc{ic}}=4r+1$ bases for rank-$r$ quantum states, and the set of random orthonormal bases of $k_{\textsc{ic}}\approx\lceil 4r(d-r)/(d-1)\rceil$ studied in Kech and Wolf (2016). This exact scaling shall be used to benchmark the experimental $k_{\textsc{ic}}$s.

To demonstrate all three schemes (see Fig. 2), we experimentally emulate a 4-qubit ($d=16$) quantum system and both entangled and product measurement bases using an OAM-based setup. In particular, we consider the Laguerre-Gauss (LG) modes with azimuthal and radial mode indices $\ell$ and $p=0$, respectively. Hence, OAM states correspond to a sub-space of the LG modes and are characterized by a helical wavefront given by \mathrm{e}^{\mbox{\footnotesize\mathrm{i}\ell\phi}}, where $\ell$ is the azimuthal index that corresponds to the OAM value, and $\phi$ is the azimuthal coordinate. The appropriate phase and intensity patterns are realized using a holographic technique called intensity masking, which is readily achieved by a programmable spatial light modulator (SLM) Bolduc et al. (2013). By doing so, we can prepare any many-body state and measurement basis. The generated photons are detected using the projective technique of intensity-flattening Bouchard et al. (2018), where any arbitrary spatial mode can be measured using an SLM followed by a single mode fiber (SMF).

A heralded single photon source is achieved by pumping a 3 mm $\beta$-barium borate type I nonlinear crystal with a quasi-continuous wave laser at a wavelength of 355 nm, producing photon pairs at 710 nm via spontaneous parametric down-conversion. A coincidence rate of 40 kHz, within a coincidence time window of 5 ns, is measured after filtering the photons to the fundamental Gaussian modes using SMF. Subsequent to the generation and detection of the photonic states, explained above, coincidence measurements are recorded using single photon detectors and a coincidence logic.

All results are summarized in Figs. 3 and 4, and the messages conveyed are succinctly stated here: For noiseless simulated data, in terms of average $k_{\textsc{ic}}$ over uniformly (Hilbert-Schmidt) distributed rank-$r$ true states, ACT is the most efficient, since it guides the measurement basis to the eigenbasis of $\rho_{r}$. The more many-body-suited pACT that adaptively generates product bases requires a larger $k_{\textsc{ic}}$ to yield IC data, but the average performance margin with ACT is narrow for low-$r$ states and is on par with the scaling of entangled Goyeneche-type bases ($k_{\textsc{ic}}=4r+1$) for larger $r$. RP turns out to be least efficient amongst all tested schemes. Even in the presence of real data noise, both ACT and pACT remain the more favorable candidates for tomography on general complex systems.

*Concluding remarks.—*The feasible concept of adaptive compressive tomography developed here provides a powerful method to reconstruct any unknown rank-deficient quantum state with optimally chosen entangled or product orthonormal measurement bases, especially for quantum sources of complex degrees of freedom, which includes many-body systems. More importantly, the adaptive scheme requires no a priori knowledge or assumptions about the state or near-proximity target states because it can self-sufficiently validate whether the measured data are informationally complete or not using semidefinite programming, so that reliable compressive tomography can now be carried out in real experimental situations with noisy data. The superior compressive efficiencies of both entangled and product versions of our adaptive schemes are confirmed experimentally and demonstrated with respect to other established protocols.

We acknowledge financial support from the BK21 Plus Program (21A20131111123) funded by the Ministry of Education (MOE, Korea) and National Research Foundation of Korea (NRF), the NRF grant funded by the Korea government (MSIP) (Grant No. 2010-0018295), the Korea Institute of Science and Technology Institutional Program (Project No. 2E27800-18-P043), the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement No. 766970, Canada Research Chairs (CRC), the Spanish MINECO (Grant No. FIS2015-67963-P), the Grant Agency of the Czech Republic (Grant No. Grant No. 18-04291S), and the IGA Project of the Palacký University (Grant No. IGA PrF 2018-003).

Appendix A Uniqueness property induced by the state-space boundary

Suppose that the eigenbasis $\{\left|{\lambda_{j}}\right>\}$ of a rank-$r$

[TABLE]

is measured and the probabilities $\sum^{r-1}_{j=0}p_{j}=1$ are collected, the estimator $\widehat{\rho}=\rho_{r}$ is one solution that is consistent with the Born probabilities $\left<{\lambda_{j}}\vphantom{\lambda_{j}}\vphantom{\lambda_{j}}\right|{\widehat{\rho}}\left|{\lambda_{j}}\vphantom{\lambda_{j}}\vphantom{\lambda_{j}}\right>=p_{j}$. If $\widehat{\rho}$ is only required to be Hermitian, then the general solution is in fact given by

[TABLE]

where $W$ is a traceless Hermitian operator outside the support of $\rho$ ($W\rho=0=\rho W$). Moreover, $p_{j\geq r}=0=\left<{\lambda_{j\geq r}}\vphantom{\lambda_{j\geq r}}\vphantom{\lambda_{j\geq r}}\right|{\widehat{\rho}}\left|{\lambda_{j\geq r}}\vphantom{\lambda_{j\geq r}}\vphantom{\lambda_{j\geq r}}\right>$ implies that $W$ is represented by a hollow (all diagonal entries equal to zero) Hermitian matrix in the basis $\{\left|{\lambda_{j\geq r}}\right>\}$ with arbitrary off-diagonal entries. The Hermitian solution subspace for $\widehat{\rho}$ has thus a nonzero volume under some metric.

It is now obvious that if one chooses $\left|{\phi}\right>$ to be the eigenket of $W$ that gives a negative eigenvalue, one expects $\left<{\phi}\vphantom{\phi}\vphantom{\phi}\right|{\widehat{\rho}}\left|{\phi}\vphantom{\phi}\vphantom{\phi}\right>=\left<{\phi}\vphantom{\phi}\vphantom{\phi}\right|{W}\left|{\phi}\vphantom{\phi}\vphantom{\phi}\right><0$. This implies that any such nonzero traceless $W$ always results in a nonpositive $\widehat{\rho}$. If the quantum positivity constraint is imposed on the solution for the Born probabilities, we must necessarily have $W=0$.

This leaves the operator $A=\sum^{r-1}_{j\neq k=0}\left|{\lambda_{j}}\right>c_{jk}\left<{\lambda_{k}}\right|$ in the right-hand side of Eq. (2). For pure states $(r=1)$, $A$ is clearly zero, so that $\widehat{\rho}=\rho_{1}=\left|{\lambda_{1}}\right>\left<{\lambda_{1}}\right|$ is the unique positive estimator after measuring $\rho_{1}$ as we expect. For $1<r\leq d$, we note that if $\widehat{\rho}$ is to be consistent with probabilities $\left<{w_{j}}\vphantom{w_{j}}\vphantom{w_{j}}\right|{\widehat{\rho}}\left|{w_{j}}\vphantom{w_{j}}\vphantom{w_{j}}\right>=\left<{w_{j}}\vphantom{w_{j}}\vphantom{w_{j}}\right|{\rho_{r}}\left|{w_{j}}\vphantom{w_{j}}\vphantom{w_{j}}\right>$ obtained from any other orthonormal measurement basis $\{\left|{w_{j}}\right>\}$, then $\left<{w_{j}}\vphantom{w_{j}}\vphantom{w_{j}}\right|{A}\left|{w_{j}}\vphantom{w_{j}}\vphantom{w_{j}}\right>=0$ for all $1\leq j\leq d-1$. As $A$ has exactly $r^{2}-r$ free parameters, it follows that measuring $k_{0}=\lceil(r^{2}-r)/(d-1)\rceil+1$ linearly independent bases results in the unique solution $A=0$ to $k_{0}(d-1)$ (pseudo-)invertible linear equations. When $r^{2}-r\leq d-1$, measuring just one basis other than the eigenbasis will result in a unique $\widehat{\rho}$. If $r=d$, we then obtain the familiar minimal bases number $k_{0}=d+1$.

To summarize, the quantum positivity constraint restricts the possible solution set consistent with probabilities derived from any given rank-$r$ state $\rho_{r}$ to a unique state estimator if the eigenbasis of $\rho_{r}$ and $\lceil(r^{2}-r)/(d-1)\rceil$ other orthonormal bases are measured. The method of ACT strives to achieve a small number $k_{\textsc{ic}}$ of optimal IC bases that is bounded from below by $k_{0}$. Looking at two extremal cases, measuring $\mathcal{B}_{\rho_{r}}$ immediately gives us the unique estimator for any pure state, whereas the minimal number of bases needed to characterize a full-rank state is certainly $k_{0}=d+1$.

Appendix B The size monotone

The size monotone $0\leq s_{\textsc{cvx},k}\leq 1$ for the data convex set $\mathcal{C}_{k}$ is a (non-strict) monotonically increasing function with its size $s_{k}$ ($s_{\textsc{cvx},k}>s_{\textsc{cvx},k+1}\implies s_{k}>s_{k+1}$).

To define a size monotone for any data convex set $\mathcal{C}_{k}$, we first pre-choose a concave function $f(\rho)$ of a unique maximum to characterize $\mathcal{C}_{k}$. In the absence of statistical fluctuation, we have the set inequality chain $\mathcal{C}_{1}\supseteq\mathcal{C}_{2}\supseteq\ldots\supseteq\mathcal{C}_{d+1}$ as linear independent bases are sequentially measured: states in the data convex set are ruled out as more information is gained through measurements. By the concavity of $f(\rho)$ we have $f_{\text{max},1}\geq f_{\text{max},2}\geq\ldots\geq f_{\text{max},d+1}$ and $f_{\text{min},1}\leq f_{\text{min},2}\leq\ldots\leq f_{\text{min},d+1}$. It is clear that if $f_{\text{max},k+1}-f_{\text{min},k+1}<f_{\text{max},k}-f_{\text{min},k}$, then $\mathcal{C}_{k+1}\subset\mathcal{C}_{k}$ or $s_{k+1}<s_{k}$.

It follows immediately that if $s_{\textsc{cvx},k}\equiv(f_{\text{max},k}-f_{\text{min},k})/(f_{\text{max},1}-f_{\text{min},1})$, then $s_{\textsc{cvx},k}$ is a size monotone that decreases with increasing $k$. When $s_{\textsc{cvx},k=k_{\textsc{ic}}}=0$, the convexity of $\mathcal{C}_{k_{\textsc{ic}}}$ implies that $s_{k_{\textsc{ic}}}=0$ as $\mathcal{C}_{k_{\textsc{ic}}}$ must contain only $\rho_{r}$ due to the unique maximum possessed by $f$. Similar arguments hold for a convex $f$. Since $f_{Z}(\rho)$ is a positive linear function it can also be used to formulate the size monotone as it facilitates the class of semidefinite programs known to give a unique maximum as well as minimum in the quantum state space.

It is easy to see that $s_{\textsc{cvx},k_{\textsc{ic}}}=0$ still implies that $s_{k_{\textsc{ic}}}=0$ with noisy data that may come from statistical fluctuation or other systematic errors. For this, we can define $\mathcal{C}_{k}$ to be the set of states that maximizes the likelihood function, of the exemplifying multinomial form $L(n_{j^{\prime}k^{\prime}}|\rho^{\prime})=\prod^{k}_{k^{\prime}=1}\prod^{d-1}_{j^{\prime}=0}{p^{\prime}_{j^{\prime}k^{\prime}}}^{n_{j^{\prime}k^{\prime}}}$ for typical physical problems involving independent single-copy sampling up to a fixed total sample size $\sum^{k}_{k^{\prime}=1}\sum^{d-1}_{j^{\prime}=0}n_{j^{\prime},k^{\prime}}=N$ derived from the observed frequencies $n_{j^{\prime}k^{\prime}}$ labeled by the outcome $j^{\prime}$ and basis $k^{\prime}$ numbers. It is clear that if the data are noiseless, the maximum of $L$ gives precisely $\widehat{\rho}_{\textsc{ml}}=\rho_{r}$, so such a definition is a valid generalization to real experimental scenarios. It is important to note that while the set inequality chain “$\mathcal{C}_{k+1}\subset\mathcal{C}_{k}$” is in general broken, that is $s_{\textsc{cvx},k}$ no longer behaves as a size monotone in $k$, the newly defined $\mathcal{C}_{k}$s are still convex sets because $\log L(n_{j^{\prime}k^{\prime}}|\rho^{\prime})$ is a concave function and hence possesses a convex plateau structure for non-IC data. The convexity of $\mathcal{C}_{k}$ arises more generally from any kind of concave $\log L(n_{j^{\prime}k^{\prime}}|\rho^{\prime})$, which behavior is common in experiments. This is the only crucial property to again conclude that $s_{\textsc{cvx},k_{\textsc{ic}}}=0\implies s_{k_{\textsc{ic}}}=0$. It therefore follows that the ICC protocol introduced in the main article for ACT is perfectly robust to noisy data, in the sense that the set of measurement bases $\mathcal{B}=\{\mathcal{B}_{1},\mathcal{B}_{2},\ldots,\mathcal{B}_{k_{\textsc{ic}}}\}$, along with their collected data, always give a unique state reconstruction for the value of $k_{\textsc{ic}}$ decided by ICC even with noisy data. No premature termination of ACT will occur. On the other hand, the final unique state estimator, of course, will have a lower statistical accuracy because of data noise.

Appendix C The spectral decomposition of states in $\mathcal{C}$

For an integer $k$ and a noiseless scenario, the data convex set $\mathcal{C}_{k}$ contains all quantum states that are consistent with the bases measurements $\mathcal{B}=\{\mathcal{B}_{1}\,\ldots,\mathcal{B}_{k}\}$. Suppose that $\mathcal{C}_{k}\neq\{\widehat{\rho}_{\textsc{ml}}\}$ We would like to check how frequently a randomly chosen state $\rho^{\prime}\in\mathcal{C}_{k}$ possesses an eigenbasis equal to $\mathcal{B}_{j}$ for some $1\leq j\leq k$.

It is easy to see that if $\rho^{\prime}=\sum_{l}\left|{\lambda_{lj}}\right>\lambda_{lj}\left<{\lambda_{lj}}\right|\in\mathcal{C}_{k}$ where $\mathcal{B}_{j}=\{\left|{\lambda_{lj}}\right>\left<{\lambda_{lj}}\right|\}$, then we must have the eigenvalues $\lambda_{lj}=p^{\prime}_{lj}=\left<{\lambda_{lj}}\vphantom{\lambda_{lj}}\vphantom{\lambda_{lj}}\right|{\rho^{\prime}}\left|{\lambda_{lj}}\vphantom{\lambda_{lj}}\vphantom{\lambda_{lj}}\right>$ according to the definition of $\mathcal{C}_{k}$. A trivial example occurs when $k=1$, where $\mathcal{C}_{1}$ contains exactly one diagonal state in the measurement basis. It follows immediately that there can exist at most $k$ states in $\mathcal{C}_{k}$ that possesses eigenbases overlapping with $\mathcal{B}$, which are clearly measure zero compared to the infinitely many states in $\mathcal{C}_{k}$. This implies that $\mathcal{C}_{k}$ for $k<k_{\textsc{ic}}$ contains states with eigenbases that are distinct from $\mathcal{B}$ as the only measurable states. We add that the actual number of states with such eigenbases is generally much lower than $k$ since for $k>1$, every state in $\mathcal{C}_{k}$ must satisfy all probability constraints.