Optimal lower bounds for quantum state tomography

TL;DR

This work resolves the copy complexity of quantum state tomography for rank-$r$ states, proving a tight lower bound of $n=\\Omega(rd/\\varepsilon^2)$ that matches the known upper bound and thus settles tomography sample complexity in trace distance. The authors analyze projector tomography, use representation-theoretic tools (Schur–Weyl duality) and the Pretty Good Measurement to achieve tight upper bounds, and develop a bootstrapping approach that transfers bounds from projector tomography to general mixed states via random rotations and a subspace reduction. A key technical contribution is a two-step bootstrapping that turns trace-distance guarantees into Bures-distance guarantees, enabling the final lower bound to apply across distance measures. The results have practical implications for efficient tomography and spectral bucketing methods, and establish the PGM as sample-optimal within projector tomography.

Abstract

We show that $n = Ω(rd/\varepsilon^2)$ copies are necessary to learn a rank $r$ mixed state $ρ\in \mathbb{C}^{d \times d}$ up to error $\varepsilon$ in trace distance. This matches the upper bound of $n = O(rd/\varepsilon^2)$ from prior work, and therefore settles the sample complexity of mixed state tomography. We prove this lower bound by studying a special case of full state tomography that we refer to as projector tomography, in which $ρ$ is promised to be of the form $ρ= P/r$, where $P \in \mathbb{C}^{d \times d}$ is a rank $r$ projector. A key technical ingredient in our proof, which may be of independent interest, is a reduction which converts any algorithm for projector tomography which learns to error $\varepsilon$ in trace distance to an algorithm which learns to error $O(\varepsilon)$ in the more stringent Bures distance.

Figures (5)

• Figure 1: An illustration of an example Jordan block decomposition. The two projectors $P$ and $Q$ can be simultaneously block-diagonalized, and each Jordan block is either $1 \times 1$ or $2 \times 2$. Within a given block, if $P$ (resp. $Q$) is nontrivial, then $P$ (resp. $Q$) acts as a projection onto a one-dimensional subspace
• Figure 2: Two partitions of $n = 8$, and their Young diagrams. Left: $\lambda = (4,3,1)$. Right: $\mu = (3,3,2)$.
• Figure 3: Illustration of \ref{['not:additional']}. Left: a partition $\lambda = (4,3,1)$. The box $(2,1)$ is shaded in dark gray, and the remaining boxes of $\mathrm{hook}_\lambda(2,1)$ are shaded light gray. The content of $(2,1)$ is $1-2 = -1$, and its hook length is $4$. Right: a partition $\mu = (5,4,4)$. We have $\lambda \subseteq \mu$, and the boxes of $\mu \setminus \lambda$ are shaded.
• Figure 4: Examples of tableaux of shape $\lambda = (4,3,1)$. Left: an SYT. Right: an SSYT for $d \geq 3$.
• Figure 5: For fixed $\lambda$, the product $\prod_{(i,j) \in \tau \setminus \lambda} \frac{r + \mathrm{cont}(i,j)}{d + \mathrm{cont}(i,j)}$ is maximized by the choice $\tau = \lambda + k \cdot e_1$, subject to the constraints $\lambda \subseteq \tau$ and $|\tau \setminus \lambda| = k$. We illustrate the reasoning here with an example. Take $d=3$. Left: $\lambda = (4,3,1)$, together with additional, shaded boxes, which represent boxes we could add to $\lambda$. The shaded boxes are labeled with their contents. To maximize the content of a new box, we should add it to the first row. Having done so, we obtain $\lambda+e_1$. Right: $\lambda+e_1 = (5,3,1)$, again with possibilities for the next box to-be-added shaded, and labeled by contents. Since content increases to the right, the maximum content of a new box will always be in the first row.

Theorems & Definitions (112)

• Theorem 1.1: Optimal tomography lower bound
• Theorem 1.2: Tight upper and lower bounds for projector tomography
• Definition 2.1: Schatten $p$-norm
• Definition 2.2: Trace distance
• Definition 2.3: Fidelity
• Lemma 2.4: Fuchs-van de Graaf inequalities, NC10
• Definition 2.5: Bures distance
• Lemma 2.6
• Definition 2.7: Affinity
• Lemma 2.8: ANSV08