Complexity of quantum tomography from genuine non-Gaussian entanglement

TL;DR

The paper ties the sample complexity of quantum state tomography in bosonic systems to the underlying entanglement structure, defining GE states as those obtainable via Gaussian protocols on separable inputs and showing they are learnable with poly($m$) copies for $m$-mode pure states. It introduces NGE states and two resource measures, NG entropy and GE cost, to quantify genuine non-Gaussian entanglement and links the tomography overhead to the GE cost. A key result is a practical, nonadaptive tomography protocol for pure GE states that uses heterodyne measurements and a Gaussian disentangling step, with precise bounds showing exponential overhead for NGE states and polynomial scaling for GE states. The work also demonstrates that NOON states with $N\ge3$ cannot be generated deterministically by Gaussian protocols, and it provides numerical validation and a framework for extending these ideas to broader CV quantum-information tasks. Overall, the study connects the nature of quantum correlations in bosonic systems to tomography efficiency, with implications for quantum sensing, error correction, and quantum advantage in CV architectures.

Abstract

Quantum state tomography, a fundamental tool for quantum physics, usually requires a number of state copies that scale exponentially with the system size, owing to the intricate quantum correlations between subsystems. We show that, in bosonic systems, the nature of correlations indeed fully determines this scaling. Motivated by the Hong-Ou-Mandel effect and Boson-sampling, we define Gaussian-entanglable (GE) states, produced by generalized interference between separable bosonic modes. GE states greatly extend the Gaussian family, encompassing arbitrary separable states, multi-mode Gottesman-Kitaev-Preskill codes, entangled cat states, and Boson-sampling outputs -- resources for error correction and quantum advantage. Nonetheless, we prove that an m-mode pure GE state is learnable with only poly(m) copies, by providing an explicit protocol involving only heterodyne detection and classical post-processing. For states outside GE, we introduce an operational monotone -- the minimum number of ancillary modes required to render them GE -- and prove that it exactly captures the exponential overhead in tomography. As a by-product, we show that deterministic generation of NOON states with N>=3 photons by two-mode interference is impossible.

Figures (7)

• Figure 1: Quantum correlations and the complexity of tomography. Gaussian-entanglable (GE) state are states generated by performing Gaussian protocols on (possibly non-Gaussian) separable input states. The class of GE states includes the output state of the Hong--Ou--Mandel experiment, i.e. two photons input to a beam-splitter where photon bunching effect emerges, and the output states of the Boson sampling protocol, where single photons together with vacuum states are input into a multi-mode linear interferometer. Other examples include entangled cat states, obtained by applying a beam-splitter to two identical cat states, and multi-mode GKP states, generated by performing Gaussian unitaries on single-mode GKP states. In addition, the class of GE states encompasses both the convex hull of Gaussian states and the entire set of separable states. Here we prove that the NOON state $|N\rangle|0\rangle+|0\rangle|N\rangle$ with $N\ge 3$, superposition of TMSV states $|\zeta_{r,0}\rangle+|\zeta_{r,\pi}\rangle$, and the two-mode arithmetic progression state (see Eq. (\ref{['NGE_state:example1']})) do not belong to the set of GE states. These states are conventionally generated by applying controlled gates followed by post-selection. At last, we show that the sample complexity in learning pure GE states scales as $\sim\textbf{poly}(m)$regarding number of modes $m$; while NGE state generally requires an exponential overhead $\sim \textbf{exp}(m)$.
• Figure 2: Sample-efficient protocol for learning Boson sampling output states. It follows a non‑adaptive procedure that uses only local measurement data: (i) Perform heterodyne measurement on all $M$ state copies to collect phase-space samples $\{\gamma_j\}$. (ii) Iterate over an $\epsilon-$net of passive-separable states $|\psi_{\rm ps}\rangle=\hat{U}_O \otimes_{j=1}^n |\psi_j\rangle$, reconstruct the corresponding local states $\{\hat{\rho}_j\}$ after every passive operation $U_{O'}^\dag$, and evaluate a local fidelity $F_j=\langle\psi_j'|\hat{\rho}_j|\psi_j'\rangle$ for each possible local state $|\psi_j'\rangle$. (iii) The first candidate with $W:=1-\sum_{j=1}^n(1-F_j)\ge 1-\epsilon^2$ is accepted as the reconstructed state; otherwise the protocol aborts.
• Figure 3: Comparison between NG entropy and entanglement entropy. The $x$-axis denotes the photon number, dashed lines indicate the entanglement entropy of the NOON state $|\text{NOON}\rangle$ and a superposition of two TMSV states $|\text{sTMSV}\rangle$, while the solid lines reflect their NG entropy.
• Figure 4: (a) A bi-partite NGE state $\hat{\rho}_{AB}$ can be extended to a tri-partite GE state $\hat{\rho}_{AB}\otimes \hat{\sigma}_C$. (b) Young diagram representation of a multipartite case of six modes. The states of subsystems $A_1A_2$ and $A_3A_4A_5$ are both NGE. By introducing the ancilla $A_1^\prime$ and $A_3^\prime$, the state is GE between $A_1A_1^\prime$, $A_2$, $A_3A_3^\prime$, $A_4$, $A_5$ and $A_6$. In addition, we can permute local systems to have a sorted column length and $\mathcal{R}(\psi)=(h_1,h_2,0,0,0,0)$.
• Figure 5: Performance of Gaussian-disentangling protocol versus direct tomography. Here, we numerically simulate the quantum state tomography process on a two-mode GE state with the true state being a thermal state correlated by a beam-splitter. The error, defined as the trace distance between the true state and the reconstructed state obtained through direct tomography using the standard algorithm lvovsky2009continuous, is shown as a function of the number of copies $M$ used in tomography and is shown by black dots. The black dashed line represents the linear fit to the black dots with a slope $k_1=-0.38$. In contrast, the bottom envelope of achievable errors in the proposed Gaussian disentangling algorithm, after minimising over possible sample number in estimating displacements and the covariance matrix, is given by red dots. Their linear fit has a slope $k_2=-0.32$.

Theorems & Definitions (37)

• Theorem 1: Decomposition of pure GE states
• Theorem 2: Sample complexity of pure GE states
• Definition 3: GE cost vector and GE cost function
• Theorem 4: Sample complexity for nondegenerate case
• Lemma 5: Generalizing Lemma 2 in Ref. mari2014quantum
• Corollary 6: Verification of pure NGE state
• Proposition 7: Sample complexity of large NGE states
• Definition 8: $\epsilon$-covering net
• Corollary 9: Volume of GE state space
• Definition S1: Gaussian protocol