Sensing time-varying signals is essential for precision measurement and for characterizing fluctuations in physical systems. In this work, we show that quantum entanglement provides an exponential advantage in the acquisition time required to learn the temporal statistics of a stationary classical signal. The task is to estimate the characteristic function of joint displacements on selected bosonic modes within a fixed-duration window, using continuous measurements of one signal realization. We present an entanglement-assisted scheme based on two-mode squeezed probes and local Bell measurements with protected idlers, and bound its acquisition time accounting for readout noise and correlations between signal windows. For a strictly stationary pulse family, it achieves polynomial acquisition time at inverse-polynomial accuracy with logarithmic squeezing and polynomial photon number. On the other hand, we prove that any local idler-free protocols require acquisition time exponential in the number of selected temporal modes per window. This lower bound holds for both sequentially prepared and arbitrarily temporally entangled probes, allowing non-Gaussian inputs and measurements and causal classical feedback, as long as each output is measured destructively within its time bin. These results establish an exponential quantum learning advantage for a single continuously monitored stationary signal, with the sensing cost measured by total acquisition time.
Quantum resources can dramatically reduce the data required to learn physical systems, with exponential improvements in sample complexity demonstrated in several quantum learning tasks. However, these advantages have typically relied on quantum resources that scale with problem complexity, most notably increasing system dimension or entanglement. This raises a fundamental question: can exponential quantum learning advantages arise within a fixed, unentangled quantum system? In this paper, we address this question by considering the learning of an unknown random-displacement distribution whose complexity is determined not by the dimensionality of the physical system, but by the Fourier resolution of its features. We show that the quantum-limited noise of vacuum probes progressively obscures high-frequency features, leading to an exponential growth in sample complexity. We establish an information-theoretic lower bound for arbitrary classical-state probes and show that squeezing overcomes this classical limit by extending the accessible Fourier bandwidth. Experimentally, we demonstrate an exponential reduction in sample complexity using squeezed vacuum probes for both binary hypothesis testing and characteristic-function reconstruction. Our results show that a single bosonic mode can exhibit exponential quantum learning advantages without entanglement or an increase in system size, identifying accessible Fourier bandwidth as a fundamentally distinct resource for quantum-enhanced learning.
While entanglement is known to enable exponential improvements in the sample complexity of quantum learning, it remains unclear which properties of entangled resources are responsible for such improvements. We address this question through the reduction criterion, a condition obeyed by all bound-entangled states. In
n-qubit Pauli-channel learning, we show that restricting either the input states or the measurement effects to satisfy this criterion rules out an exponential advantage for incoherent adaptive protocols. An exponential lower bound persists for the one-sided coherent adaptive protocols considered here, even when the unrestricted side retains quantum correlations across channel uses. Using conditional min-entropy, we further quantify how the sample-complexity lower bounds weaken as larger violations of the reduction criterion are allowed. Finally, we show that the same obstruction appears in conjugate-state learning: restricted joint measurements cannot reproduce the logarithmic-sample advantage of unrestricted joint measurements on
ρ⊗ρ∗. These results identify violation of the reduction criterion as a necessary condition for an exponential advantage in the learning tasks considered here.
We show that entangled measurements provide an exponential advantage in sample complexity for Pauli channel estimation, which is both a fundamental problem and a practically important subroutine for benchmarking near-term quantum devices. The specific task we consider is to simultaneously learn all the eigenvalues of an
n-qubit Pauli channel to $\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}$ precision. We give an estimation protocol with an
n-qubit ancilla that succeeds with high probability using only $O(n/{\ensuremath{\varepsilon}}^{2})$ copies of the Pauli channel, while proving that any ancilla-free protocol (possibly with adaptive control and channel concatenation) would need at least $\mathrm{\ensuremath{\Omega}}({2}^{n/3})$ rounds of measurement. We further study the advantages provided by a small number of ancillas. For the case that a
k-qubit ancilla ($k\ensuremath{\le}n$) is available, we obtain a sample complexity lower bound of $\mathrm{\ensuremath{\Omega}}({2}^{(n\ensuremath{-}k)/3})$ for any nonconcatenating protocol, and a stronger lower bound of $\mathrm{\ensuremath{\Omega}}(n{2}^{n\ensuremath{-}k})$ for any nonadaptive, nonconcatenating protocol, which is shown to be tight. We also show how to apply the ancilla-assisted estimation protocol to a practical quantum benchmarking task in a noise-resilient and sample-efficient manner, given reasonable noise assumptions. Our results provide a practically interesting example for quantum advantages in learning and also bring insights for quantum benchmarking.
Quantum channels can be characterized by their action on an orthogonal operator basis, where these operators are related to observable properties of the quantum system. For qudit and multimode bosonic systems, this is encoded respectively in the Heisenberg--Weyl transfer matrix estimated from the Choi-state, and in the characteristic-function transfer map estimated from a two-mode squeezed vacuum based Choi-state. We derive sample-complexity bounds for estimating entries of the transfer matrix/map to additive accuracy
ε with success probability
≥1−δ, under different resources: access to the complex-conjugate channel
E∗ and/or parallel access to
c copies. In all settings, the learner uses parallel channel calls with adaptively chosen, ancilla-assisted input states and measurements. Absolute values of transfer-matrix entries can be learned efficiently with simultaneous access to
E and
E∗, with tight scaling
ε−4. Without conjugate access, any
c<d copies are insufficient for efficient learning, requiring sample complexity exponential in the number of (
d-level) qudits
n (for prime
d). Efficiency is recovered at
c=d, with tight scaling
ε−2d. For bosonic systems, exponential sample complexity holds in terms of an effective dimension induced by an energy constraint for all
c=O(1/ε). Although the task is learning a particular state, these bounds carry stronger implications than standard state-learning bounds since the learner controls the inputs and has ancillary assistance. This establishes a hierarchy of channel-learning resources: self-complex-conjugate channels require two-copy ancilla-assisted access for efficient learning, while for every square-free
d, some channels require
d-copy access. As a corollary, we derive tighter lower bounds for state learning with limited multi-copy access.
We study how entangled measurements, reflected states, and input-state classicality determine the sample complexity of learning bosonic continuous-variable quantum states through their characteristic function. We first analyze a simple entanglement-free protocol and show that its sample complexity is exponential in the number of modes n for general inputs. We then prove that this exponential scaling is intrinsic through three complementary lower bounds: one excluding entangled measurements, one for reflection-symmetric inputs, and one excluding reflected states even when entangled measurements on finitely many copies are allowed. Next, we show that the classicality parameter s max quantitatively determines when this barrier can be softened: positive classicality can reduce the entanglement-free sample cost, and for fully classical states heterodyne learning matches the ϵ -scaling of a Bell-measurement (BM) scheme. Finally, combining reflected states with a two-copy entangled measurement yields sample complexity independent of n . These results identify when entanglement-free strategies already suffice and when, within the resource models analyzed here, efficient continuous-variable state learning requires the full BM resource combination of entangled measurements and reflected-state access.