We consider two related tasks: (a) estimating a parameterisation of a given Gibbs state and expectation values of Lipschitz observables on this state; and (b) learning the expectation values of local observables within a thermal or quantum phase of matter. In both cases, we wish to minimise the number of samples we use to learn these properties to a given precision. For the first task, we develop new techniques to learn parameterisations of classes of systems, including quantum Gibbs states of non-commuting Hamiltonians with exponential decay of correlations and the approximate Markov property. We show it is possible to infer the expectation values of all extensive properties of the state from a number of copies that not only scales polylogarithmically with the system size, but polynomially in the observable's locality -- an exponential improvement. This set of properties includes expected values of quasi-local observables and entropies. For the second task, we develop efficient algorithms for learning observables in a phase of matter of a quantum system. By exploiting the locality of the Hamiltonian, we show that
M local observables can be learned with probability
1−δ to precision
ε with using only
N=O(log(δM)epolylog(ε−1)) samples -- an exponential improvement on the precision over previous bounds. Our results apply to both families of ground states of Hamiltonians displaying local topological quantum order, and thermal phases of matter with exponential decay of correlations. In addition, our sample complexity applies to the worse case setting whereas previous results only applied on average. Furthermore, we develop tools of independent interest, such as robust shadow tomography algorithms, Gibbs approximations to ground states, and generalisations of transportation cost inequalities for Gibbs states.
The detection and characterization of quantum coherence is of fundamental importance both in the foundations of quantum theory as well as for the rapidly developing field of quantum technologies, where coherence has been linked to quantum advantage. Typical approaches for detecting coherence employ {\it coherence witnesses} -- observable quantities whose expectation value can be used to certify the presence of coherence. By design, coherence witnesses are only able to detect coherence for some, but not all, possible states of a quantum system. In this work we construct protocols capable of detecting the presence of coherence in an {\it unknown} pure quantum state
∣ψ⟩. Having access to
m copies of an unknown pure state
∣ψ⟩ we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success
≥2/3 is
Θ(c(∣ψ⟩)−1), where
c(∣ψ⟩) is the geometric measure of coherence of
∣ψ⟩. However, assuming access to the unitary
Uψ which prepares the unknown state
∣ψ⟩, and its inverse
Uψ†, we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number
O(c(∣ψ⟩)−1/2) of samples. Furthermore, by augmenting amplitude amplification with phase estimation we obtain an experimental estimation of upper bounds on the geometric measure of coherence within additive error
ε with a sample complexity that scales as
O(1/ε) as compared to the
O(1/ε2) sample complexity of Monte Carlo estimation methods. The average number of samples needed in our amplitude estimation protocol provides a new operational interpretation for the geometric measure of coherence. Finally, we also derive bounds on the amount of noise our protocols are able to tolerate.
We describe a quantum algorithm to estimate the $\ensuremath{\alpha}$-Renyi entropy of an unknown density matrix $\ensuremath{\rho}\ensuremath{\in}{\mathbb{C}}^{d\ifmmode\times\else\texttimes\fi{}d}$ for $\ensuremath{\alpha}\ensuremath{\ne}1$ by combining the recent technique of quantum singular value transformations with the method of estimating normalized traces in the one clean qubit model. We consider an oracular input model where the input state is prepared via a quantum oracle that outputs a purified version of the state, assumed to be nonsingular. Our method outputs an estimate of the $\ensuremath{\alpha}$-Renyi entropy to additive precision $\ensuremath{\epsilon}$, using an expected total number $O(\frac{1}{{(x\ensuremath{\epsilon})}^{2}})$ of independent applications of a quantum circuit which coherently queries the input unitary $O(\frac{1}{\ensuremath{\delta}}log\frac{d}{\ensuremath{\epsilon}})$ times, in each case measuring a single output qubit. Here $\ensuremath{\delta}$ is a lower cutoff on the smallest eigenvalue of $\ensuremath{\rho}$ and $x=\frac{1}{d}\mathrm{Tr}({\ensuremath{\rho}}^{\ensuremath{\alpha}})$. The expected number of measurements made in this method can be compared to results in the sample complexity model that generally require $\mathrm{\ensuremath{\Theta}}(\frac{{d}^{2}}{{\ensuremath{\epsilon}}^{2}})$ samples. Furthermore, we also show that multiplicative approximations can be obtained by iteratively using additive approximations, with an overhead logarithmic in the dimension
d.
We extend our previous study of squeezed two-component Schrödinger cat states to a phase-referenced single-mode family with an arbitrary number
N of coherent components and general relative phases. Within the established circular-state construction, we derive a closed complex overlap kernel for the ordered action of displacement, phase rotation, and squeezing, retaining exact finite-amplitude normalization. We distinguish finite-displacement orthogonality from local phase-estimation sensitivity quantified by quantum Fisher information (QFI). For equal-phase states, we compare
N=2,…,9 at a common mean photon number and include coherent-state and squeezed-vacuum benchmarks. A bound on the higher photon-number sectors establishes the fixed-energy asymptote
FQ,N=4nˉ(N−nˉ)+o(1). Its leading term agrees with the known QFI of a vacuum--
N-photon superposition. The resulting decrease of the ideal local quantum Cramér--Rao bound does not establish unlimited attainable precision: prior phase information, sample size, phase-reference resources, optical loss, state preparation, and measurement choice remain relevant.