Quantum state ensembles are important in quantum information processing. For example, quantum $t$-designs model highly entangled states in complex systems, while projected ensembles appear in generative quantum machine learning and studies of thermalization. With their sample state accompanied by a classical label, these ensembles contain operational information beyond their average density operators. Yet an ensemble differs from a classical-quantum state because it is invariant under permutations of labels. We formulate binary hypothesis testing between finite quantum ensembles and derive fundamental limits on error probability. Given an observed label pattern, we show that the joint sampled state can be described by power-weighted ensemble moments. This yields the Bayes-optimal measurement and exact finite-sample error, revealing that discrimination is governed by the full moment hierarchy up to the number of samples. In the many-sample limit, we derive Chernoff bounds and obtain exact error exponents for finite uniform pure-state ensembles. We apply these results to optical communication and $t$-designs. For finite uniform pure-state $t$-designs with large $t$, the maximal discrimination exponent scales sharply as $\sim t^{-2}$, while equal-prior fixed-error testing requires $\sim t^2$ samples.
The output state of a 2D geometrically local shallow random quantum circuit does not have long range correlations due to its lightcone structure. But this changes if one measures a subset of the qubits: long-range entanglement can be induced by the measurement process, leading to conditional correlations between distant qubits. In this paper we investigate the structure of conditional dependence in these circuits and its consequences for quantum advantage. For a tripartition $ABC$ of the qubits, we consider the ensemble of post-measurement states on $A$ that is conditioned on a specific measurement outcome on $B$ and ranges over all possible measurement outcomes on $C$. For circuit depths exceeding a constant critical value $d^*$, we conjecture that this ensemble is well approximated by a certain generalization of the Haar ensemble, called the Scrooge ensemble~[Jozsa \textit{et al.}, \href{https://doi.org/10.1103/PhysRevA.49.668}{Phys. Rev. A \textbf{49}, 668 (1994)}]; we also provide supporting numerical and analytical evidence. Our conjecture describes a precise sense in which the state retains its lightcone structure on the remaining unmeasured qubits, but also develops some globally random features arising from the measurement. A consequence is that $n$-qubit shallow random quantum circuits in two dimensions are classically efficiently simulable in the presence of a tiny depolarizing noise rate $\Omega(\log(n)/n)$.
Yinchen Liu, Max McGinley, T. Schuster et al.· 1 citation
Qualification of quantum states judges which of two sets of quantum states an unknown state lies in, where the two sets are labeled by two distinct parameter regions. We formulate this problem as a composite quantum hypothesis test and uncover universal scaling laws for the minimum error probability for $N$ copies. Taking polarization-direction qualification and purity qualification as examples, we show that the $N$-copy permutation symmetry and the geometric symmetries of the parameter regions identify the optimal measurements and the"worst pairwise states". The minimum error probability scales as $N^{-3/2}\exp(-N\xi)$ for disjoint regions and as $(NF)^{-1/2}$ for adjacent regions, where $\xi$ and $F$ are the quantum Chernoff divergence and quantum Fisher information associated with the"worst pairwise states", respectively. With the minimum error probability serving as an order parameter, the transition between the scaling behaviors becomes a second-order phase transition as $N\to\infty$. Our approach determines whether a quantum state belongs to a given set without full state tomography, thereby enabling qualification of large ensembles using finite samples.
Zhaoyu Fei, Yaotian Li, Weicheng Huang et al.· 0 citations
Coherent states bridge the gap between quantum and classical physics, but their overcomplete and nonorthogonal nature makes it difficult to identify the minimal discrete set needed to reconstruct quantum information. Finite spin-coherent tomography and discrete coherent-state operator bases are known, but here we address the more specific rank-resolved problem of preserving the canonical contravariant-symbol representation. We show that the canonical finite coherent-state formula reconstructs every operator in the rank-$S$ sector exactly if and only if the sampling points form a spherical $(2J+S)$-design. We call the associated configurations spin-coherent quantum designs. We further give a fully explicit positive-weight Gauss-Legendre construction that avoids the need for an equal-weight spherical design. Together, these results establish a unified framework for reading out physical observables from a handful of measurement samples, playing for spin systems the role that the so-called von Neumann lattice plays for canonical coherent states. Finally, we derive practical protocols for estimating moments of spin operators from these constructions, with direct applications to polarimetry, magnetometry, and quantum state tomography.
Marcin Rudzinski, A. Goldberg, A. B. Klimov et al.· 0 citations
This work proves convergence of the channel's outputs to a QGP and derive the associated closed-form kernel under a uniform (Lebesgue measure) prior over quantum channels and proposes an empirical Bayes heuristic that replaces the dimensional factor with a learnable scale parameter while retaining the kernel's state-overlap correlation structure.
Jonas Jäger, Yaroslav Khmelnitskiy, Paolo Braccia et al.· 0 citations
The minimum change principle provides an information-theoretic characterization of the Bayes reversal channel in classical probability theory and has recently been proposed as a framework for extending Bayes'rule to quantum information theory. Using quantum relative entropy, we investigate a minimum change principle for the setting of quantum statistical inference. Specifically, we consider a forward process based on a classical-to-quantum preparation channel and a reverse process based on a quantum-to-classical measurement channel. We establish a closed-form characterization of measurements that are optimal for this principle, and this optimal measurement can be found via a dual formulation involving a single unconstrained Hermitian variable. This perspective allows us to recover some notable measurements within the same framework, including pretty good measurements and Fermi-Dirac thermal measurements, and we use it to discover a novel family that we call softmin thermal measurements. We further show that softmin thermal measurements arise as optimal solutions to entropy-regularized semidefinite optimization problems, demonstrating that they play a role for measurements analogous to that of thermal states in statistical mechanics. Finally, we prove an additivity property for the relative-entropy minimum change principle and investigate the performance of Fermi-Dirac thermal measurements for quantum hypothesis testing.
How a quantum state responds to small parameter changes is central to quantum criticality, state distinguishability, and metrological sensitivity. Pauli-string statistics provide a natural many-body representation of quantum states, but it is not evident how much of their local quantum geometry is retained by the corresponding classical probability distribution. We show that, for any smooth family of pure $N$-qubit states, the classical Fisher matrix of the complete labeled Pauli distribution is exactly twice the quantum Fisher information matrix (QFIM). The Pauli spectrum is therefore locally metric-complete, while scalar R\'enyi functionals retain only compressed information about its parameter-dependent motion. The same distribution has a direct physical realization: pairwise Bell measurements between corresponding sites of a state and its complex conjugate sample the Pauli spectrum and attain the full QFIM of the conjugate-pair state with a fixed transversal readout. For generic mixed states, physical Bell statistics separate from Hilbert--Schmidt-normalized squared Pauli coordinates, and we derive an exact positive decomposition of the resulting Bell-information gap. These results identify the labeled Pauli spectrum as a common statistical structure underlying many-body response, nonstabilizerness, and multiparameter quantum geometry.
E. A. Ramirez Trino, M. A. Rajabpour· 0 citations
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