The recovery of quantum information after subsystem loss is a central challenge in quantum information processing. However, some states remain beyond the reach of any recovery strategies. Here we identify the algebraic origin of irrecoverability, the ghost information---correlations encoded in the global state that leave no trace on any accessible subsystem. We introduce a scalar measure quantifying its magnitude and prove a universal error floor below which no virtual recovery map can operate, irrespective of resource investment. We further uncover a spectral phase transition in the sampling cost: bounded when the underlying linear map exhibits a spectrum gap, and divergent with a universal exponent in the gapless regime. Together with the universal error floor, this dichotomy organizes all multipartite quantum states into four classes. Moreover, it is revealed that conditional mutual information---the standard entropic diagnostic---is fundamentally irrelevant to virtual recoverability. As an implication, we show that the error floor imposes a detection threshold for loss-tolerant quantum metrology.
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 $\rho\otimes\rho^*$. These results identify violation of the reduction criterion as a necessary condition for an exponential advantage in the learning tasks considered here.
Quantum resource testing is a fundamental primitive of quantum information processing, profoundly connected to resource manipulation. Its goal is to discriminate $n$ copies of a given resourceful state $\rho$ from all free (i.e., resourceless) states; key instances for applications are entanglement testing and quantum magic testing. The asymptotic characterisation relies on the recently proven Generalised Quantum Stein's Lemma (GQSL), which establishes the rate of decay of the false negative error probability for a fixed false positive error probability. This result, however, is intrinsically asymptotic and thus can provide no finite-resource guarantees, which makes its practical implications unclear. Here, we establish the first rigorous finite-$n$ bounds on quantum resource testing and hence quantum resource manipulation, thus strengthening the GQSL and providing explicit estimates on the number of copies needed to achieve a prescribed performance. As notable consequences, we obtain (a) the convergence of the regularised R\'enyi relative entropies of a resource, which settles the important open problem from [Fang/Hayashi, arXiv:2508.12901, IEEE ToIT 72:6, 2026]; and (b) the first sample-complexity bound for asymmetric resource testing: for any fixed false positive error probability, a false negative error probability of at most $\delta$ can be achieved with $n=O\left(\frac{\log(1/\delta)}{D^\infty(\rho\|F)}\right)$ copies of $\rho$, in the limit where $\delta \to 0$.
Extracting total correlations from a quantum system usually requires reconstructing its state, whereas many experiments access only a few measurement settings. A possible shortcut is to add the mutual informations obtained from complementary measurements; in dimensions above two, however, this procedure can count the same classical correlation twice. We establish that qubits are protected from such overcounting. For every two-qubit state, the correlations observed in two complementary local bases are bounded by the premeasurement quantum mutual information. The proof traces this protection to binary-entropy curvature on the Bloch ball and combines a qubit information-exclusion tradeoff with data processing under local dephasing. Consequently, two correlation tables give a tomography-free lower bound on total correlation. A score above one bit also certifies a quantitative one-way entanglement-distillation rate; when applied to the Choi state of a qubit channel, the same data lower bound its quantum capacity. The theorem therefore identifies both an operational use of complementarity and the trusted two-dimensional setting in which its correlation accounting is valid.
We introduce a measure of information loss for any quantum process that may be modeled by a prepare-evolve-measure scenario: Alice prepares an ensemble of states that gets sent via a quantum channel to Bob, who then measures the output. As a quantum channel models open system dynamics, our measure of information loss quantifies Bob's inability to retrodict with certainty which state Alice sent through the channel. By minimizing this measure over all possible pure state ensemble decompositions of a fixed state $\rho$, and over all POVMs on the output of a channel $\mathcal{E}$, we arrive at an intrinsic notion of information loss for any state-channel pair $(\rho,\mathcal{E})$. We show that the vanishing of information loss with respect to all states supported on a fixed codespace $\mathcal{H}_{\text{code}}$ is equivalent to a condition we term \emph{universal pristineness}, which ensures that orthogonal pure states in $\mathcal{H}_{\text{code}}$ get sent via the channel $\mathcal{E}$ to possibly mixed states whose supports are orthogonal. Moreover, we prove universal pristineness is equivalent to the Knill-Laflamme conditions in quantum error correction, which are necessary and sufficient for the existence of a perfect recovery channel for all states supported on $\mathcal{H}_{\text{code}}$. As an application, we apply our framework to the Hayden-Preskill model of black hole evaporation, demonstrating that the evaporation channel becomes asymptotically universally pristine, thereby providing a purely channel-theoretic formulation of Page-time information retrieval.
James Fullwood, Wu-zhong Guo, Boyu Yang· 0 citations
We analyze a temporal-mode extension of the two-way LM05 quantum communication primitive, in which a message block is encoded into a fixed-weight binary occupation vector (the support) alongside classical ordering information. We demonstrate that applying an independent-slot recovery model to this correlated source is fundamentally inconsistent. For an ideal erasure channel, we derive the exact maximum-a-posteriori (MAP) recovery probability, showing that source-aware decoding outperforms independent guessing by orders of magnitude. We extend this analysis to a realistic threshold-detector model, quantifying the degradation caused by dark counts and inefficiency. Furthermore, we clarify the quantum-classical information split: in the ideal single-photon limit, preparation averaging renders Eve's state independent of the support, yielding zero Holevo information. We establish conditional security bounds using entropic uncertainty relations and identify photon-number-splitting as the primary practical vulnerability. These results provide a rigorous source-aware recovery benchmark and define the physical conditions under which the quantum layer offers a key-consumption advantage over classical constant-weight coding.
Programmable quantum devices nominally act on a Hilbert space whose dimension grows exponentially with the number of constituents, but the presence of noise makes it unlikely that they remain coherent across all of this immense Hilbert space. Then, what is the effective coherent quantum dimension that should be associated with such imperfect devices? To answer this question we here introduce an operational basis-independent framework which imposes a dimension bottleneck on the programmable transformations. Concretely we ask how strongly the quantum information they process can be compressed. Formalizing this idea we identify three inequivalent notions, termed $d$-compressibility, $d$-simulability and $d$-embeddability, which differ in the causal structure used to impose the bottleneck and form a strict hierarchy. The framework unifies several existing notions: joint measurability and simulability of quantum measurements, and the absolute dimensionality of state ensembles, are recovered as special cases. We illustrate the hierarchy with noisy qubit measurements in complementary bases, and we determine the white-noise thresholds at which the set of all noisy unitary channels in dimension $n$, a noisy universal quantum processor, becomes $d$-compressible, $d$-simulable and $d$-embeddable. The thresholds confirm the expectation -- maintaining coherence across the full Hilbert space becomes increasingly demanding as the nominal dimension $n$ increases.