Given an unknown state of a qudit that has already been measured optimally, can one still extract any information about the original unknown state? Clearly, after a maximally informative measurement, the state of the system collapses into a postmeasurement state from which the same observer cannot obtain further information about the original state of the system. However, the system still encodes a significant amount of information about the original preparation for a second observer who is unaware of the actions of the first one. We study how a series of independent observers can obtain, or can scavenge, information about the unknown state of a system (quantified by the fidelity) when they sequentially measure it. We give closed-form expressions for the estimation fidelity when one or several qudits are available to carry information about the single-qudit state, and we study the classical limit when an arbitrarily large number of observers can obtain (nearly) complete information on the system. In addition to the case where all observers perform most informative measurements, we study the scenario where a finite number of observers estimates the state with equal fidelity, regardless of their position in the measurement sequence and the scenario where all observers use identical measurement apparatuses (up to a mutually unknown orientation) chosen so that a particular observer's estimation fidelity is maximized.
Given a finite number of copies of an unknown qubit state that have already been measured optimally, can one still extract any information about the original unknown state? We give a positive answer to this question and quantify the information obtainable by a given observer as a function of the number of copies in the ensemble, and of the number of independent observers that, one after the other, have independently measured the same ensemble of qubits before him. The optimality of the protocol is proven and extensions to other states and encodings are also studied. According to the general lore, the state after a measurement has no information about the state before the measurement. Our results manifestly show that this statement has to be taken with a grain of salt, specially in situations where the quantum states encode confidential information.
Quantum limits to parameter estimation Introduction Many experiments can be thought of as comprising two steps: (i) a preparation procedure in which the system to be measured is isolated and prepared, and the apparatus is initialized; and (ii) a measurement step in which the system is coupled to an apparatus and the measurement result recorded. The preparation procedure can be specified by a set of classical parameters, or settings of a physical device. The measurement results are random classical variables that will be correlated with the preparation procedure. In this chapter we are concerned with the case in which the classical parameters specifying the preparation of the state are imperfectly known. Then, assuming that the physical system is well understood, these correlations allow the unknown parameters to be estimated from the measurement results. As we saw in the last chapter, in quantum mechanics the results of measurements are generally statistical, even when one has complete knowledge of the preparation procedure. A single preparation step and measurement step might not be sufficient to estimate a parameter well. Thus it is common to repeat the two steps of preparation and measurement on a large number of systems, either all at one time or sequentially. Whether measuring one quantum system or many, one is faced with a number of questions. How should one prepare the system state? What sort of measurement should one make on the system?
Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address. We present several schemes for a reconstruction of states of quantum systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states on incomplete observation levels. (2) We show how Bayesian inference can be used for reconstruction of quantum states when only a finite number of identically prepared systems are measured. (3) We describe the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.
The extraction of information from a quantum system unavoidably implies a modification of the measured system itself. In this framework partial measurements can be carried out in order to extract only a portion of the information encoded in a quantum system, at the cost of inducing a limited amount of disturbance. Here we analyze experimentally the dynamics of sequential partial measurements carried out on a quantum system, focusing on the trade-off between the maximal information extractable and the disturbance. In particular we implement two sequential measurements observing that, by exploiting an adaptive strategy, is possible to find an optimal trade-off between the two quantities.
Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown
d-dimensional quantum state
ρ and a known collection of observables
E1,…,EM, the goal is to estimate all expectation values
{Tr(ρEi)}i=1M to additive accuracy
ε with probability at least
1−δ.
An elusive open question from the seminal shadow tomography work of Aaronson is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on
M, as suggested by the best-known lower bounds. In this work, we propose two different quantum protocols for shadow tomography with the best sample complexity
O(ε2log(M)log(M/δ)), which is polylogarithmic in the number of observables and independent of the dimension of the unknown state, thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography from Sinha (STOC 2025) and, more recently, Chen, O'Donnell, Pelecanos, and Wright.
Our approach first reduces the general shadow tomography problem to a finite-ensemble estimation problem via a minimax argument. We then develop an observable-independent protocol that repeatedly applies the pretty-good measurement while updating the prior distribution over the finite ensemble according to the measurement outcomes. A tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables and a cubic-logarithmic upper bound. We also introduce a refined recovery-label measurement for the same finite ensemble, which yields the bound in our main theorem.