When discriminating between two pure quantum states, there exists a quantitative tradeoff between the information retrieved by the measurement and the disturbance caused on the unknown state. We derive the optimal tradeoff and provide the corresponding quantum measurement. Such an optimal measurement smoothly interpolates between the two limiting cases of maximal information extraction and no measurement at all.
We demonstrate a fundamental principle of disturbance tradeoff for quantum measurements, along the lines of the celebrated uncertainty principle: The disturbances associated with measurements performed on distinct yet identically prepared ensembles of systems in a pure state cannot all be made arbitrarily small. Indeed, we show that the average of the disturbances associated with a set of projective measurements is strictly greater than zero whenever the associated observables do not have a common eigenvector. For such measurements, we show an equivalence between disturbance tradeoff measured …
We present a simple information-disturbance tradeoff relation valid for any general measurement apparatus: The disturbance between input and output states is lower bounded by the information the apparatus provides in distinguishing these two states.
Heisenberg's uncertainty principle is quantified by error-disturbance tradeoff relations, which have been tested experimentally in various scenarios. Here we shall report improved new versions of various error-disturbance tradeoff relations by decomposing the measurement errors into two different components, namely, operator bias and fuzziness. Our improved uncertainty relations reveal the tradeoffs between these two components of errors, and imply various conditionally valid error-tradeoff relations for the unbiased and projective measurements. We also design a quantum circuit to measure the …
We derive the amount of information retrieved by a quantum measurement in estimating an unknown maximally entangled state, along with the pertaining disturbance on the state itself. The optimal tradeoff between information and disturbance is obtained, and a corresponding optimal measurement is provided.
We show the flaws found in the customary fidelity-based definitions of disturbance in quantum measurements and evolutions. We introduce the "entropic disturbance" D and show that it adequately measures the degree of disturbance, intended essentially as an irreversible change in the state of the system. We also find that it complies with an information-disturbance tradeoff, namely the mutual information between the eigenvalues of the initial state and the measurement results is less than or equal to D.
We show that the global balance of information dynamics for general quantum measurements given in [F. Buscemi, M. Hayashi, and M. Horodecki, Phys.Rev.Lett. 100, 210504 (2008)] makes it possible to unify various and generally inequivalent approaches adopted in order to derive information-disturbance tradeoffs in quantum theory. We focus in particular on those tradeoffs, constituting the vast majority of the literature on the subject, where disturbance is defined either in terms of average output fidelity or of entanglement fidelity.
Unambiguous non-orthogonal state discrimination has fundamental importance in quantum information and quantum cryptography. The discrimination is carried out by POVM generalized measurements. For this process, we find a tradeoff between the needed energy resources and the evolution time, and express it in terms of action-like cost inequality. We find the realization that minimizes this action-like cost and show that in this case the cost is determined by the maximal population transfer from the system to the ancilla needed for the POVM implementation.
We show how to quantify the optimal tradeoff between the amount of information retrieved by a quantum measurement in estimating an unknown spin coherent state and the disturbance on the state itself, and how to derive the corresponding minimum-disturbing measurement.
The uncertainty principle is often interpreted by the tradeoff between the error of a measurement and the consequential disturbance to the followed ones, which originated long ago from Heisenberg himself but now falls into reexamination and even heated debate. Here we show that the tradeoff is switched on or off by the quantum uncertainties of two involved non-commuting observables: if one is more certain than the other, there is no tradeoff; otherwise, they do have tradeoff and the Jensen-Shannon divergence gives it a good characterization.
We provide a general framework for quantifying the optimal tradeoff between the information retrieved by a quantum measurement and the disturbance on the quantum state in covariant quantum state estimation.
We consider measurement disturbance tradeoffs in quantum machine learning protocols which seek to learn about quantum data. We study the simplest example of a binary classification task, in the unsupervised regime. Specifically, we investigate how a classification of two qubits, that can each be in one of two unknown states, affects our ability to perform a subsequent classification on three qubits when a third is added. Surprisingly, we find a range of strategies in which a non-trivial first classification does not affect the success rate of the second classification. There is, however, a non…
For sending unknown direction information, antiparallel spins contains more direction information than parallel spins(Gisin and Popescu, 1999, \textit{Phys. Rev. Lett.} 83, 432).In this paper, the optimal information-disturbance tradeoff bound for antiparallel spins is derived. The quantum measurements which attain the optimal tradeoff bound are obtained. This result can be of practical relevance for posing some general limits on Eve's eavesdropping process. Finally, we also present a comparison between the bound for antiparallel spins and the bound for parallel spins.
We address the problem of energy-error tradeoff in the discrimination between two linear passive quantum optical devices with a single use. We provide an analytical derivation of the optimal strategy for beamsplitters and an iterative algorithm converging to the optimum in the general case. We then compare the optimal strategy with a simpler strategy using coherent input states and homodyne detection. It turns out that the former requires much less energy in order to achieve the same performances.
We discuss the disturbance by measurements which unambiguously discriminate between given candidate states. We prove that such an optimal measurement necessarily changes distinguishable states indistinguishable when the inconclusive outcome is obtained. The result was previously shown by Chefles~[Phys. Lett. A 239, 339 (1998)] under restrictions on the class of quantum measurements and on the definition of optimality. Our theorems remove these restrictions and are also applicable to infinitely many candidate states. Combining with our previous results, one can obtain concrete mathematical cond…
When we extract information from a system by performing a quantum measurement, the state of the system is disturbed due to the backaction of the measurement. Numerous studies have been performed to quantitatively formulate tradeoff relations between information and disturbance. We formulate a tradeoff relation between information and disturbance from an estimation-theoretic point of view, and derive an inequality between them. The information is defined as the classical Fisher information obtained by the measurement, and the disturbance is defined as the average loss of the quantum Fisher info…
We address the problem of the information-disturbance trade-off associated to the estimation of a quantum transformation, and show how the extraction of information about the a black box causes a perturbation of the corresponding input-output evolution. In the case of a black box performing a unitary transformation, randomly distributed according to the invariant measure, we give a complete solution of the problem, deriving the optimal trade-off curve and presenting an explicit construction of the optimal quantum network.
One of the characteristic features of quantum mechanics is that every measurement that extracts information about a general quantum system necessarily causes an unavoidable disturbance to the state of this system. A plethora of different approaches has been developed to characterize and optimize this tradeoff. Here, we apply the framework of quantum instruments to investigate the optimal tradeoff and to derive a class of procedures that is optimal with respect to most meaningful measures. We focus our analysis on binary measurements on qubits as commonly used in communication and computation p…
We address the class of positive operator-valued measures (POVMs) for qubit systems that are obtained by coupling the signal qubit with a probe qubit and then performing a projective measurement on the sole probe system. These POVMs, which represent the simplest class of qubit POVMs, depends on 3+3+2=8 free parameters describing the initial preparation of the probe qubit, the Cartan representative of the unitary coupling, and the projective measurement at the output, respectively. We analyze in some details the properties of the POVM matrix elements, and investigate their values for given rang…
Stinespring's dilation theorem is the basic structure theorem for quantum channels: it states that any quantum channel arises from a unitary evolution on a larger system. Here we prove a continuity theorem for Stinespring's dilation: if two quantum channels are close in cb-norm, then it is always possible to find unitary implementations which are close in operator norm, with dimension-independent bounds. This result generalizes Uhlmann's theorem from states to channels and allows to derive a formulation of the information-disturbance tradeoff in terms of quantum channels, as well as a continui…
In general, classical measurement statistics of a quantum measurement is disturbed by performing an additional incompatible quantum measurement beforehand. Using this observation, we introduce a state-independent definition of disturbance by relating it to the distinguishability problem between two classical statistical distributions --- one resulting from a single quantum measurement and the other from a succession of two quantum measurements. Interestingly, we find an error-disturbance trade-off relation for any measurements in two-dimensional Hilbert space and for measurements with mutually…
I consider the tradeoff between the information gained about an initially unknown quantum state, and the disturbance caused to that state by the measurement process. I show that for any distribution of initial states, the information-disturbance frontier is convex, and disturbance is nondecreasing with information gain. I consider the most general model of quantum measurements, and all post-measurement dynamics compatible with a given measurement. For the uniform initial distribution over states, I show that the least-disturbing way of making any measurement is with conditional dynamics satisf…
Information-theoretic definitions for the noise associated with a quantum measurement and the corresponding disturbance to the state of the system have recently been introduced [F. Buscemi et al., Phys. Rev. Lett. 112, 050401 (2014)]. These definitions are invariant under relabelling of measurement outcomes, and lend themselves readily to the formulation of state-independent uncertainty relations both for the joint estimate of observables (noise-noise relations) and the noise-disturbance tradeoff. Here we derive such relations for incompatible qubit observables, which we prove to be tight in t…
We introduce information-theoretic definitions for noise and disturbance in quantum measurements and prove a state-independent noise-disturbance tradeoff relation that these quantities have to satisfy in any conceivable setup. Contrary to previous approaches, the information-theoretic quantities we define are invariant under relabelling of outcomes, and allow for the possibility of using quantum or classical operations to `correct' for the disturbance. We also show how our bound implies strong tradeoff relations for mean square deviations.
We derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems. The relations are formulated in terms of a directly operational quantity, namely the probability of distinguishing the actual operation of a device from its hypothetical ideal, by any possible testing procedure whatsoever. Moreover, they may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information …
We investigate the tradeoff between the quality of an approximate version of a given measurement and the disturbance it induces in the measured quantum system. We prove that if the target measurement is a non-degenerate von Neumann measurement, then the optimal tradeoff can always be achieved within a two-parameter family of quantum devices that is independent of the chosen distance measures. This form of almost universal optimality holds under mild assumptions on the distance measures such as convexity and basis-independence, which are satisfied for all the usual cases that are based on norms…
When an observer wants to identify a quantum state, which is known to be one of a given set of non-orthogonal states, the act of observation causes a disturbance to that state. We investigate the tradeoff between the information gain and that disturbance. This issue has important applications in quantum cryptography. The optimal detection method, for a given tolerated disturbance, is explicitly found in the case of two equiprobable non-orthogonal pure states.
The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the act…
We investigate the optimal tradeoff between information gained about an unknown coherent state and the state disturbance caused by the measurement process. We propose several optical schemes that can enable this task, and we implement one of them, a scheme which relies on only linear optics and homodyne detection. Experimentally we reach near optimal performance, limited only by detection inefficiencies. In addition we show that such a scheme can be used to enhance the transmission fidelity of a class of noisy channels.
Heisenberg's intuition was that there should be a tradeoff between measuring a particle's position with greater precision and disturbing its momentum. Recent formulations of this idea have focused on the question of how well two complementary observables can be jointly measured. Here, we provide an alternative approach based on how enhancing the predictability of one observable necessarily disturbs a complementary one. Our measurement-disturbance relation refers to a clear operational scenario and is expressed by entropic quantities with clear statistical meaning. We show that our relation is …
In quantum physics, measurement error and disturbance were first naively thought to be simply constrained by the Heisenberg uncertainty relation. Later, more rigorous analysis showed that the error and disturbance satisfy more subtle inequalities. Several versions of universally valid error-disturbance relations (EDR) have already been obtained and experimentally verified in the regimes where naive applications of the Heisenberg uncertainty relation failed. However, these EDRs were formulated for discrete measurements. In this paper, we consider continuous measurement processes and obtain new …
In 1927, Heisenberg heuristically disclosed the tradeoff between the error in the measurement and the caused disturbance on another complementary observable. In the quantum theory, most of uncertainty relations are proposed to reveal the amount of unavoidable uncertainty in the measuring process. In this paper, we study the error-disturbance relation from the information viewpoint. We ask how much information, rather than how much uncertainty, can be obtained during the two sequential measurements. To achieve optimal information gain, we argue that the strategy for the "intelligent" prior appa…
Heisenberg's original uncertainty relation is related to measurement effect, which is different from the preparation uncertainty relation. However, it has been shown that Heisenberg's error-disturbance uncertainty relation can be violated in some cases. We experimentally test the error-tradeoff uncertainty relation by using a continuous-variable Einstein-Podolsky-Rosen (EPR) entangled state. Based on the quantum correlation between the two entangled optical beams, the errors on amplitude and phase quadratures of one EPR optical beam coming from joint measurement are estimated respectively, whi…
A quantum string seal encodes the value of a (bit) string as a quantum state in such a way that everyone can extract a non-negligible amount of available information on the string by a suitable measurement. Moreover, such measurement must disturb the quantum state and is likely to be detected by an authorized verifier. In this way, the intactness of the encoded quantum state plays the role of a wax seal in the digital world. Here I analyze the security of quantum string seal by studying the information disturbance tradeoff of a measurement. This information disturbance tradeoff analysis extend…
We study macroscopic observables defined as the total value of a physical quantity over a collection of quantum systems. We show that previous results obtained for infinite ensemble of identically prepared systems lead to incorrect conclusions for finite ensembles. In particular, exact measurement of a macroscopic observable significantly disturbs the state of any finite ensemble. However, we show how this disturbance can be made arbitrarily small when the measurement are of finite accuracy. We demonstrate a tradeoff between state disturbance and measurement coarseness as a function of the siz…
In this paper we give a new way to quantify the folklore notion that quantum measurements bring a disturbance to the system being measured. We consider two observers who initially assign identical mixed-state density operators to a two-state quantum system. The question we address is to what extent one observer can, by measurement, increase the purity of his density operator without affecting the purity of the other observer's. If there were no restrictions on the first observer's measurements, then he could carry this out trivially by measuring the initial density operator's eigenbasis. If, h…
We propose two experimental schemes for quantum state discrimination that achieve the optimal tradeoff between the probability of correct identification and the disturbance on the quantum state.
Heisenberg formulated a noise-disturbance principle stating that there is a tradeoff between noise and disturbance when a measurement of position and a measurement of momentum are performed sequentially, and another principle imposing a limitation on the product of the uncertainties in a joint measurement of position and momentum. We prove that the former, the Heisenberg sequential noise-disturbance principle, holds when the detectors are assumed to be initially uncorrelated from each other, but that it can be violated for some properly correlated initial preparations of the detectors.
We present two new quantum union bounds for sequential projective measurements. These bounds estimate the disturbance accumulation and probability of outcomes when the measurements are performed sequentially. These results are based on a trigonometric representation of quantum states and should have wide application in quantum information theory for information processing tasks such as communication and state discrimination, and perhaps even in the analysis of quantum algorithms.
Information gain and measurement disturbance for quantum agents
University of Toronto（多伦多大学）；National Research Council of Canada（加拿大国家研究委员会）；University of Ottawa（渥太华大学）；IonQ, Inc.（IonQ公司）；Canadian Institute for Advanced Research（加拿大高级研究院）
General Probabilistic Theories provide the most general mathematical framework for the theory of probability in an operationally natural manner, and generalize classical and quantum theories. In this article, we study state-discrimination problems in general probabilistic theories using a Bayesian strategy. After re-formulation of the theories with mathematical rigor, we first prove that an optimal observable to discriminate any (finite) number of states always exists in the most general setting. Next, we revisit our recently proposed geometric approach for the problem and show that, for two-s…
Time-frequency (TF) filtering of analog signals has played a crucial role in the development of radio-frequency communications, and is currently being recognized as an essential capability for communications, both classical and quantum, in the optical frequency domain. How best to design optical time-frequency (TF) filters to pass a targeted temporal mode (TM), and to reject background (noise) photons in the TF detection window? The solution for coherent TF filtering is known, the quantum pulse gate, whereas the conventional, more common method is implemented by a sequence of incoherent spectr…
A continuous key distribution scheme is proposed that relies on a pair of canonically conjugate quantum variables. It allows two remote parties to share a secret Gaussian key by encoding it into one of the two quadrature components of a single-mode electromagnetic field. The resulting quantum cryptographic information vs disturbance tradeoff is investigated for an individual attack based on the optimal continuous cloning machine. It is shown that the information gained by the eavesdropper then simply equals the information lost by the receiver.
By exploiting a generalization of recent results on environment-assisted channel correction, we show that, whenever a quantum system undergoes a channel realized as an interaction with a probe, the more efficiently the information about the input state can be erased from the probe, the higher is the corresponding entanglement fidelity of the corrected channel, and vice-versa. The present analysis applies also to channels for which perfect quantum erasure is impossible, thus extending the original quantum eraser arrangement, and naturally embodies a general information-disturbance tradeoff.
Quantum measurement is a basic tool to manifest intrinsic quantum effects from fundamental tests to quantum information applications. While a measurement is typically performed to gain information on a quantum state, its role in quantum technology is indeed manifold. For instance, quantum measurement is a crucial process element in measurement-based quantum computation. It is also used to detect and correct errors thereby protecting quantum information in error-correcting frameworks. It is therefore important to fully characterize the roles of quantum measurement encompassing information gain,…
In this paper, we propose a resources-optimal linear-optical scheme for quantum nondemolition detection of single-photon presence. By measuring the state of ancillary photons, the presence of a photon in signal mode is revealed with a success probability of 1/2 without any disturbance to its state. We also show how to tune the setup to perform quantum nondemolition measurement of the signal photon state, and we provide tradeoff between the extracted information and the signal state disturbance. Moreover, the optimality of resources and methods by which to increase the success probability are d…
Time and energy of quantum processes are a tradeoff against each other. We propose to ascribe to any given quantum process a time-energy cost to quantify how much computation it performs. Here, we analyze the time-energy costs for general quantum measurements, along a similar line as our previous work for quantum channels, and prove exact and lower bound formulae for the costs. We use these formulae to evaluate the efficiencies of actual measurement implementations. We find that one implementation for a Bell measurement is optimal in time-energy. We also analyze the time-energy cost for unambi…
In this work we present a general mathematical framework to deal with Quantum Networks, i.e. networks resulting from the interconnection of elementary quantum circuits. The cornerstone of our approach is a generalization of the Choi isomorphism that allows one to efficiently represent any given Quantum Network in terms of a single positive operator. Our formalism allows one to face and solve many quantum information processing problems that would be hardly manageable otherwise, the most relevant of which are reviewed in this work: quantum process tomography, quantum cloning and learning of tra…
From the perspective of quantum information theory, a system so simple as one restricted to just two nonorthogonal states can be surprisingly rich in physics. In this paper, we explore the extent of this statement through a review of three topics: (1) ``nonlocality without entanglement'' as exhibited in binary quantum communication channels, (2) the tradeoff between information gain and state disturbance for two prescribed states, and (3) the quantitative clonability of those states. Each topic in its own way quantifies the extent to which two states are ``quantum'' with respect to each other,…
We present a security analysis of the recently introduced Quantum Private Query (QPQ) protocol. It is a cheat sensitive quantum protocol to perform a private search on a classical database. It allows a user to retrieve an item from the database without revealing which item was retrieved, and at the same time it ensures data privacy of the database (the information that the user can retrieve in a query is bounded and does not depend on the size of the database). The security analysis is based on information-disturbance tradeoffs which show that whenever the provider tries to obtain information …
