The action of qubit channels on projective measurements on a qubit state is used to establish an equivalence between channels and properties of generalized measurements characterized by bias and sharpness parameters. This can be interpreted as shifting the description of measurement dynamics from the Schrodinger to the Heisenberg picture. In particular, unital (non-unital) quantum channels are shown to induce unbiased (biased) measurements. The Markovian channels are found to be equivalent to measurements for which sharpness is a monotonically decreasing function of time. These results are illustrated by considering various noise channels. Further, the effect of bias and sharpness parameters on the energy cost of a measurement and its interplay with the non-Markovian dynamics is also discussed.
The action of qubit channels on projective measurements on a qubit state is used to establish an equivalence between channels and properties of generalized measurements characterized by bias and sharpness parameters. This can be interpreted as shifting the description of measurement dynamics from the Schrodinger to the Heisenberg picture. In particular, unital (non-unital) quantum channels are shown to induce unbiased (biased) measurements. The Markovian channels are found to be equivalent to measurements for which sharpness is a monotonically decreasing function of time. These results are illustrated by considering various noise channels. Further, the effect of bias and sharpness parameters on the energy cost of a measurement and its interplay with the non-Markovian dynamics is also discussed.
The action of qubit channels on projective measurements on a qubit state is used to establish an equivalence between channels and properties of generalized measurements characterized by bias and sharpness parameters. This can be interpreted as shifting the description of measurement dynamics from the Schrodinger to the Heisenberg picture. In particular, unital quantum channels are shown to induce unbiased measurements. The Markovian channels are found to be equivalent to measurements for which sharpness is a monotonically decreasing function of time. These results are illustrated by considering various noise channels. The effect of bias and sharpness parameters on various quantum correlations and on the energy cost of measurements is discussed. Further, we illustrate the quantum circuit implementation of the two element positive operator-valued measures characterized by the bias and sharpness parameters.
Quantum non-Markovianity has been widely studied and connected to the existence of memory effects in the dynamics of open systems. Surprisingly, working in the Schrödinger or in the Heisenberg picture can provide inequivalent description non-Markovianity: a process can appear to be memoryless in one picture, while displaying memory effects in the other. Here, we investigate which kind of memory is relevant for different quantum information tasks. Some of them, such as sending information via a noisy channel, require memory in both pictures in order to exhibit revivals in the task performance. For others, only one type of memory is sufficient. We also provide necessary conditions for non-Markovianity in both pictures by only considering the dynamics in one picture, showing for instance that the previously considered witness of Schrödinger non-Markovianity in terms of the volume of accessible states does indeed witness non-Markovianity in both pictures at the same time.
Obtaining information from a quantum system through a measurement typically disturbs its state. The postmeasurement states for a given measurement, however, are not unique and highly rely on the chosen measurement model, complicating the puzzle of information disturbance. Two distinct questions are then in order. First, what is the minimum disturbance a measurement may induce? Second, when a fixed disturbance occurs, how informative is the possible measurement in the best-case scenario? Here we propose various approaches to tackle these questions and provide explicit solutions for the set of unbiased binary qubit measurements and postmeasurement state spaces that are equivalent to the image of a unital qubit channel. In particular, we show there are different tradeoff relations between the sharpness of this measurement and the average fidelity of the premeasurement and postmeasurement state spaces as well as the sharpness and quantum resources preserved in the postmeasurement states in terms of coherence and discordlike correlation once the measurement is applied locally.
A resource theory of quantum measurement can be addressed in terms of quantum coherence and measurement sharpness, respectively. The former analyzes the off-diagonal structure of POVM elements in a predetermined basis while the latter analyzes the deviation from trivial, state-independent, measurements. We establish a direct connection between the two resource theories by identifying measurement sharpness as the maximal coherence that is achievable under all possible unitary changes of the reference basis. For a broad class of POVMs whose elements share a common eigenbasis, we show that the maximal distance-based coherence of measurement coincides exactly with the corresponding distance-based sharpness monotone. We further extend this equivalence, with element-additive distances, to POVMs whose elements admit a common mutually unbiased basis structure. These results provide a measurement-theoretic analogue of the maximal-coherence \& purity correspondence for quantum states. We also show that the maximal coherence of measurement is faithful with respect to trivial measurements and is monotonic under fuzzifying operations for dichotomic measurements, as well as under mixed-unitary and unitarily covariant preprocessing channels. Finally, we illustrate the operational meaning and limitations of the equivalence through qubit POVMs, single-photon phase sensing, and noisy photon-number resolving detection. In particular, the maximal Fisher information in a Mach-Zehnder interferometer is shown to be determined by the squared maximal coherence of the measurement, while in an imperfect photon-number resolving detector the maximal coherence behaves as a proper sharpness monotone, unlike conventional PVM-based unsharpness measures.
We consider a qubit coupled to another system (its environment) and discuss the relationship between the effects of subjecting the qubit to either a dynamical decoupling sequence of unitary operations or a sequence of projective measurements. We give a formal statement concerning the equivalence of a sequence of coherent operations on a qubit, precisely operations from a minimal set ${{\mathbb{1}}_{Q},{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}}$, and a sequence of projective measurements of observable ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}$. Using it we show that when the qubit is subjected to n such successive projective measurements at certain times, the expectation value of the last measurement can be expressed as a linear combination of expectation values of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}$ observed after subjecting the qubit to dynamical decoupling sequences of $\ensuremath{\pi}$ pulses, with $k\ensuremath{\le}n$ of them applied at subsets of these times. Performing a sequence of measurements on the qubit gives then access to the same properties of the environment and qubit-environment coupling that are affecting the coherence observed in a dynamical decoupling experiment. Analyzing the latter has been widely used to characterize the environmental dynamics (perform so-called noise spectroscopy), so our result shows how the results obtained with dynamical-decoupling-based protocols are related to those that can be obtained just by performing multiple measurements on the qubit. We also discuss in more detail the application of the general result to the case of the qubit undergoing pure dephasing and outline possible extensions to higher-dimensional systems (a qudit or multiple qubits).
Quantum measurements are an essential ingredient in numerous protocols and quantum technologies.As a characteristic trait, quantum measurements are typically employed a vast number of times to allow for higher accuracy, better control or just increased performance of quantum devices.In this thesis, we pursue a resource-theoretic approach to quantum measurements, aiming at the quantification of their energetic cost.While there exists a plethora of literature on this topic already, we develop a general framework that allows us to unify several of these previous results and to improve them systematically.We highlight an issue common to most of these previous works, where a notion of quantum measurement is employed which is not detailed enough to be applicable to several realistic and important applications for quantum measurement.More concretely, previous work typically involves a state transformation concept, similar to thermodynamic considerations, where only a specific initial and final state of the measured system are specified to characterise the process.For real applications however, quantum measurement devices are typically required to perform the measurement for all possible initial states of the measured system.In this thesis we analyse this requirement and obtain as a consequence much stronger energy constraints on quantum measurements than previously known.For particular forms of measurement, namely dephasing channels, projective quantum instruments, and projection valued measures, we even determine the exact energy costs in contrast to the common lower bound statements in previous literature.Remarkably, the exact cost of projective quantum instruments constitutes the strongest energy constraint on quantum measurement known so far.We demonstrate the implications of these results in two important applications, where quantum measurements play a significant role, namely quantum Zeno stabilisation and quantum error correction.The exact energetic requirements we compute for these two protocols show that any restriction on the amount of available energy immediately places fundamental constraints on the performance of these protocols.