We explore the possibility of using ``weak'' measurements to carry out quantum state tomography via numerical simulations. Given a fixed number of copies of identically prepared states of a qubit, we perform state tomography using weak as well as projective measurements. Due to the collapse of the state after measurement, we cannot reuse the state after a projective measurement. If the coupling strength between the quantum system and the measurement device is made weaker, the disturbance caused to the state can be lowered. This then allows us to reuse the same member of the ensemble for further measurements and thus extract more information from the system. However, this happens at the cost of getting imprecise information from the first measurement. We implement this scheme for a single qubit and show that under certain circumstances, it can outperform the projective measurement-based tomography scheme. This opens up the possibility of new ways of extracting information from quantum ensembles. We study the efficacy of this scheme for different coupling strengths and different ensemble sizes.
We explore the possibility of using ``weak'' measurements to carry out quantum state tomography via numerical simulations. Given a fixed number of copies of identically prepared states of a qubit, we perform state tomography using weak as well as projective measurements. Due to the collapse of the state after measurement, we cannot reuse the state after a projective measurement. If the coupling strength between the quantum system and the measurement device is made weaker, the disturbance caused to the state can be lowered. This then allows us to reuse the same member of the ensemble for further measurements and thus extract more information from the system. However, this happens at the cost of getting imprecise information from the first measurement. We implement this scheme for a single qubit and show that under certain circumstances, it can outperform the projective measurement-based tomography scheme. This opens up the possibility of new ways of extracting information from quantum ensembles. We study the efficacy of this scheme for different coupling strengths and different ensemble sizes.
We explore the use of weak quantum measurements for single-qubit quantum state tomography processes. Weak measurements are those where the coupling between the qubit and the measurement apparatus is weak; this results in the quantum state being disturbed less than in the case of a projective measurement. We employ a weak measurement tomography protocol developed by Das and Arvind, which they claim offers a new method of extracting information from quantum systems. We test the Das-Arvind scheme for various measurement strengths, and ensemble sizes, and reproduce their results using a sequential stochastic simulation. Lastly, we place these results in the context of current understanding of weak and projective measurements.
We explore the possibility of using "weak measurements" without "weak value" for quantum state estimation. Since for weak measurements the disturbance caused during each measurement is small, we can rescue the state, unlike for the case of projective measurements. We use this property of weak measurements and design schemes for quantum state estimation for qubits and for Gaussian states. We show, via numerical simulations, that under certain circumstances, our method can outperform the estimation by projective measurements both for qubits and for Gaussian states. It turns out that ensemble size plays an important role and the scheme based on recycling works better for small ensembles.
Abstract We present a scheme to estimate Gaussian states of one-dimensional continuous variable systems, based on weak (unsharp) quantum measurements. The estimation of a Gaussian state requires us to find position ( q ), momentum ( p ) and their second order moments. We measure q weakly and follow it up with a projective measurement of p on half of the ensemble, and on the other half we measure p weakly followed by a projective measurement of q . In each case we use the state twice before discarding it. We compare our results with projective measurements and demonstrate that under certain conditions such weak measurement-based estimation schemes, where recycling of the states is possible, can outperform projective measurement-based state estimation schemes. We establish beyond statistical fluctuations that our method works better for small ensemble sizes.
PACS. 03.65.Ta – Foundations of quantum mechanics; measurement theory. PACS. 03.65.Wj – State reconstruction, quantum tomography. Abstract. – We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, XXZ-, or XY-type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data. Quantum information processing requires the effective measurement of quantum states. However, a single quantum measurement can only obtain partial information of a quantum state. The reconstruction of a quantum state requires measuring a complete set of observables on an ensemble of identically prepared copies of the system. This method called quantum state tomography [1], is very important because any unknown state can be ascertained by tomographic measurements. Moreover, the full description of qubit states can increase the accuracy of quantum operations. Tomographic measurements have been experimentally implemented for, e.g., the nuclear spin state of an NMR system [2], the electromagnetic field and photon
Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigen state will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. In this work, I revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. I describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigen states. In this effective theory framework for the ensemble of quantum trajectories, the measurement interaction is specific to each system-apparatus pair—a context necessary for understanding weak measurements. Also in this framework, the constraint to reproduce projective measurement as per the Born rule in the appropriate limit, requires that the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation dissipation relation. This relation implies that both the noise and the attraction have a common origin in the underlying measurement interaction between the system and the apparatus. I analyse the quantum trajectory ensemble for the scenarios of quantum diffusion and binary quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter.
I test the trajectory ensemble distribution predicted by the quantum diffusion model against the experimental data for weak measurement of superconducting transmon qubits. There is a good fit between theory and experiment for different initial states and several weak measurement couplings. This test vindicates the continuous stochastic measurement framework for quantum state collapse, where the rate of collapse is a characteristic parameter for each system-apparatus pair and is not a universal constant. Furthermore, it implies that the environment can influence the measurement outcomes only via the apparatus and not directly. These are important clues in construction of a complete theory of quantum measurement.
The framework of weak measurements can also be used to construct quantum error correction protocols that protect a quantum state from external disturbances. Unlike projective measurements, one can extract only partial information about the error syndrome
from the encoded state using weak measurements. I construct a feedback protocol that probabilistically corrects the error based on the extracted information. Using numerical simulations of one-qubit error correction codes, I show that the error correction succeeds for a range of the weak measurement strength, where (a) the error rate is below the threshold beyond which multiple errors dominate, and (b) the…
Conventionally, unknown quantum states are characterized using quantum-state tomography based on strong or weak measurements carried out on an ensemble of identically prepared systems. By contrast, the use of protective measurements offers the possibility of determining quantum states from a series of weak, long measurements performed on a single system. Because the fidelity of a protectively measured quantum state is determined by the amount of state disturbance incurred during each protective measurement, it is crucial that the initial quantum state of the system is disturbed as little as possible. Here we show how to systematically minimize the state disturbance in the course of a protective measurement, thus enabling the maximization of the fidelity of the quantum-state measurement. Our approach is based on a careful tuning of the time dependence of the measurement interaction and is shown to be dramatically more effective in reducing the state disturbance than the previously considered strategy of weakening the measurement strength and increasing the measurement time. We describe a method for designing the measurement interaction such that the state disturbance exhibits polynomial decay to arbitrary order in the inverse measurement time 1/T. We also show how one can achieve even faster, subexponential decay, and we find that it represents the smallest possible state disturbance in a protective measurement. In this way, our results show how to optimally measure the state of a single quantum system using protective measurements.