Generally, the measurement process consists in coupling a system to a detector that can give a continuous output. However, it may be interesting to use as a detector a system with a discrete spectrum, especially in view of applications to quantum information. Here, we study 1) a two-level system measuring another two-level system (qubit); 2) a generic system measuring a qubit; 3) a qubit measuring a generic system. The results include the case when a postselection on the measured system is made. We provide the exact solution, and also a controlled expansion in the coupling parameter, giving formulas valid in the weak measurement regime for arbitrary preparation and postselection. The concept of generalized Wigner functions is introduced.
Generally, the measurement process consists in coupling a system to a detector that can give a continuous output. However, it may be interesting to use as a detector a system with a discrete spectrum, especially in view of applications to quantum information. Here, we study 1) a two-level system measuring another two-level system (qubit); 2) a generic system measuring a qubit; 3) a qubit measuring a generic system. The results include the case when a postselection on the measured system is made. We provide the exact solution, and also a controlled expansion in the coupling parameter, giving formulas valid in the weak measurement regime for arbitrary preparation and postselection. The concept of generalized Wigner functions is introduced.
Weak measurement is a novel quantum measurement scheme, which is usually characterized by the weak value formalism. To guarantee the validity of the weak value formalism, the fidelity between the preselection and the postselection should not be too small generally. In this work, we study the weak measurement on a qubit system with exactly or asymptotically orthogonal pre- and postselections. We shall establish a general rigorous framework for the weak measurement beyond the weak value formalism, and obtain the average output of a weak measurement when the pre- and postselections are exactly orthogonal. We shall also study the asymptotic behavior of a weak measurement in the limiting process that the pre- and postselections tend to be orthogonal.
We analyze the operation of a switching-based detector that probes a qubit's observable that does not commute with the qubit's Hamiltonian, leading to a nontrivial interplay between the measurement and free-qubit dynamics. In order to obtain analytical results and develop intuitive understanding of the different possible regimes of operation, we use a theoretical model where the detector is a quantum two-level system that is constantly monitored by a macroscopic system. We analyze how to interpret the outcome of the measurement and how the state of the qubit evolves while it is measured. We find that the answers to the above questions depend on the relation between the different parameters in the problem. In addition to the traditional strong-measurement regime, we identify a number of regimes associated with weak qubit-detector coupling. An incoherent detector whose switching time is measurable with high accuracy can provide high-fidelity information, but the measurement basis is determined only upon switching of the detector. An incoherent detector whose switching time can be known only with low accuracy provides a measurement in the qubit's energy eigenbasis with reduced measurement fidelity. A coherent detector measures the qubit in its energy eigenbasis and, under certain conditions, can provide high-fidelity information.
We calculate the noise spectrum of the output signal of a quantum detector during continuous measurement of a two-level system (qubit). We generalize the previous results obtained for the regime of high voltages (when eV is much larger than the qubit's energy level splitting Δ) to the case of arbitrary voltages and temperatures. When V∼Δ the output spectrum is essentially asymmetric in frequency, i.e., the output signal is no longer classical. In the emission (negative frequency) part of the spectrum the peak due to the qubit's coherent oscillations can be 8 times higher that the background pedestal. For V<Δ and T=0 the coherent peaks do not appear at all.
Full text: Along with the well-known concept of projective measurements, quantum mechanics allows various kinds of generalized measurement operators. Two important examples are: joint measurements on two or more quantum systems that cannot be achieved by local operations (LOCC); and weak measurements that obtain less information about a system than does a projective measurement, but with correspondingly less disturbance. Unlike the result of a strong measurement, the average value of a weak measurement of an observable (its weak value), when followed by projective postselection in a complementary basis, can lie outside the range of eigenvalues. This discrepancy is not observed in analogous classical measurements. Weak values aid the resolution of quantum paradoxes, and can simplify analysis of weakly coupled systems. We use a generalized measurement device to measure the weak value of a photon's polarization in the horizontal/vertical basis (the Stokes operator S1 = |H> weak up to 47, outside the usual range -1 {<=} S1 {<=} 1. Unlike previous observations of weak values, our measurement works by entangling two separate systems, and thus can only be described by quantum theory, not a classical wave theory. Also, we have used a two-qubit joint measurement based on a controlled-NOT gate by which certain twoqubit unentangled states can be more reliably distinguished than by using LOCC. We quantify this using a payoff function, for which the optimal LOCC measurement attains 2/3, and our experimental measurement attains 0.72 {+-} 0.02, close to the global optimum of 3/4. (author)
We investigate the role of continuous measurement and postselection in the dynamics and entanglement of a transmon-cavity-transmon coupled system. In the dispersive regime, characterized by a large detuning between the transmons and the cavity, the two transmons interact via virtual excitation of the cavity, giving rise to an effective transmon-transmon coupling. In addition to this coherent interaction, each transmon undergoes spontaneous emission, which is continuously monitored through independent detection channels. By incorporating realistic detector inefficiencies, we analyze both efficient and imperfect monitoring scenarios and demonstrate that postselection significantly slows down the decay of entanglement compared to the unmonitored case. We formulate the stochastic master equation for the coupled system, derive the corresponding postselected master equation, and investigate the dynamics through the Liouvillian superoperator spectrum. In the interaction frame, we identify the emergence of an exceptional point and characterize the associated broken and unbroken PT-symmetric phases. We show how these phases influence the system dynamics and the corresponding entanglement behavior. Our results provide insight into how continuous measurement and postselection affect entanglement in dissipative quantum systems, with potential applications in quantum information processing.
We now have all the basic tools we need to describe systems of one and two qubits. In this final chapter of Part I we look at some of the consequences of the peculiar properties of qubits when they are used to encode information. Many of these results can be traced to the properties of quantum measurement. Measuring a single qubit A key result in quantum information theory is that it is impossible to accurately characterize a single qubit. In other words, there is no experiment, or sequence of experiments, which allows us to find out the state of a single quantum bit. The reason for this problem is twofold. Firstly, we have to make some sort of decision about the basis we will use for the measurement. For example, when measuring a single qubit the most popular choice is to make a measurement in the computational basis. This is equivalent to asking the qubit whether it is in state |0⧽ or state |1⧽. If the qubit is indeed in one of the basis states the measurement process is simple, and we will get the obvious answer of 0 if it is in |0⧽ and 1 if it is in |1⧽. If, however, the qubit is in a superposition, such as α|0⧽+β|1⧽, then the situation is more difficult. Characterizing the state now means determining the values of the two complex numbers α and β, but the measurement can only return the answer 0 or 1, and for a superposition state one of these two answers will be returned at random, with probabilities |α| 2 and |β| 2 , respectively.