1.4
Quantum Measurement
1.4.1 Properties of Quantum Measurements
One of the most important properties of quantum mechanics is the difference between quantum and classical measurement. A classical measurement can be thought of as record- ing the state of a classical system before the measurement. It generally leaves the state of the system unperturbed and it is possible to measure the state of all observables si- multaneously. In addition, if the exact state of the system is known, it is possible to deterministically predict the outcome of all measurements. Quantum measurements on the other hand follow none of these properties, except in the case where the system is in an eigenstate of the measurement operator.
One of the fundamental principles of quantum measurements is that they disturb the measured system. The measurement outcome tells us the state of the system after measure- ment, but doesn’t tell us what the state was before measurement. Rather than recording the state of the system prior to the measurement, a quantum measurement projects the state onto a new state. In addition, an observable of a quantum state only has a definite value if the quantum state is in an eigenstate of the measurement operator corresponding to that observable. This means that performing a second measurement can change the state so that the new state is no longer in an eigenstate of the first measurement, which means that the result of the first measurement is no longer valid. In other words, it is im- possible to simultaneously observe different observables, unless those observables commute with each other. Another consequence of this property is that even if the state of a system is known before a measurement, it is only possible to probabilistically predict the mea- surement outcome, unless the initial state was an eigenstate of the measurement. These properties play a crucial role in the security of many quantum cryptographic schemes. In particular, they limit the potential performance of an adversary, which can lead to the guaranteed security of quantum information protocols. The interested reader is referred to one of the many excellent comprehensive books on quantum measurement, for example [31].
Mathematically, a quantum measurement is described by a set of operators {Ei} sat-
isfying the completeness relationP
iE
†
iEi =I, whereI is the identity operator. EachEi
corresponds to a possible measurement outcome i. For a measurement performed on an input state ˆρthat gives an outcome i, the state is projected into the new state
ˆ
ρi =
EiρEˆ i†
pi
, pi= tr( ˆρEi†Ei), (1.73)
wherepi is the probability of measuring the outcomei. If we only care about the result of
a measurement, and not the state after the measurment, we can introduce Πi ≡Ei†Ei and
describe the measurement as a positive operator-valued measure (POVM) [123] described by the new set of operators{Πi}. For a continuous variable system where quantum mea-
surements can have a continuous outcome,pi becomes a probability distribution and sums
are replaced by integrals. Here we are particularly interested in Gaussian measurements, which are defined as having a Gaussian probability distribution of outcomes. In fact, if a Gaussian measurement is made onN modes of an N+M mode Gaussian state, the clas- sical measurement outcomes follow a Gaussian distribution and the remaining M modes remain in a (generally different) Gaussian state.
Figure 1.2: The signal is mixed with the local oscillator at a balanced beamsplitter. The pho- tocurrent difference of the outgoing beams is proportional to the quadrature ˆxθwith the phase set
by the local oscillator.
1.4.2 Quantum Optical Measurements
The most important Gaussian measurement in quantum optics is balanced homodyne de- tection [228], which is effectively a measurement of the rotated quadrature ˆxθ = ˆxcosθ−
ˆ
psinθ, where θ = 0 corresponds to the ˆx-quadrature and θ = 3π/2 corresponds to the
p-quadrature. Other values of θ allow any rotated quadrature to be measured. The measurement operators of homodyne detection are projectors onto the required quadra- ture basis, e.g. |xihx|, and the resulting outcome has a probability distribution given by the appropriate marginal distribution of the Wigner function, e.g. P(x) = R
W(x, p)dp. Practically, homodyne detection is realised following the procedure in Fig. 1.2 [1]. The signal state is interfered on a balanced beamsplitter with a coherent laser beam. The laser beam is known as the local oscillator and must be intense enough to give a precise phase reference, and be powerful enough to be treated classically by ignoring the quantum fluc- tuations. After the beamsplitter, the photocurrentsI1 andI2of the outputs are measured
and then subtracted from each other to give the photocurrent differenceI21. It is assumed
that the signal and local oscillator have a fixed phase reference, which is normally a safe assumption since they generally come from the same source, but must be ensured in an experiment. The bosonic operators of the output modes are given by
ˆ a01 = √1 2(ˆa1−αLO), ˆa 0 2 = 1 √ 2(ˆa1+αLO), (1.74) where ˆa1 is the amplitude of the signal and αLO is the complex amplitude of the local
oscillator. The photocurrent differenceI21is proportional to the photon number difference
given by
ˆ
n21= ˆn2−nˆ1=α∗LOaˆ+αLOˆa†. (1.75)
Using the definition of ˆxθ = ˆxcosθ−pˆsinθ from Eq. (1.39) we see that the measured
photocurrent difference I21 is proportional to ˆxθ since it can be shown from the above
equation that
ˆ
n21=
√
2|αLO|xˆθ. (1.76)
A homodyne detector thus measures the quadrature component ˆxθ, where the phase θ is