Chapter 2Basic Concepts of the QuantumTheory (II): Projection Postulateand Symmetrization Postulate2.1 Quantum measurement2.1.1 Open system vs. closed systemF tex( )reservoirdissipation & fluctuationunknown forcereadoutquantum probequantum systemFigure 2.1: A theoretical model for quantum measurement process.If a quantum system, used for the readout of an unknown force, couples to its reservoirsstrongly as shown in Fig. 2.1, the dissipation process restores the steady state of thequantum system before the next quantum measurement will be attempted. In such acase, we can say an ensemble of identical quantum systems is prepared from a singlephysical system and the standard probability interpretation applies.On the other hand, if a quantum system is well decoupled from reservoirs, the stateof the quantum system at a time of the next quantum measurement is governed by theresults of previous measurements. We must develop a new theory for how a quantumsystem evolves with the simultaneous actions of an unknown external force and quantummeasurements.12.1.2 Exact measurementAn exact quantum measurement with no measurement error is fully characterized by thethree questions :1. What is a measurement result?It is one of the eigenvalues of a measured observable ˆq defined byˆq|qni = qn|qni . (2.1)2. What is the probability of finding a specific result?It is given by the q-distribution of the state,P (qn) = Tr(|qnihqn|ˆρs) , (2.2)where |qnihqn| is a projection operator. If the system consists of a single system,ˆρs= |ψihψ| (pure state) orXψpψ|ψihψ| (mixed state), the above formula is reducedtop(qn) =|hqn|ψi|2: pure stateXψpψ|hqn|ψi|2: mixed state .(2.3)If the system consists of many sub-systems,ˆHs=ˆH1⊗ˆH2⊗ ···, we need to takethe trace operations over all sub-systems.3. What is the post-measurement state?It is given byˆρs(qn) =1p(qn)|qnihqn|ˆρs|qnihqn| , (2.4)where qnis a measurement result. If the system consists of a single system, theabove formula is reduced to a trivial result, ˆρs(qn) = |qnihqn|.This is the von Neumann’s recipe for an exact measurement [1]. Any self-adjoint operatorwhich is a decomposition of unity, such asR∞−∞|qnihqn|dqn=ˆI in the above example,describes an exact measurement of a single observable. Three important measurementsin quantum optics are summarized in Table. 2.1. None of them satisfy the above criteriaof the von Neuman’s projection postulate. The post-measurement states for those threemeasurements are not given by (2.4) but always a vacuum state. In the case of optical het-erodgne detection, there is a finite measurement error, since two conjugate observables aresimultaneously measured. This example highlights the fact that a practical measurementis far from the ideal quantum measurement describ ed by the von Neuman’s projectionpostulate.2Observable Projection operator Physical detectorphoton number ˆa+ˆa |nihn| photon counterquadrature amplitude12(ˆa + ˆa†) |α1ihα1| homodyne detectoror12i(ˆa − ˆa+) |α2ihα2| phase sensitive amplifiertwo quadrature amplitudes |αihα| heterodyne detector(simultaneous meas.) phase insensitive amplifierˆa = ˆa1+ iˆa2Table 2.1: Three quantum measurements frequently used in quantum optics experiments.2.1.3 Approximate measurementAny indirect measurement shown in Fig. 2.1 has a finite measurement error b ecause aquantum probe carries its own quantum uncertainty in the readout observable. We there-fore need to develop a new formulation for such an approximate measurement.ˆ q step 1Macroscopic Meter(many degrees of freedom)ˆ p ˆ x ˆ y ˆ α kˆ β kstep 2System ProbeΙ= hχˆ q ˆ y ΙΙ= hξkk∑ˆ x ˆ β kˆ x xn= xnxn“pointer-basis”Figure 2.2: A mathematical model for indirect measurements.The first step in the indirect measurement is the unitary evolution of a coupled system-probe governed by the interaction Hamiltonian:ˆHI= ~χˆqˆy . (2.5)The Heisenberg equations of motion for the system observables are given byddtˆx =1i~[ˆx,ˆHI] = −χˆq (measurement) , (2.6)ddtˆp =1i~[ˆp,ˆHI] = −χˆy (back action) . (2.7)To derive (2.6) and (2.7), we use the commutation relations [ˆy, ˆx] = i~ and [ˆq, ˆp] = i~,respectively. In the Schr¨odinger picture, this unitary evolution is expressed byˆρf=ˆU ˆρs⊗ ˆρpˆU+, (2.8)3whereˆU = exp·1i~ˆHIt¸= exp [−iχˆqˆyt] . (2.9)The second step is the (destructive) readout of the probe observable ˆx (von Neumannmeasurement). After switching off the interaction HamiltonianˆHI, the probe-meter cou-plingˆHIIis switched on. We can measure the probe observable ˆx exactly but randomizethe conjugate observable ˆy completely in this second step. After this exact measurement,the probe is posteriori known to have b een in an eigen-state |xnihxn|, where xnis themeasurement result. Since there is one-to-one correspondence between the measurementresult xnof the quantum probe and the inferred value ˜qnof the quantum system, we use| ˜qnipph ˜qn| as a projection operator for this exact measurement for the probe.Probability of obtaining a specific measurement result ˜q is now expressed asp(˜q) = Trp[|˜qipph˜q|ˆρp,red] , (2.10)where ˆρp,red= T rs(ˆρf) is the reduced density operator of the probe, which is the ensembleof probe objects for all possible measurement results for the system. We can rewrite (2.10)asp(˜q) = Trph|˜qipph˜q|Trs³ˆUˆρs⊗ ˆρpˆU+´i(2.11)= TrshTrp(ˆU+|˜qipph˜q|ˆUˆρp) × ˆρsi= TrshˆX(˜q)ˆρsi,whereˆX(˜q) = Trp³ˆU+|˜qipph˜q|ˆUˆρp´is the generalized projection operator.If the probe is prepared in a pure state |ϕip,ˆX(˜q) is simplified as,ˆX(˜q) =phϕ|ˆU+|˜qipph˜q|ˆU|ϕip(2.12)=Z∞−∞|qis¯¯¯ph˜q|ˆU(q)|ϕip¯¯¯2shq|dq ,whereR∞−∞|q|isshq|dq =ˆI is used. ω(˜q, q) = |ph˜q|ˆU(q)|ϕip|2is the conditional probabilitythat ˜q is obtained when the system is in an eigenstate |qis. SinceR∞−∞ω(˜q, q)d˜q = 1, weFigure 2.3: The conditional probability in an approximate measurement.4haveZ∞−∞ˆX(˜q)d˜q =Z∞−∞|qisshq|dq =ˆI . (2.13)ˆX(˜q) is a self-adjoint operator which is the decomposition of unity, and thus describes aphysically realizable but approximate measurement.The post-measurement state is given by the von Neumann recipe:ˆρs(˜q) = Trp·1p(˜q)|˜qipph˜q|ˆρf|˜qipph˜q|¸(2.14)=1p(˜q)ph˜q|ˆU ˆρs⊗ ˆρpˆU+|˜qip.If the probe is in a pure state |ϕip,ˆρs(˜q) =1p(˜q)ˆY (˜q)ˆρsˆY