Iterated Stochastic Measurements

Introduction

Informal and formal similarities between Bayesian inference and quantum mechanics have been noted quite some time ago, see e.g. . Bayesian inference may be seen as a way to update trial probability distributions by taking into account the partial information one has gained on the system under study. Indirect quantum measurement consists in obtaining partial information on a quantum system by letting it interact with another quantum system, called a probe, and performing a direct Von Neumann measurement on this probe. Iterating the process of system-probe interaction and probe measurement increases the information on the system because of system-probe entanglements.

This has been experimentally implemented in electrodynamics in cavities , but also in superconductor circuits . As shown by these experiments, repeating a large number of times (formally, infinitely many times) indirect non-demolition measurements reproduces macroscopic direct measurements with collapse of the system quantum wave function. Each collapse is stochastic and progressive, becoming sharper and sharper as the number of indirect measurements increases.

Controlling quantum systems by repeating measurements is, in some way, as old as quantum mechanics, but it has recently been further developed aiming at quantum state manipulations and quantum information processing . At a theoretical level, the concept of quantum trajectories emerges from the need to describe quantum jumps and randomness inherent to repeated measurements. In parallel, studies of open quantum systems led to the theory of quantum feedback and quantum continual measurements . Belavkin equations are stochastic non-linear generalizations of the Schrödinger equation adapted to quantum systems under continual measurements.

Contact between experiments of the type described in ref. and classical stochastic processes was made in ref., showing in particular that the approach to the collapse is controlled by a relevant relative entropy. The aim of this note is to follow and complement the study of ref., by, in some way, reversing the logic. We start by forgetting quantum mechanics for a while and we study a random process obtained by discretely and randomly updating a system state probability distribution using Bayes’ rules. Iterated stochastic measurements refer to this random recursive updating. We describe why and how this leads to a stochastic measurement principle allowing to measure the initial system state probability distribution but which implements a random collapse of the system state distribution at each individual complete system measurement. The initial system state distribution is nevertheless reconstructed by repeating the complete system measurements. We point out a connection between De Fenetti’s theorem on exchangeable random variables, see e.g. ref., and iterated stochastic measurements. We also show that these discrete measurement devices admit continuous formulations with continual updating. There are two limits: a Brownian diffusive limit in which the random data used to update the system state distribution are coded into Brownian motions, this case was studied in ref., and a Poissonian jumpy limit in which these random data are coded in point processes. The construction of the continuous time process relies on deforming an a priori probability measure on the updating data. The key tool is Girsanov’s theorem. Then we transport these results, in an almost automatic way, to quantum mechanics, and we show that quantum mechanical systems under repeated non-demolition indirect measurements admit a continuous time limit described by Belavkin equations (18,19). This completes results proved in ref. and makes contact with those described in ref..

Iterated indirect stochastic measurements

Let ${\cal S}$ be the system under study and $A$ be a chosen countable set of system states $\alpha\in A$ that we shall call pointer states According to the quantum terminology, but the concept of states is here more general as it simply refers to a complete list of labels characterizing the system behavior.. The model apparatus is going to measure the probability distribution $Q_{0}(\alpha)$, with $\sum_{\alpha}Q_{0}(\alpha)=1$, for the system ${\cal S}$ to be in one of the pointer state.

The model apparatus is made of an infinite series of indirect partial measurements. Let $I$ denote the set of possible results of one partial measurements, which we assume to be finite or countable. For each system complete measurement, the output datum is thus an infinite sequence of data $(i_{1},i_{2},\cdots)$, $i_{k}\in I$, associated to the series of successive partial measurements. The output data are random. The probability distribution $Q_{0}(\alpha)$ is to be reconstructed from the sequences $(i_{1},i_{2},\cdots)$.

To be concrete one may keep in mind that the indirect partial measurements arise from direct measurements on probes which have been coupled to the system. The model apparatus is then made of an infinite set of in-going probes – which, for simplicity, are supposed to be all identical – passing through the system ${\cal S}$ and interacting with it one after the other. Measurements are done on the out-going probes.

Specifications of the model apparatus depend on the chosen set of pointer states. One of its manufacturing characteristics is a collection of probability distributions $p(i|\alpha)$, $\sum_{i}p(i|\alpha)=1$, for the output partial measurement to be $i\in I$ conditioned on the system ${\cal S}$ be in the state ${\alpha}\in A$. For simplicity, we shall assume a non-degeneracy hypothesis which amounts to suppose that all probability distributions $p(\cdot|\alpha)$ are distinct, i.e. for any pair of distinct pointer states $\alpha$ and $\beta$ there exists $i\in I$ such that $p(i|\alpha)\not=p(i|\beta)$.

In the model apparatus, a complete measurement is made of an infinite series of partial measurements such that each output of these partial measurements provides a gain of information on the system. Our first aim is to decipher which informations one is gaining from the $n^{\rm th}$ first partial measurements. This will allow us to spell out the way the model apparatus is working as a measurement device.

$\bullet$ Series of partial measurements and specification of the model apparatus. Suppose that the first partial measurement gives result $i_{1}\in I$. Bayes’ law then tells us that the probability for the system ${\cal S}$ to be in the state $\alpha$ conditioned on the first measurement be $i_{1}$ is $Q_{1}(\alpha|i_{1})=Q_{0}(\alpha)p(i_{1}|\alpha)/\pi_{0}(i_{1})$, with $\pi_{0}(i):=\sum_{\alpha}Q_{0}(\alpha)p(i|\alpha)$, if $Q_{0}(\alpha)$ is the initial probability for the system ${\cal S}$ to be in the state $\alpha$ (this probability is yet unknown but shall be recovered from the series of partial measurements making a complete measurement). Let us now ask ourselves what is the probability to get $i_{2}$ as second output partial measurement? By the law of conditioned probabilities, $\pi_{1}(i_{2}|i_{1})=\sum_{\alpha}p(i_{1},i_{2}|\alpha)\,Q_{0}(\alpha)/\pi_{0}(i_{1})$ with $p(i_{1},i_{2}|\alpha)$ the probability to measure $i_{1}$ and $i_{2}$ on the two first partial measurements conditioned on the system to be in the state $\alpha$. At this point we need to make an assumption: we assume that the output partial measurements are independent and identically distributed (i.i.d.) provided that the system ${\cal S}$ is in one of the pointer state $\alpha\in A$. This translates into the relation

which implies that $\pi_{1}(i_{2}|i_{1})=\sum_{\alpha}p(i_{2}|\alpha)\,Q_{1}(\alpha|i_{1})$. That is the probability $\pi_{1}(i_{2}|i_{1})$ is identical to the probability to get $i_{2}$ as output partial measurement assuming that the system distribution is $Q_{1}(\alpha|i_{1})$.

Hence, as a defining characteristic property of our model apparatus, we assume that the output of the $n^{\rm th}$ partial measurements is independent of those of the $(n-1)$-first outputs provided the system ${\cal S}$ is in one of the pointer state $\alpha\in A$, that is:

This specifies our model apparatus. This specification is clearly attached to the chosen set of pointer states.

Conversely, the pointer states associated to this device are those system states for which the values of the output partial measurements are independent, i.e. conditioned on the system to be in a pointer state, the output variables $i_{1},\,i_{2},\cdots$ are independent and identically distributed. If the system is initially in a pointer state $\alpha$, that is its probability distribution is peaked, $Q_{0}(\cdot)=\delta_{\cdot;\alpha}$, the occurrence frequency $\nu(i)$ of the value $i$ in the output sequence $(i_{1},i_{2},\cdots)$ is $p(i|\alpha)$. As we shall see later, one may then identify the pointer states as the system states for which independent infinite series of partial measurements (i.e. independent complete measurements) provide identical occurrence frequencies $\nu(\cdot)$, and this gives a way to calibrate the device and to determine the conditioned probabilities $p(\cdot|\alpha)$.

If the system is not in a pointer state, its initial distribution $Q_{0}(\alpha)$ – to be determined – is un-peaked. Let $Q_{n}(\alpha|i_{1},\cdots,i_{n})$ be the probability for the system to be in the state $\alpha$ conditioned on the $n$-first output partial measurements be $i_{1},i_{2},\cdots,i_{n}$. From our hypothesis (1), the probability to get $i$ as the $n^{\rm th}$ output conditioned on the ${(n-1)}^{\rm th}$ first outputs be $(i_{1},\cdots,i_{n-1})$ is

By Bayes’ law, the probability for the system to be in the state $\alpha$ conditioned on the $n$-first measurements be $i_{1},i_{2},\cdots,i_{n}$ is then recursively computed by

where $\pi_{n-1}$ is the probability to get $i_{n}$ as the $n^{\rm th}$ output. To simplify notations we denote $Q_{n}(\alpha|i_{1},\cdots,i_{n})$ by $Q_{n}(\alpha)$ and $\pi_{n-1}(i|i_{1},\cdots,i_{n-1})$ by $\pi_{n-1}(i)$. Eq.(3) can be solved explicitely:

with $N_{n}(i)$ the number of times the value $i$ appears in the $n^{\rm th}$ first outputs.

Let us point out an interesting reformulation of the above conditions on the outputs of the model apparatus. A sequence $(i_{1},i_{2},\cdots)$ of random variables is called exchangeable if the distribution of $(i_{1},i_{2},\cdots,i_{n})$ is the same as the distribution of $(i_{\sigma_{1}},i_{\sigma_{2}},\cdots,i_{\sigma_{n}})$ for each $n$ and each permutation $\sigma$ of $[1,2,\cdots,n]$. A remarkable theorem due to De Finetti (see e.g. ref. or the last two items of ref.) asserts that an infinite sequence $(i_{1},i_{2},\cdots)$ of random variables is exchangeable if and only if there is a random variable $A$ such that, conditionally on $A$, $(i_{1},i_{2},\cdots)$ is a sequence of independent identically distributed random variables. In our construction, the values taken by $A$ are nothing but the pointer states and the measure on $A$ is $Q_{0}$. So the hypotheses on the model apparatus can be rephrased as the fact that the order of partial measurements is immaterial.

Let us then quote properties of the random probability distribution $Q_{n}(\cdot)$, which will be keys for specifying the model measurement device:

with target pointer state $\gamma_{\omega}$ depending on the event $\omega$. The probability for the target to be a given pointer state $\alpha$ is the initial probability distribution:

(iii) The asymptotic occurrence frequencies $\nu(i):=\lim_{n}N_{n}(i)/n$, with $N_{n}(i)$ the number of times the value $i$ appears in the $n^{\rm th}$ first outputs, are those of the target pointer state. That is:

(iv) The convergence is exponentially fast:

for $n$ large enough, with $S(\gamma_{\omega}|\alpha)$ the relative entropy of $p(\cdot|\gamma_{\omega})$ relative to $p(\cdot|\alpha)$.

$\bullet$ How to read-off a complete measurement and consequences. Let us summarize how the model apparatus is (concretely) working and how data are analysed, see Fig.1. For a given system measurement, the data is an infinite sequence $\omega=(i_{1},i_{2},\cdots)$ of output partial measurements. From its asymptotic behaviour, the apparatus computes the asymptotic frequencies $\nu(i)$ of occurrences of the values $i$ in the sequence $\omega$, and it compares it to one of the apparatus data-base distributions $p(i|\alpha)$. By the non-degeneracy hypothesis and the above convergence theorem , each of the asymptotic frequencies coincide with one of the data-base distributions, so that the comparison identifies uniquely the target pointer state and that identified state is by definition the result of a complete system measurement. Since by the above theorem the distribution of the target pointer states is the initial distribution $Q_{0}(\cdot)$, the histogram of repeated independent complete system measurements yields the initial distribution.

Notice that by the end of a complete measurement the system state distribution has collapsed into one of the pointer states. The need for an infinite series of partial measurement reflects the need for a macroscopic apparatus to implement the collapse. If the system measurement is stopped after a finite number of partial measurements the collapse is only partial, i.e. the probability distribution $Q_{n}(\cdot)$ is still smeared around the target pointer state. The target pointer state may nevertheless be identified with high fidelity if the differences between the data-base probability distributions $p(\cdot|\alpha)$ are bigger than the fluctuations of the frequencies $\nu_{n}(\cdot)$ which generically scale like $n^{-1/2}$.

2 Continuous time limit

We now describe continuous time limits of the previous model apparatus in which the partial measurements are done continuously in time. There are different continuous time limits, depending on the behaviour of the data-base conditioned probability distributions $p(\cdot|\alpha)$: a Brownian diffusive limit, a Poissonian jumpy limit, or a mixture of them.

Recall now the recursion relation (3) that we may rewrite as

with $(\Delta Q)_{n}(\alpha):=Q_{n}(\alpha)-Q_{n-1}(\alpha)$ and $(\Delta X)_{n}(i):=X_{n}(i)-X_{n-1}(i)$. We thus have rewritten the recursion relation (3) as a discrete non-linear difference equation for the probability distribution $Q_{n}(\cdot)$ driven by discrete differences of the martingales $X_{n}(i)$. This will be the starting point of the continuous time limits.

The Brownian diffusive limit occurs when the conditioned probability $p(\cdot|\alpha)$ depends on an extra small parameter $\delta$ such that

with all $p_{0}(i)$’s non vanishing and $\alpha$-independent. Since $\sum_{i}p_{0}(i)=1$, the $p_{0}(\cdot)$’s define an $\alpha$-independent probability measure on $I$. Note that $\sum_{i}p_{0}(i)\Gamma(i|\alpha)=0$ for all $\alpha$ since $\sum_{i}p(i|\alpha)=1$ for all $\delta$. By the non-degeneracy assumption, the functions $\Gamma(\cdot|\alpha)$ on $I$ are all different.

The continuous time limit is then obtained by performing the scaling limit $\delta\to 0$, $n\to\infty$ with $t:=n/\delta$ fixed.

So, let us define the scaling diffusive limit of the state distribution and the Doob martingales:

The discrete difference equation (6) naively translates into the non-linear stochastic equation for $Q_{t}(\alpha)$. Recall that in the diffusive limit, $p(i|\alpha)\simeq p_{0}(i)[1+\sqrt{\delta}\,\Gamma(i|\alpha)+\cdots]$ as $\delta$ goes to zero, so that $\pi_{n-1}(i)\simeq p_{0}(i)[1+\sqrt{\delta}\,\langle\Gamma(i)\rangle_{t}+\cdots]$ with $\langle\Gamma(i)\rangle_{t}:=\sum_{\alpha}\Gamma(i|\alpha)\,Q_{t}(\alpha)$. In the continuous time limit, eq.(6) then becomes:

with Itô’s convention. We used $\sum_{i}X_{t}(i)=0$ and $\sum_{i}p_{0}(i)\Gamma(i|\alpha)=0$ to take this limit. Remark that this equation preserves the normalisation condition $\sum_{\alpha}Q_{t}(\alpha)=1$. This equation is that which governs the evolution of the system probability distribution under continuous Bayes’ updating in the diffusive limit. The random fields $X_{t}(i)$ code for the information of the continuous time series of partial measurements. Not all of these fields are independent since $\sum_{i}X_{t}(i)=0$. As we shall see, the main feature of the Brownian diffusive limit is that the $X_{t}(i)$’s are Gaussian processes with zero mean and covariance

Alternatively, the fields $X_{t}(i)$ are zero mean Gaussian martingales with quadratic variation

which is of course compatible with the relation $\sum_{i}X_{t}(i)=0$. Actually the proofs of equation (7) and of the correctness of (8) are a bit tricky, see for details. We shall here present an alternative less rigorous but quicker and simpler argument.

Let us now argue for eq.(8). Recall the Doob decomposition of the counting process $N_{n}(i)=X_{n}(i)+A_{n}(i)$. Its naive scaling limit reads

2.2 Poissonian jumpy limit

The Poissonian limit occurs when the conditioned probabilities $p(i|\alpha)$ vanish as a small parameter $\delta$ vanishes. Not all $p(i|\alpha)$’s may vanish simultaneously as they sum up to $1$. So let us single out one value $i^{*}$ for which $p(i|\alpha)$ goes to $1$ as $\delta\to 0$ for all $\alpha$ and assume that all other $p(i|\alpha)$ vanish in this limit:

By consistency $p(i^{*}|\alpha)=1-\delta(\sum_{i\not=i^{*}}\theta(i|\alpha))+o(\delta)$ and all $\theta(i|\alpha)$ are positive and assumed to be non-vanishing. In the limit $\delta\to 0$, the output of the partial measurements is most frequently $i^{*}$ with sporadic jumps to another value $i$ different from $i^{*}$ We may generalize this by assuming that more than one conditioned probabilities remain finite as $\delta$ goes to zero. In these cases, the continuous time limit will be a mixture between the Brownian and Poissonian limits..

The continuous time limit is obtained by performing the scaling limit $\delta\to 0$, $n\to\infty$ with $t=n/\delta$ fixed.

Similar computations, based on the general formula

The properties of $N_{n}(i^{*})$ and their limits are reconstructed using the sum rule, $\sum_{i}N_{n}(i)=n$. In particular, for small $\delta$, $N_{[t/\delta]}(i^{*})\simeq t/\delta$ up to order $1$ random corrections.

So, let us define the scaling Poisson limits of the state distribution and of the Doob martingales $X_{n}(i)$’s,

Again, the naive scaling limit of the difference equation (6) yields a stochastic equation for the system state distribution. In the Poissonian limit, one has $p(i|\alpha)\simeq\delta\theta(i|\alpha)+\cdots$ for $i\not=i^{*}$ as $\delta\to 0$, so that $\pi_{n-1}(i)\simeq\delta\,\langle\theta(i)\rangle_{t}+\cdots$ with $\langle\theta(i)\rangle_{t}:=\sum_{\alpha}\theta(i|\alpha)Q_{t}(\alpha)$, for $i\not=i^{*}$, whereas both $p(i^{*}|\alpha)$ and $\pi_{n-1}(i^{*})$ approach $1$ as $\delta$ goes to zero. The continuous time limit of eq.(6) is then

where we used $dY_{t}(i^{*})=-\sum_{i\not=i^{*}}dY_{t}(i)$ to deal with the term associated to $i^{*}$ in eq.(6). As we shall show just below, the $Y_{t}(i)$’s are related to the counting processes by

We shall furthermore argue that the processes $dN_{t}(i)$, $i\not=i^{*}$, are point processes with intensities $\langle\theta(i)\rangle_{t}\,dt$. This intensity is sample dependent – a point that we shall explain –, but predictable. Equations (10,11) are those which governs the evolution of the system probability distribution under continuous Bayes’ updating in the Poissonian limit. The random counting processes $N_{t}(i)$ code for the informations on the continuous time series of partial measurements.

Let us now argue for eq.(11). Consider again the Doob decomposition $N_{n}(i)=X_{n}(i)+A_{n}(i)$. Because $\pi_{[t/\delta]}(i)\simeq\delta\,\langle\theta(i)\rangle_{t}$ for small $\delta$, its naive scaling reads

Its infinitesimal version is eq.(11), as announced. Since $p(i^{*}|\alpha)\simeq 1-\delta\sigma(i^{*}|\alpha)$ with $\sigma(i^{*}|\alpha):=\sum_{i\not=i^{*}}\theta(i|\alpha)$, the counting function $N_{n}(i^{*})$ slightly deviates from $n$, and $M_{t}(i^{*})=Y_{t}(i^{*})-\int_{0}^{t}ds\,\langle\sigma(i^{*})\rangle_{s}$ with $\langle\sigma(i^{*})\rangle_{s}:=\sum_{\alpha}\sigma(i^{*}|\alpha)Q_{s}(\alpha)$.

Iterated indirect quantum measurements

Although purely probabilistic – involving classical probability only – the previous description of iterated stochastic measurements finds applications in the quantum world, in particular in the framework of repeated indirect non-demolition measurements . Recall that an indirect quantum measurement consists in letting a quantum system interact with another quantum system, called the probe, and implementing a direct Von Neumann measurement on the probe. One then gains information on the system because the probe and the system have been entangled. Repeating the cycle of entanglement and measurement progressively increases the information on the system as in the model apparatus we described above.

Let $\mathcal{S}$ be the quantum system and $\mathcal{H}_{s}$ be its Hilbert space of states. Pick a basis of states $\{|\alpha\rangle\}$ in $\mathcal{H}_{s}$, which are going to play the role of pointer states. Let $\mathcal{P}$ be the probe and $\mathcal{H}_{p}$ be its Hilbert space. We assume that the probe-system interaction preserves the pointer states: a system initially prepared in one of the pointer state remains in this state after having interacted with the probes. This requires a peculiar form for the unitary operator $U$ of the probe-system interaction:

with $U_{\alpha}$ an unitary operators on $\mathcal{H}_{p}$. Alternatively, $U|\alpha\rangle\otimes|\nu\rangle=|\alpha\rangle\otimes U_{\alpha}|\nu\rangle$ for any $|\nu\rangle\in\mathcal{H}_{p}$, a property coding for the fact that the pointer states $|\alpha\rangle$ are preserved by this interaction.

We imagine sending identical copies of the probe, denoted $\mathcal{P}_{1},\mathcal{P}_{2},\cdots$, one after the other through the system and measuring an observable on each probe after the interaction. We assume that the in-going probes have all been prepared in the same state $|\psi\rangle\in\mathcal{H}_{p}$, and that the observables measured in the out-going channel are all identical with non-degenerate spectrum $I$. Let $\{|i\rangle\}\in\mathcal{H}_{p}$, $i\in I$, be the basis of eigenstates of the measured observable. We denote by $i_{k}$ the output of the measurement on the $k^{\rm th}$ out-going probe. In analogy with previous section, we call the cycle entanglement and measurement on a probe a partial measurement. The results of repetitions of theses cycles of partial measurements are random sequences $(i_{1},i_{2},\cdots)$, $i_{k}\in I$. As before, such infinite series of partial measurements will be called a complete measurement. The unitary operator $U$ codes for the probability of measuring a given value $i$ on the out-going probe. Suppose that the in-going probe has been prepared in the state $|\psi\rangle$ and the system $\mathcal{S}$ in the state $|\alpha\rangle$. After interaction, the system-probe state is $|\alpha\rangle\otimes U_{\alpha}|\psi\rangle$ and the probability to measure the value $i$ of the probe observable is

by the rule of quantum mechanics. So $|\langle i|U_{\alpha}|\psi\rangle|^{2}$ is the probability to measure $i$ in the out-going channel conditioned on the system state be $|\alpha\rangle$. The analogy with the previous section should start to become clear.

Let $\rho$ be the system density matrix. The system state probability distribution is $Q(\alpha)=\langle\alpha|\rho|\alpha\rangle$. The aim of this section is to describe how the system state distribution and the density matrix evolve when the cycles of entanglement and measurement are repeated, and to make explicit contact with previous sections.

Assume that the system is initially prepared in a density matrix state $\rho_{0}$, and let us look at what happens during a cycle of entanglement and interaction. Recall that the probe is assumed to be prepared in the density matrix state $|\psi\rangle\langle\psi|$. After interaction, the joint system-probe density matrix is $U\rho_{0}\otimes|\psi\rangle\langle\psi|U^{{\dagger}}$. The observable, with spectrum $I$, is then measured on the probe. If $i_{1}$ is the output value of this measurement, the joint system-probe state is projected into $\rho_{1}\otimes|i_{1}\rangle\langle i_{1}|$ with

This occurs with probability \pi_{0}(i_{1})={\rm Tr}\big{(}\,\langle i_{1}|U|\psi\rangle\,\rho_{0}\,\langle\psi|U^{{\dagger}}|i_{1}\rangle\,\big{)}. Using the assumed property of $U$, eq.(14), this can rewritten as

How this cycle is to be repeated is clear. Let $\rho_{n-1}$ be the system density matrix after the $n-1$ first partial measurements – this density matrix depends on the random values of these measurements, so that $\rho_{n-1}=\rho_{n-1}(i_{1},\cdots,i_{n-1})$, but we simplify the notation by not writing explicitly the values of the measurements. We let the system interact with the $n^{\rm th}$ probe and do a measurement on this probe. If $i_{n}$ is the output value of this $n^{\rm th}$ partial measurement, the system state is projected into

where again we simplified the notation by not writing the values of the partial measurements – $\rho_{n}$ should have been written as $\rho_{n}(i_{n}|i_{1},\cdots,i_{n-1})$ and similarly for $\pi_{n-1}$. This projection occurs with probability $\pi_{n-1}(i_{n})$, with

The diagonal matrix elements of the density matrix are the probabilities for the system be in a pointer state, that is $Q_{n}(\alpha)=\langle\alpha|\rho_{n}|\alpha\rangle$. From eq.(15) we read that

The two above equations exactly coincide with eqs.(2,3) defining iterated stochastic measurements. So everything we wrote in the previous sections applies. In particular the collapse of the system probability distribution is a discrete implementation of the wave function collapse in Von Neumann measurement. The quantum system observable measured by the iteration of cycles of entanglement and indirect measurement is that with eigenstate basis $\{|\alpha\rangle\}$. The collapse happens only for an infinite sequence of partial measurement reflecting the fact that the iterated stochastic measurement apparatus is macroscopic only if an infinite sequence of partial measurements is implemented, see ref..

2 Continuous time limit

The aim of this section is to take the continuous time limit of the discrete recurrence equation (15) for the quantum density matrix using the results of the previous section. Doing this we will make contact with the so-called Belavkin equations , describing continuous time measurements in quantum mechanics and which are non-linear stochastic Schrödinger equations .

The small parameter $\delta$ is the time duration of the system-probe interaction, so that the unitary operator is $U=\exp(-\imath\delta H)$ with $H$ the system-probe hamiltonian We use the notation $\imath$, without a dot, to code for the square root of $-1$.. As is well known, the dynamics of a quantum system under continuous measurements is frozen by continuous wave packet reductions, a fact named the quantum Zeno effect. To avoid it, we have to rescale the system-probe interaction at the same time we decrease the interaction time duration. So we assume the following form of the hamiltonian $H$,

where $H_{s}$ is the system hamiltonian, $H_{p}$ the probe hamiltonian and $H_{I}$ the interaction hamiltonian.

For the pointer state to be stable under the action of $U=e^{-\imath\delta H}$, eq.(14), we should assume that $H_{s}$ is diagonal in the pointer basis, $H_{s}=\sum_{\alpha}|\alpha\rangle E_{\alpha}\langle\alpha|$ for some energies $E_{\alpha}$, – this is linked to the non-demolition character of the measurement – and that

with $H_{\alpha}$ acting on $\mathcal{H}_{p}$ but $\alpha$ dependent. The conditioned probabilities $p(i|\alpha)$ are then

so that the Brownian diffusive case corresponds $\langle i|\psi\rangle\not=0$ and the Poissonian jumpy case to $\langle i|\psi\rangle=0$.

In both cases, the continuous time limit is then obtained by performing the scaling limit $\delta\to 0$, $n\to\infty$ with $t:=n/\delta$ fixed as above.

The Brownian diffusive limit occurs when $\langle i|\psi\rangle\not=0$ for all $i$. Then $p(i|\alpha)\simeq p_{0}(i)(1+\sqrt{\delta}\,\Gamma(i|\alpha)+\cdots)$ for $\delta$ small with,

This is the situation we encountered in the previous section on the classical diffusive limit, so that we can borrow all results obtained there.

It is then a simple matter to naively take the continuous time limit of the difference equations (17). This limit exists only if $\langle\psi|H_{I}|\psi\rangle=0$, which is equivalent to

a criteria which we assume to hold true. Recall that this scaling limit consists in $\delta\to 0$ with $t=n\delta$ fixed. Let us first expand the term $(D\rho)_{n-1}$ in power of $\sqrt{\delta}$. The term of order $\sqrt{\delta}$ vanishes due to the condition $\langle\psi|H_{I}|\psi\rangle=0$, and for the term of order $\delta$ we get: $(D\rho)_{[t/\delta]}\simeq_{\delta\to 0}L_{d}(\rho_{t})\delta$, with Linbladian

where we defined the operators $C_{i}$’s acting on $\mathcal{H}_{s}$ by $C_{i}:=-\imath\frac{\langle i|H_{I}|\psi\rangle}{\langle i|\psi\rangle}$, or equivalently

using the decomposition of $H_{I}$ on pointer states. Remark that $\sum_{i}p_{0}(i)C_{i}=0$ thanks to the assumed condition $\langle\psi|H_{\alpha}|\psi\rangle=0$. Similarly, expanding the term $(\Delta\rho)_{n}$ using $\pi_{n-1}(i)\simeq p_{0}(i)[1+\delta{\rm Tr}[(C_{j}+C_{j}^{\dagger})\rho_{t}]+\cdots]$, we get $\lim_{\delta\to 0}(\Delta\rho)_{[t/\delta]}=\sum_{j}\mathcal{D}_{j}(\rho_{t})\,dX_{t}(j)$, with

Note that computing these limits only uses the decomposition (16) of the hamiltonian $H$ and not the existence of a preferred pointer state basisThe existence of the pointer state basis was however used in determining the statistical properties of the fields $X_{t}(i)$.. Gathering shows that the Brownian limit of the difference equation (15) is

where the $X_{t}(j)$’s are the Gaussian centred processes, with quadratic variation

defined in eq.(8) and in the discussion around this equation. This is an example of the diffusive Belavkin equation . It is important to recall that $\sum_{i}p_{0}(i)C_{i}=0$ since without this condition, but with $\sum_{i}X_{t}(i)=0$ as we do have, eq.(18) would not be positivity preserving . Contact with previous sections can be made explicit by recalling that the state probability distribution is $Q_{t}(\alpha)=\langle\alpha|\rho_{t}|\alpha\rangle$ and by noticing that ${\rm Tr}[(C_{j}+C_{j}^{\dagger})\rho_{t}]=\langle\Gamma(i)\rangle_{t}$. We only took a naive limit of the difference equation (15), to mathematically prove the diffusive Belavkin equation for the system density matrix in the scaling limit would require more delicate arguments.

2.2 Poissonian jumpy limit

The Poissonian limit occurs when $\langle i|\psi\rangle=0$. This cannot happen for all $i$ as $\{|i\rangle\}$ forms an orthonormal basis of $\mathcal{H}_{p}$ and $|\psi\rangle$ is non zero. So, we assume, for simplicity, that one element of this basis is $|\psi\rangle$, say $|i^{*}\rangle=|\psi\rangle$, and all others are orthogonal to $|\psi\rangle$, i.e. $\langle i|\psi\rangle=0$ for all $i\not=i^{*}$. Then, $p_{0}(i^{*}|\alpha)\simeq_{\delta\to 0}1$ and $p_{0}(i|\alpha)\simeq_{\delta\to 0}\delta\,\theta(i|\alpha)$, for $i\not=i^{*}$, with

This is the situation we encountered in the previous section on the classical Poisson jumpy limit, so that we can borrow all results obtained there.

As in the diffusive case, it is a simple matter to naively take the continuous time limit of the difference equation (17). This only uses the decomposition the hamiltonian $H$ but the limit exists only if $\langle\psi|H_{I}|\psi\rangle=0$, and we assume this to be true. Expanding the first term $(D\rho)_{n-1}$ to second order in $\sqrt{\delta}$, we get $(D\rho)_{[t/\delta]}\simeq_{\delta\to 0}L_{p}(\rho_{t})\delta$, with Linbladian

where we defined the operators $D_{i}:=-\imath\,{\langle i|H_{I}|\psi\rangle}$ acting on $\mathcal{H}_{s}$, that is

To compute the limit of the second term $(\Delta\rho)_{n}$, we notice that, to leading order in $\delta$, $\langle i|U|\psi\rangle\rho\langle\psi|U|i\rangle\simeq\delta\,D_{i}\rho D_{i}^{\dagger}$ and $\pi_{n-1}(i)\simeq\delta\,{\rm Tr}[D_{i}\rho_{t}D_{i}^{\dagger}]$ for $i\not=i^{*}$, and we get $\lim_{\delta\to 0}(\Delta\rho)_{[t/\delta]}=\sum_{i\not=i^{*}}\widehat{\mathcal{D}}_{i}(\rho_{t})\,dY_{t}(i)$, with

where the last term $-\rho_{t}$ comes from using $dY_{t}(i^{*})=-\sum_{i\not=i^{*}}dY_{t}(i)$ when computing the contribution of the $i^{*}$-term in $(\Delta\rho)_{n}$, as in previous section. Gathering shows that the Poissonian limit of the difference equation (15) is

where the $Y_{t}(j)$’s are the Poisson-like compensated martingales defined in eq.(11) above. That is,

where $dN_{t}(i)$’s are the point processes with intensities $\langle\theta(i)\rangle_{t}\,dt$ defined in previous section, see e.g. eq.(12,13). Note that, using the decomposition of the interaction hamiltonian $H_{I}$ on the pointer state basis, $H_{I}=\sum_{\alpha}|\alpha\rangle\langle\alpha|\otimes H_{\alpha}$, we have

so that the previous equation indeed coincides with eq.(11). Equation (19) is an example of a jumpy Belavkin equation . Let us finally point out that the stochastic processes (18,19) are not the most general one because we assumed that they preserve the pointer state basis We made this assumption when computing the probability distributions of the processes $X_{t}(i)$ and $Y_{t}(i)$ but not when formally taking the continuous time limit of the discrete eq.(15). so that the operator $H_{s}$, $C_{i}$ or $D_{j}$ are diagonal in the pointer basis. This is of course related to the non-demolition property. Eqs.(18,19) are also peculiar examples of more general class of models for continuous quantum measurements whose long time behavior leads to purification of mixed states, see e.g. .

Acknowledgements: This work was in part supported by ANR contract ANR-2010-BLANC-0414.01 and ANR-2010-BLANC-0414.02.

References