Unitarity, Feedback, Interactions – Dynamics Emergent from Repeated Measurements

# Unitarity, Feedback, Interactions – Dynamics Emergent from Repeated Measurements

Natacha Altamirano Perimeter Institute, 31 Caroline St. N. Waterloo Ontario, N2L 2Y5, Canada Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1    Paulina Corona-Ugalde Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1    Robert B. Mann Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1    Magdalena Zych Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Queensland 4072, Australia
July 6, 2019
###### Abstract

Motivated by the recent efforts to describe the gravitational interaction as a classical channel arising from continuous quantum measurements, we study what types of dynamics can emerge from a collisional model of repeated interactions between a system and a set of ancillae. We show that contingent on the model parameters the resulting dynamics ranges from exact unitarity to arbitrarily fast decoherence (quantum Zeno effect). For a series of measurements the effective dynamics includes feedback-control, which for a composite system yields effective interactions between the subsystems. We quantify the amount of decoherence accompanying such induced interactions, generalizing the lower bound found for the gravitational example. However, by allowing multipartite measurements, we show that interactions can be induced with arbitrarily low decoherence. These results have implications for gravity-inspired decoherence models. Moreover, we show how the framework can include terms beyond the usual second-order approximation, which can spark new quantum control or simulation protocols. Finally, within our simple approach we re-derive the quantum filtering equations for the different regimes of effective dynamics, which can facilitate new connections between different formulations of open systems.

###### pacs:
03.65.Ta 03.65.Yz 04.60.-m

## I Introduction

Modern measurement theory dispenses with the description of a measurement as a projection onto one of the complete set of orthogonal eigensubspaces of a Hermitian operator (an observable) with the results (the observable’s eigenvalues) distributed according to a probability measure VonNeumann:1932 (); Lueders:1951 (). Rather, the measurement is understood as an operation, whereby the system’s final state is determined by an action of a completely positive trace non-increasing map, corresponding to a given result, and the outcomes are described by linear operators on the system, distributed according to a positive-operator valued measure (POVM) Kraus:1983 (). This generalized description of a measurement allows achievement of tasks that are impossible with projective measurements NielsenBook2000 () and is in fact necessary in most practical situations, where measurements are made with inefficient detectors, additional noise, or provide limited information about the system WisemanMilburn:Book:2010 (); jacobs2014BookQuantum ().

Of key importance is that the POVM approach unifies the theory of measurements with a general description of dynamics, the theory of open quantum systems ref:BreuerBook (). It follows from Stinespring’s dilation theorem Stinespring:1955 () that any POVM operator can be constructed from a projective measurement on an enlarged Hilbert space: where the system of interest and an additional ancilla evolve under a joint unitary and then the ancilla is measured. In the context of measurement theory, the ancillae can be regarded as the measuring apparatus, whereas in the theory of open systems they can model the system’s environment. Engineering a particular measurement and engineering a particular dynamics for the system are thus two complementary aspects of the same conceptual framework. This correspondence is directly applied in quantum simulations Lloyd1996 (), quantum control WisemanMilburn:Book:2010 (), quantum computation Ladd2010QCNature (); Du.2009decoupling () – in all scenarios where a particular Hamiltonian for the system is desired, or when an existing system-environment interaction needs to be suppressed ViolaKnillLlyod:1999DD ().

Recently, a particular model was developed where repeated position measurements result in an effective long-range interaction between systems measured by common ancillae 2013arXiv1311.4558K (). The interactions arise with dissipation of just the right magnitude to render the resulting dynamics classical – unable to increase entanglement. The picture of interactions as mediated by quantum systems, WeinbergQFT:1995 (), is still missing for the gravitational case, despite a variety of efforts Kiefer:2014sfr (). The above result is thus of high interest Kafri:2014zsa (); Kafri:2015iha (); Tilloy:2015zya () for gravitational quantum physics. So far, an approximately Newtonian interaction was constructed from this model Kafri:2014zsa (); Kafri:2015iha (), where decoherence does not only keep the resulting force classical, but is also claimed to be equivalent Tilloy:2015zya (); Kafri:2014zsa (); Wehner:2016UnivesalTest () to the Diosi-Penrose decoherence model Diosi:1986nu (); ref:Diosi1989 (). However, it is also well known that any local dynamics can be efficiently simulated by suitably chosen interactions with ancillae NielsenBook2000 (). In particular, repeated interactions employed in the research described above correspond to a collisional model of an open system Rau1963 (); AlickiLendi1987Book (); Ziman2005AllQubit (); Ziman2005 (), which can reproduce any Markovian dynamics Ziman2005AllQubit (); Ziman2005 () (including recently revisited examples of effectively unitarity Layden2015unitarity () as well as fully decoherent Layden2015QZE () evolutions). The questions thus arise: What are the assumptions necessary to obtain any particular type of dynamics from the continuous quantum measurement? Is it possible to induce the interactions but with less decoherence? Is it possible to generate an exact Newtonian, or even post-Newtonian, interaction from such a model?

Here we study what types of dynamics can in general emerge from a simple model of repeated measurement, where refs. 2013arXiv1311.4558K (); Kafri:2014zsa (); Kafri:2015iha (); Tilloy:2015zya () are a particular example. We show that the interaction terms found in those studies, arise for a particular choice of the model parameters. We discuss the necessary conditions and highlight all relevant assumptions required for their emergence. Furthermore, we quantify the amount of decoherence arising with the effective interactions. We provide a very simple proof that for bipartite measurements dissipation accompanying effective interactions is indeed lower bounded, generalizing the observation made in the gravitational example. However, we also show how effective interactions can emerge with arbitrarily low decoherence – if one allows for measurements realized through, admittedly less appealing, multipartite system-ancillae interactions.

While our results are motivated by position measurements in the gravitational sector, they also have applicability beyond these particular considerations. The very simple approach applied throughout this work shows which assumptions can be modified, and how, in order to obtain a larger class of effective evolutions; for example, it provides a means to construct collisional models that would give Markovian master equations beyond the usual Born-Markov approximation and suggests how these can be used to recover exact Newtonian (or post-Newtonian) interaction terms from the repeated measurements. By deriving quantum filtering equations corresponding to all the different regimes of emergent dynamics our work can also provide new connections between the stochastic calculus and other approaches to open quantum systems. In this context we also note concurrent work DanDavid () investigating emergent open dynamics of a quantum system undergoing rapid repeated unitary interactions with a sequence of ancillary systems. Our results, are commensurate with these, though the work of ref. DanDavid () is concerned with understanding how thermalization, purification, and dephasing can emerge whereas our concern is with the continuum limit and the nature of the emergent interactions arising in such models.

The structure of this paper is as follows: in Sec. II we revise a general model of a repeated interaction between a system and a set of independent ancillae. We show how – contingent on the relationship between the strength and duration of the interaction and the state of ancillae (moments of its probability distribution) – any type of system dynamics can emerge: from exact unitary evolution (related to “decoherence free subspaces” Lidar2003 ()), effectively unitary evolution under an “external potential” recently re-investigated in ref. Layden2015unitarity () and decoherence, with the quantum Zeno effect (QZE) Misra:1976by (); 2002PhRvL..89h0401F (); Layden2015QZE () in the extreme case. In Sec. III we generalise the model to a sequence of repeated interactions. In particular, we identify conditions under which coherent quantum feedback Lloyd2000Feedback (); WisemanMilburn:Book:2010 (); jacobs2014BookQuantum () arises. In Sec. IV we consider composite systems under sequences of interactions. We identify conditions under which an effective interaction between two systems emerges and quantify the accompanying decoherence. For a particular choice of measurements we recover the emergence of the Newtonian gravitational interaction of ref. Kafri:2014zsa (). Finally, we discuss the applied method, results and outlook in Sec. V, where we also discuss the connection to stochastic calculus.

## Ii Continuous Quantum Measurement

We consider a system and a set of identically prepared ancillae , . Initially, the system is uncorrelated with the ancillae, couples to the first one for a time , decouples, then couples to the second one for time , decouples, etc. This process repeats times, as illustrated in Figure 1. This is equivalent to a collisional model Rau1963 (); AlickiLendi1987Book (); Ziman2005AllQubit (); Ziman2005 () of an open system, modelling interaction with a Markovian environment which has relaxation time .

During an cycle the joint system evolves under the Hamiltonian

 ^Hsmr=^H0+g(t)^HI=^S0+^M0+gr(t)^S⊗^M, (1)

where acts on the system only, – only on the ancilla and we thus call the total free Hamiltonian, is the interaction Hamiltonian. The latter is identical at each cycle: the same operators and act on and for each and the interaction strength satisfies , where and . After the interaction the joint state of the system and the respective ancilla reads

 ρsmr(tr+1)=^Ur(τ)ρsmr(tr)^U†r(τ), (2)

where

 ^Ur(τ)=Texp(−iℏ∫tr+τtrHsmr(t)dt). (3)

For each interaction we assume the same initial state of the ancilla and as a result the final state of the system is described by iterations of a superoperator , where denotes the partial trace over the ancilla degrees of freedom, and is the initial state of the ancilla. We are interested in the dynamics of in the limit of a continuous interaction, given by

 n→∞,τ→0,suchthatlimn→∞,τ→0nτ=T, (4)

where is a fixed (and finite) time interval. Given the initial state of the system , the state at time is fully described by the map:

 ρs(T)=limn→∞Vn(Tn)[ρs(0)], (5)

which is completely positive and trace preserving, but in general not unitary. The resulting dynamics in the continuous limit gives rise to a Markovian master equation, which we derive next.

If the interaction strength is continuous and differentiable in the interval , the mean value theorem allows to obtain

 ∫tr+1tr^Hsmdt=(^H0+¯g^HI)τ, (6)

where and the Hamiltonian is time independent. Under certain restrictions on the interaction strength , outlined in Appendix A, we can write the density matrix at a time as

 ^ρs(tr)=(^I+∞∑m=1Pm)[^ρs(tr−1)], (7)

where is the super-operator consisting of commutators:

 Pm[^ρs(tr)]=1m!(−iτℏ)m⟨[^H,[^H,[...,[^H,^ρs(tr−1)]]]]⟩Mr, (8)

where denotes the trace over the degrees of freedom of the r-ancilla. Note that Eq. (7) holds in particular for symmetric in time switching functions, e.g. modelling interactions that are constant in time, and applies to typical scenarios involving photons, but also to toy models of gravitons in the recent gravitational decoherence models Kafri:2014zsa (); Kafri:2015iha (); Tilloy:2015zya () – where we aim to apply results of this work. Using Eq.(8) to expand (7) yields

 ρs(tn) = ρs(tn−1)−iℏτ[^S0+¯g⟨^M⟩^S,ρs(tn−1)]+iτ22ℏ2¯g⟨i[^M,^M0]⟩[^S,ρs(tn−1)]+ (9) − τ22ℏ2([^S0,[^S0,ρs(tn−1)]]+¯g⟨^M⟩[^S,[^S0,ρs(tn−1)]]+¯g⟨^M⟩[^S0,[^S,ρs(tn−1)]])+ − τ22ℏ2¯g2⟨^M2⟩[^S,[^S,ρs(tn−1)]]+⋯

where for . Note that is a Hermitian operator which can contribute to the effective unitary evolution of the system (see Sec. II.3). Analogous terms appear also at higher orders, and we discuss their potential contributions to the final master equation in Appendix A. The equations of motion for the system at time are finally obtained from

 ˙ρs(T)=limτ→0,n→∞ρs(tn)−ρs(tn−1)τ. (10)

Equations (9) and (10) define a general quantum master equation that describes the effect of repeated interactions with ancillae on the reduced state of the system. While such collisional models are well studied in the context of open quantum systems and decoherence (see e.g. refs. Ziman2005AllQubit (); Ziman2005 ()), the scope of the present work is to analyze the types of unitary contributions effectively arising in such models and to quantify their strength relative to the noise.

Equivalently, a repeated interaction of the form describes a repeated measurement of the observable on the system made by the ancillae. The ancillae play the role of “meters” (measuring apparatus) whose “pointer states” span a basis conjugate to the basis of the eigenstates of . The limit in Eq. (4) corresponds to a continuous measurement made over time . Note, that since we work with a collisional model, we shall not consider measurement channels that have no short-time expansion.

The types of dynamics arising from such a continuous measurement in general depend on the interaction strength , the relation between the free and the interaction terms in the total Hamiltonian Eq. (1) and on the state of the ancillae. We shall discuss the different possibilities in the following section.

### ii.1 Exact unitary evolution

For an arbitrary initial state of the system the evolution under the Hamiltonian (1) is exactly unitary if and only if: the initial state of the ancilla is supported on a linear subspace of eigenstates of with a common eigenvalue and the subspace is invariant under . This is an analogous condition to the one derived in the context of decoherence free subspaces Lidar2003 () or error correction PlenioVedralKnight.PRA.55.67 (), with the crucial difference that here we present conditions on the state of the ancillae, rather than the system. The proof is sketched in Appendix B.

The conditions above, and the proof, naturally extend to the most general case of a bipartite interaction . The evolution of the system is exactly unitary if the joint state of the system and ancilla is supported on a subspace where the total Hamiltonian can be written in block-diagonal form, where the system is in a joint eigenstate of a subset of operators , with the corresponding eigenvalues , and the ancilla is in an eigenstate of the operators in the remaining interaction terms, with eigenvalues . The interaction then effectively reads . One also further requires that the free dynamics of the system and the ancilla preserve the above eigensubspaces. This generalizes the results discussed in Lidar2003 () to an arbitrary interaction. The case of a general interaction, for a non-factorizable initial state, has also been studied in Napoli.PRA.89.062104 ().

From the viewpoint of the measurement interpretation of interactions such a scenario is somewhat unusual, since the allowed states of the system are constrained to a specific subspace (with the exception of a single interaction term, when only the state of the ancilla is constrained). The measurement interpretation can still be applied in the sense that time evolution of the measurement apparatus (ancillae) depends on the state of the system – it is given by a Hamiltonian . Analogously, the system evolves under the Hamiltonian . The system’s evolution is therefore on the one hand “interaction-free” – exactly unitary – and on the other, it still depends on the state of another system, through the eigenvalues .

### ii.2 Effective unitarity

Unitary evolution of a system interacting with some environment typically emerges only as an approximate description – when one assumes finite precision of any measurements made on the system to probe its dynamics. The quantitative condition for such an effective unitarity is that the terms for in Eq. (9) remain small compared to the first order ones, which is the case if

 limτ→0τk¯gk⟨^Mk⟩τ¯g⟨^M⟩=0,k=2,3,⋯ (11)

These conditions are not automatically satisfied because the quantities are moments of an in principle arbitrary probability distribution over the eigenstates of defined by the state of the ancilla111For finite-dimensional systems the number of independent moments is of course finite, as well as for continuous variable systems in e.g. a Gaussian state, which yields distribution with only two independent moments.. If these conditions are met, from Eqs. (9) and (10) the following master equation is obtained222Terms containing at least one or in Eq. (9) automatically have the required limiting behaviour: the expressions with for are small compared to lower order ones, , in the limit .

 ˙ρs=−iℏ[^S0+Ξ^S,ρs], (12)

where we defined

 Ξ:=limτ→0¯g⟨^M⟩. (13)

The system is effectively subject to an external potential induced by the interactions and evolves approximately unitarily under an effective Hamiltonain . The latter entails that in this regime the system-ancilla interaction is non-entangling.Similar results have been found in the context of classical control theory of quantum systems Milburn2012PhilTransA (), where the system interacts with ancillae that are themselves an open quantum system. The effective Hamiltonian and the Hamiltonian in the case of the exact unitary dynamics , Sec. II.1, have the same general structure but the key difference is that is valid only approximately, while holds exactly but only for a particular state of ancillae.

From the viewpoint of the measurement interpretation, the regime of effective unitarity is tantamount to a limiting case of a weak measurement (or unsharp measurement WisemanMilburn:Book:2010 ()) – where the interaction between the system and the measuring apparatus is non-negligible only to lowest order. Decoherence induced by such a measurement is vanishingly small, but so is the information about the system that could be gained from the apparatus, since each of the ancillae only evolves by a global phase – as expected from general complementarity relations between information gain and state disturbance Busch2009 ().

An effective unitary evolution is a generic feature of a weak interaction regime: for and a generic state of the ancilla – with fixed but arbitrary moments – the reduced state of the system evolves according to Eq. (12). In this regime higher order corrections in can be made arbitrarily small by taking a suitably short time step (and therefore can be neglected provided that all subsequent measurements have finite resolution).

Importantly, effective unitarity can also emerge in the strong interaction regime – which we model by taking with for simplicity333For any value of a function satisfying can be obtained from any suitably normalised family of functions that converge to a Dirac delta distribution. – for specific states of the ancillae. The conditions in Eq. (11) now reduce to , and we also need to ensure that stays finite. An example of a suitable ancillae state is a Gaussian distribution over the eigenvalues of with mean and variance , where are fixed parameters. The effective potential arising from this example is .

### ii.3 Quantum Zeno effect

When at least one of the conditions in Eq. (11) is not satisfied, the reduced dynamics of the system is not unitary. This can arise both in weak or strong interaction regimes, depending on the state of the ancillae. We first focus on the regime of strong interactions, where non-unitarity will be shown to be a generic feature.

As in the section above, the strong interaction regime is understood as . We consider a generic state of the ancillae, where the moments are in principle arbitrary but fixed, independent of . The terms in Eq. (9) will then dominate over all others, and remain non-negligible in Eq. (9) for arbitrary high . Summing them all and denoting the magnitude of an arbitrary matrix element of the system (in the basis of ) by , where , yields

 ˙ρij=ρijlimτ→01τ(|⟨e−iΔsij^Mℏ⟩|−1), (14)

with . The right hand side remains finite only in two cases: for diagonal elements of the system, , or for the ancilla in an exact eigenstate of . For a generic state of the ancilla the suppression of the off-diagonal elements of the system becomes “infinitely” fast. More precisely, is a characteristic function of the probability distribution over the eigenvalues of defined by the state of the ancilla, and the moments of this distribution characterize the rate of decoherence. For a particular example of a Gaussian distribution where we set for simplicity, and where is the variance of , an exact solution to Eq. (14) reads

 ρij(t)=ρij(0)limτ→0e−tτ(1−e−σ2Δs2/2ℏ2), (15)

for . Approximating (15) yields

 ρij(t)≈ρij(0)limτ→0(1−σ2Δs22ℏ2tτ), (16)

to first non-vanishing order in . Eq. (16) is a generic result at this order – valid for any state of ancilla when keeping up to second moments of its distribution.

In this regime, the interaction is diagonalizing the system in the eigenbasis of at an arbitrarily fast rate . Unitary evolution, stemming from the typically leading order term , becomes irrelevant – it only acts non-trivially on the off-diagonal elements, but these are “instantaneously” suppressed. From the viewpoint of the measurement-interpretation, this “infinite” decoherence is simply the QZE effect: the measurements become repeated infinitely often () and projective (interaction strength diverges, ) and the system “freezes” in the measurement basis.

Finally, note that the reduced dynamics of the system is non-unitary – is discontinuous – even for ancillae in an eigenstate of (with a non-zero eigenvalue), since is then divergent. If only finite-precision measurements can be made on the system, decoherence and the QZE will arise also in that case. Superposition states of the system will accumulate a relative phase at a divergent rate and thus any coarse-graining will entirely suppress their coherence. Furthermore, assuming finite precision in the preparation of the ancillae (any non-vanishing variance) decoherence will always be non-negligible in the strong interaction case – and in this sense is a generic feature of the strong interaction regime.

QZE has been realized with continuous (as well as pulsed) measurements e.g. with Bose-Einstein condensates PhysRevLett.97.260402 (), and has been theoretically studied in a number of contexts, including freezing the evolution of a two-level Jaynes-Cummings atom interacting with a resonant cavity mode PhysRevA.45.5228 (), controlling decoherence doi:10.1142/S0217979206034017 () producing effective hard-core repulsions in cold atomic gases Syassen1329 (); PhysRevLett.102.040402 (), preparing and stabilizing the Pfaffian state in rotating harmonic traps loaded with cold bosonic atoms PhysRevLett.104.096803 (), and inducing topological states of fermionic matter via suitably engineered dissipative dynamics 1367-2630-15-8-085001 ().

### ii.4 Finite decoherence

We now consider conditions under which only terms up to second order remain relevant. The model of repeated measurements reduces then to the usual Born-Markov master equation ref:BreuerBook (). In analogy to defined in Eq. (II.2) we introduce

 Γ:=limτ→0τ¯g2⟨^M2⟩,~M:=limτ→0¯g⟨i[^M,^M0]⟩2ℏ, (17)

and assume that all higher order terms vanish in the considered limit (see also Appendix A). This is indeed the case e.g. (a) in a strong interaction regime () for ancillae in a Gaussian state with mean and variance ; (b) in a weak interaction regime (fixed ) and ancillae in a Gaussian state with fixed and . Importantly, both in (a) and (b) the quantities , remain finite in the limit .

Eqs. (9), (10), (17) yield the following master equation

 ˙ρ(t)=−iℏ[^S0+(Ξ−~M)^S,ρ]−Γ2ℏ2[^S,[^S,ρ]], (18)

which features two different second order contributions: The term contributes to the unitary system dynamics and simply adds to the effective potential already present in Eq. (12), and a non-unitary term , which results in decoherence at a finite rate. For the example (a) above, the off-diagonal elements of the system are suppressed according to

 ρij(t)≈ρij(0)(1−ΓΔs22ℏ2t), (19)

(neglecting for simplicity) in agreement with Eq. (16). Decoherence vanishes provided that (implying ) i.e. for an exact eigenstate of with the eigenvalue 0, in agreement with the condition found in the QZE case (since the mean in the present example vanishes faster than the variance, unless ).

The regime where Eq. (18) applies and finite decoherence is observed corresponds to the typical case of continous weak measurements: the interactions between the system and the measuring apparatus are finite but the contributions stemming from are considered non-negligible. In that context one often considers ancillae with trivial free evolution, . The ensuing system dynamics features finite decoherence, due to noise introduced by the measurements, but with no modifications to the unitary part.

As an exemplary application of the above, for a strong interaction and a particular choice of operators: , (momentum operator of the ancillae), (position operator of the system), and for a Gaussian state of the ancilla with and , where is a fixed parameter, our Eq. (18) reduces to a continuous position measurement derived in ref. PhysRevA.36.5543 ().

## Iii Continuous measurement of multiple observables

Here we generalize our discussion to the case when several observables are repeatedly measured on the system. This situation can be accommodated by considering that each interaction in Sec. II is composed of sub-interactions, each of duration , as shown in Figure 2.

The total Hamiltonian in the cycle, Eq. (1), now generalizes to

 ^H(p)smr=^H0+p∑i=1gi(t)^HIi=^S0+^M0+p∑i=1gi(t)^Si⊗^Mi. (20)

The operators , act only on , and act on the ancillae, is the switching function, now supported in the sub-step (of length ), continuous in the interval where applied. The density matrix of the joint system at time is given by

 ρsm(tr+1)=1∏i=p^Ui(τ′)ρsm(tr)p∏i=1^Ui(τ′)†, (21)

where . We apply the mean value theorem, as in Eq. (6), and define . Expanding Eq. (21) in powers of at time and tracing over the ancillae degrees of freedom gives

 ρ(tn) = ρ−p∑i=1{iℏτ′[^S0+¯gi⟨^Mi⟩^Si,ρ]+ (22) − τ′22ℏ2i∑j=1(2−δij)[[^S0,[^S0,ρ]]+¯gj⟨^Mj⟩[^S0,[^Sj,ρ]]+¯gi⟨^Mi⟩[^Si,[^S0,ρ]]+¯gi⟨[^Mi,^M0]⟩[^Si,ρ] + ¯gi¯gj2(⟨[^Mi,^Mj]⟩[^Si,^Sjρ+ρ^Sj]+⟨{^Mi,^Mj}⟩[^Si,[^Sj,ρ]])]}⋯,

where and . Equation (22) generalises Eq. (9) to the series of repeated measurements. It introduces a new type of term

 ⟨[^Mi,^Mj]⟩[^Si,^Sjρ+ρ^Sj], (23)

which can contribute to the unitary part of the system dynamics. In particular, it can allow for feedback control of the system, discussed in Sec. III.3.

### iii.1 Exact and effective unitarity

The conditions for exact unitary evolution of the system under arbitrary bipartite interactions with ancillae were discussed in Sec. II.1 and they thus apply also to the present case, where the different interactions are applied sequentially.

The conditions for effective unitarity, Eq. (11), directly generalize to the series of interactions. The resulting effective dynamics reads

 ˙ρ(t)=−iℏ[^S0+1pp∑i=1Ξi^Si,ρ(t)], (24)

where

 Ξi:=limτ′→0¯gi⟨^Mi⟩. (25)

This is a straightforward generalization of Eq. (12). The examples of interaction strengths and ancilla states discussed in Sec. II.2 apply to the present case as well. Thus, for multiple measurement/interactions effective unitary dynamics are also a generic feature of a weak interaction regime, for , but can also arise in the strong interaction regime for -dependent state preparation of the ancillae.

### iii.2 Generalized QZE

Here we consider the case when arbitrary high order terms contribute to the reduced dynamics of the system. Such situation arises in the regime of strong interactions, , , for a generic state of the ancillae. For clarity, below we restrict to sub interactions.

In a full analogy to the QZE discussed in Sec. (II.3), the free evolution can be neglected compared to the interaction terms. Thus, the time evolution of the matrix elements of the system reads

 ˙ρij=limτ→01τ(|Trm{⟨si|e−iτ′ℏ¯g2^HI2e−iτ′ℏ¯g1^HI1ρm⊗ρeiτ′ℏ¯g1^HI1eiτ′ℏ¯g2^HI2|sj⟩}|−ρij). (26)

As an illustrative example one can consider a repeated measurement of the same operator on the system via two conjugate operators for the ancilla . Eq. (26) then reduces to

 ˙ρij=ρijlimτ→01τ(|⟨e−iΔsij^M′ℏ⟩|−1) (27)

which is just (14) for . Another simple example is when conjugate observables are measured on the system (i.e. ) via the same ancilla operator . Eq. (26) then reads

 ˙ρi′j′=ρi′j′limτ→01τ(|⟨e−iΔs′ij^Mℏ⟩|−1), (28)

where the off-diagonal elements are taken in the eigenbasis of , defining where and . In the most general case the decoherence basis is established from the full expression

 e−iτ′ℏ¯g2^HI2e−iτ′ℏ¯g1^HI1=e−iτ′ℏ(¯g1^HI1+¯g2^HI2)−12τ′2ℏ2¯g1¯g2[^HI2,^HI1]+⋯.

Decoherence rates in this regime are again (cf. Sec. II.3) formally divergent.

### iii.3 Feedback

Next we analyze conditions under which only terms up to second order contribute to the system dynamics. At the end of this section we discuss sufficient conditions for the emergence of feedback-control from the repeated measurement model.

To simplify the notation, along with , Eq. (25), we define

 Γij:=limτ′→014τ′¯gi¯gj⟨{^Mi,^Mj}⟩~Mij:=limτ′→014ℏτ′¯gi¯gj⟨i[^Mi,^Mj]⟩. (29)

From the Eqs. (10) and (22) (with ) we obtain a general master equation for a system subject to two sequential measurements:

 ˙ρ(t) = −iℏ[^S0+(12Ξ1−~M10)^S1+(12Ξ2−3~M20)^S2,ρ]+iℏ~M12[^S2,^S1ρ+ρ^S1]+ (30) − 12ℏ2∑i=1,2Γii[^Si,[^Siρ]]−1ℏ2Γ12[^S2,[^S1,ρ]]),

where in defining we introduced the convention .

We now discuss how coherent feedback can result from the terms . Note that the first measurement induces a translation of the state of the ancillae in the basis complementary to the eigenbasis of . The magnitude of this translation depends on the state of the system (on its -eigenvalue). The resulting state of the ancillae then determines the effective potential which arises for the system from the next interaction . Thus, for a suitable choice of the interactions and the state of the ancillae, an operation on the system is effectively performed that depends on its quantum state – that is coherent feedback Lloyd2000Feedback (); WisemanMilburn:Book:2010 (); Zhang2014quantum (); jacobs2014BookQuantum (). This makes clear why a necessary condition for feedback is . A sufficient condition is related with the question whether feedback is possible without introducing some decoherence. The answer is negative in the present model of ancillae. The reason is that the feedback term is at most of the same order as the decoherence terms – a direct consequence of the inequality

 ⟨(¯g1^M1−i¯g2^M2)(¯g1^M1+i¯g2^M2)⟩≥0. (31)

Therefore, independently of the weak or strong interaction regime, the state of ancillae or the repetition rate of the measurements, with the present model of ancillae-system interactions, feedback-control of the system cannot be realized without introducing dissipation lower bounded according to Eq. (31). See also refs. Yamamoto:2014Feedback (); Jacobs:2014feedback () for a comparison between coherent quantum feedback and the measurement-based feedback.

An example of a feedback-enabled control of a quantum system is a restoring force resulting from a quadratic potential . It can be achieved by taking in the model (30). More generally, feedback can take the form of a dissipative force, for for . Taking canonically conjugate pair of ancillae operators , results in feedback-control that is independent of the state of the ancillae.

Eq. (30) is valid when the quantities , , remain finite in the limit , while contributions from higher moments vanish. A particular example of the ancillae state and operators that satisfy these conditions is a series of weak continuous position measurements first given in ref. PhysRevA.36.5543 (), see also Appendix C. In this case, a harmonic potential arises as feedback and the accompanying decoherence keeps the momentum of the system finite. Experimental realization of feedback-control has been achieved with various systems, e.g. in cooling of optomechanical devices Poggio:2007cooling.cantilever (), trapped ions Bushev:2006coolions () or single atoms Koch:2010coolatom ().

Finally, a tacit assumption was made in the above: that only measurements that are linear in the system operators can be realized by the ancillae. Relaxing this assumption would allow for noise-free feedback in the following sense: If an arbitrary measurement/interaction was allowed – of the form , for arbitrary – one could induce an arbitrary potential term already in the regime of effective unitarity, Sec. III.1. For example, a quadratic potential arising due to weak measurement of the system position in ref. PhysRevA.36.5543 () could be implemented unitarily if the ancillae would measure directly . We note, however, that many experimental schemes (including optical devices BachorRalphBook2004QO (), mechanical oscillators BowenMilburnBook2015QOM (), atomic ensembles Hammerer:2010RevModPhys ()) indeed allow only for such linear measurements/interactions.

## Iv Measurement-induced dynamics for composite systems

Here we consider the scenario from a previous section, but for a composite system. We allow that the different subsystems can have different interactions with the ancillae. We restrict our attention to a bipartite system subject to two continuously repeated interactions/measurements.

For a system comprising subsystems the system operator describing the interaction in Eq. (20) most generally can be written as

 ^Si=∑jcj^Ss1i,j⊗^Ss2i,j, (32)

with real coefficients , and where is an operator acting on subsystem . As in the previous section, we are looking for a continuous limit of a protocol whose one step of duration is composed of two sub-steps, each of length . Thus, the master equation for such a case can directly be obtained from Eq. (22) for system operators given in Eq. (32), and where the density matrix describes the state of both subsystems.

The discussions in Sec. III of the various regimes: unitarity (exact and approximate), QZE effect, finite decoherence, directly applies here. However the physical meaning of terms describing the induced potential, feedback, and decoherence is different: Since the operators connect different subsystems, in general they entail emergence of interactions between them. Moreover, the decoherence basis will in general not be a product of the bases of the subsystems – they can decohere into correlated states. This follows from the discussion of the decoherence basis in the QZE case of Eq. (28) for system operators given by Eq. (32).

Below we focus on a particular case where only bipartite interactions involving the ancillae are allowed – i.e. the ancillae only interact with one subsystem at a time. This assumption has been made in the gravitational case studied in refs. Kafri:2014zsa (); Kafri:2015iha (); Tilloy:2015zya () – and is in fact crucial for the main results reported therein, as we will show at the end of this section. Under the above assumption the system operators describing the interactions take the form:

 ^S1=^Ss11⊗^Is2,^S2=^Is1⊗^Ss22, (33)

where is the identity operator on the Hilbert space of subsystem . Recall that acts in the first sub-step and in the second. For simplicity, below we take (since would give terms analogous to those discussed in Sec. III.2). The total Hamiltonian acting during the entire interaction now reads

 ^H(p)s1s2mr=^S0+g1(t)^S1s1⊗^Is2⊗^M1+g2(t)^Is1⊗^S2s2⊗^M2, (34)

analogously to the case of a single system in Eq. (20). Operators act on the ancillae in the sub-step.

For the gravitational case it is natural to consider a symmetrized version of the above scenario: a second ancillae is added, which interacts with in the first sub-step and with in the second sub-step. However, since this only doubles the terms already resulting from Eq. (34) we defer the presentation of the symmetric case to the appendix D. In general, we can visualize the resulting process through the circuit in Figure 3.

From the Hamiltonian in Eq. (34), and with defined in Eqs. (25),(29) we obtain the master equation for a composite system under two consecutive continuous measurements:

 ˙ρs1,s2(T) = −iℏ[^S0+∑i=1,212Ξi^Ssii,ρs1,s2]+iℏ~M12[^Ss22,^Ss11ρs1,s2+ρs1,s2^Ss11]+ (35) − 12ℏ2∑i=1,2Γii[^Ss1i,[^Ssiiρs1,s2]]−1ℏ2Γ12[^Ss22,[^Ss11,ρs1,s2]]),

where is the joint state of . Note, that this is a particular case of Eq. (30) for the system operators defined in Eq. (33).

We find in Eq. (65) terms that are similar to those in Eq. (30) for the single-system case. The terms describe effective potentials contributing to the unitary development that can arise with negligible decoherence, see Sec.III.1. The terms are decoherence terms for each subsystem. The term in Eq. (65), analogous to the corresponding term in Eq. (30), expresses the fact that the decoherence basis in general is given by some combination of the operators acting on the system in the different sub-steps. Note, however, that for to the operators in Eq. (33), the resulting decoherence basis is still of a product form. The term , which more explicitly reads

 ∝¯g1¯g2⟨[^M2,^M1]⟩[^Ss22,^Ss11ρs1,s2+ρs1,s2^Ss11], (36)

is analogous to the feedback term in Eq. (23). However, the terms in Eq. (36) connect two subsystems and they thus introduce effective interactions that can generate forces between them. The example of an approximately Newtonian interaction first derived in ref. Kafri:2014zsa () is presented in Appendix D.

For the bipartite system-ancillae measurements, Eq. (33), the effective interactions can only arise at the second (or higher) order. As a result, the interactions terms are not larger than the decoherence terms arising from the double commutators, in full analogy to the case of feedback. The interactions are also lower-bounded by decoherence in the same way as feedback, as a consequence of the same inequality (31). In fact, these effective interactions can also be interpreted as feedback: the result of a measurement made by an ancillae on one system determines the strength of the effective potential acting on another system, which interacts with the same ancillae.

However, the conclusion about the necessary noise does not arise if one allows more general measurements. A particular example is that of measurements realized simultaneously on both subsystems, described by system operators in Eq. (32). In such a case the potential terms in Eq. (65) would read and could induce interactions even entangling the two systems. With such measurements the interaction terms could arise in the regime of effective unitarity, Sec. III.1, and would thus differ from the feedback schemes where the conditional state exhibits entanglement or where non-unitary terms are present WisemanMilburn:Book:2010 (); Zhang2014quantum (); PhysRevA.71.042309 (); jacobs2014BookQuantum (); PhysRevLett.98.190501 (). The above assumption regarding interactions, and its role, is fully analogous to the linearity assumption in the case of feedback discussed at the end of Sec. III.3. Finally, we note that the above scheme differs from measurement-induced entanglement generation, where quantum correlations are created between systems interacting with a common environment by post-selecting on a particular state of the environment PhysRevLett.91.097905 (); chou2005measurement (); PhysRevLett.112.170501 (); bernien2013heralded (). (Here, the environment is assumed to be inaccessible and is always averaged over.)

## V Discussion

The very simple approach we have applied highlights several key aspects of the formalism. First, it stresses that a system subject to a repeated measurement/interaction with ancillae can still evolve unitarily or be subject to decoherence depending on the state of the ancillae, interaction strength and repetition rate (relaxation time of the environment). Second, in linear systems and under only bipartite system-ancillae interactions, the emergence of feedback-control and of induced interactions, respectively, is accompanied by a finite amount of decoherence, lower bounded by the magnitude of the induced unitary terms. However, decoherence can be made arbitrarily low if more general interactions with the ancillae are permitted. For inducing interactions between different systems this would, however, require non-local (multipartite) interactions.

Our approach has implications for generating gravitational interactions and gravitational decoherence. It has recently been shown that decoherence terms first introduced ad-hoc in gravity-inspired models such as Diosi:1986nu (), could be derived from repeated interactions with the ancillae together with an approximately Newtonian interaction Kafri:2014zsa (); Tilloy:2015zya (). The intriguing aspect of this relation is that those two approaches are aimed at enforcing distinct notions of classicality. In decoherence models the desired classical regime is that where large spatial superpositions of massive systems are suppressed. Whereas in recent works, the notion of classicality is applied to interactions 2013arXiv1311.4558K (), the classical regime being understood as eliminating the ability of interactions to generate entanglement. The approach of the present work can help clarify the extent to which these two notions of classicality have common consequences.

It would be quite remarkable if the exact Newtonian or post-Newtonian interaction could be reproduced from the repeated-measurements model. The resulting theory could be seen as a toy-model for quantum gravitational degrees of freedom – constructed not by quantizing their classical dynamics but by reconstructing the effective forces they generate. To this end, one would need to retain higher order terms than just first and second moments of the ancillae distribution (see appendix C). Our approach suggests a viable route in this direction: one can ask whether a physical state of the ancillae exists that will have higher order non-vanishing moments such that the resulting unitary corrections to the system dynamics would sum to the Newtonian potential444For example, a skew Gaussian distributions have three independent moments. One can begin by asking what dynamics emerges if ancillae are prepared in such a skew-Gaussian state? Can one reconstruct a third-order approximation to the Newtonian potential within the measurement-based approach?. As a further step one could extend the present approach beyond Markovian processes Pollock2015NM (); Breuer:2016RMP () by incorporating, for example, initially correlated ancillae Rybar2012 (), interactions between the ancillae Ciccarello2013 () or initially correlated system-ancillae states Modi:2012SciRep () – particularly desirable for modelling gravitational degrees of freedom. Finally, instead of constructing a classical channel from quantum degrees of freedom, one could ask if an entangling channel can arise from interactions with the ancillae in a scenario where the reduced state of the ancillae can nevertheless be described classically. The motivation here is that while the description of quantum states of matter in a general, even curved, space-time is well understood BIR82 (), the problem lies in giving a consistent quantization of the latter.

If it is indeed possible to generate general-relativistic gravity as an effective interaction with ancillae, the resulting toy-model of quantum-gravitational degrees of freedom could shed new light on the pernicious problems associated with quantum gravity. The above questions and in particular the practical question of detection of the interaction-induced decoherence and its implications for precision tests of gravity remain interesting subjects for further study.

Departing from the main scope of the present work and allowing the ancillae to be accessible to an experimenter, we can describe the state of the system conditioned on the results of measurements performed on the ancillae. The resulting conditional dynamics takes the form of a quantum filtering equation. Usually quantum filters are studied using the quantum extension of the classical Ito calculus Bouten2007introduction (); Zhang2014quantum ();