Repeated measurements on non-replicable systems and their consequences for Unruh-DeWitt detectors

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Repeated measurements on non-replicable systems and their

consequences for Unruh-DeWitt detectors

Nicola Pranzini1,2,3, Guillermo Garc

ıa-P´

erez2,3,4,5, Esko Keski-Vakkuri1,6,3, and

Sabrina Maniscalco2,3,4,7,8

1Department of Physics, P.O.Box 64, FIN-00014 University of Helsinki, Finland

2QTF Centre of Excellence, Department of Physics, University of Helsinki, P.O. Box 43, FI-00014 Helsinki, Finland

3InstituteQ - the Finnish Quantum Institute, University of Helsinki, Finland

4Algorithmiq Ltd, Kanavakatu 3C 00160 Helsinki, Finland

5Complex Systems Research Group, Department of Mathematics and Statistics, University of Turku, FI-20014 Turun Yliopisto, Finland

6Helsinki Institute of Physics, P.O.Box 64, FIN-00014 University of Helsinki, Finland

7QTF Centre of Excellence, Department of Applied Physics, School of Science, Aalto University, FI-00076 Aalto, Finland

8InstituteQ - the Finnish Quantum Institute, Aalto University, Finland

The Born rule describes the probability of

obtaining an outcome when measuring an ob-

servable of a quantum system. As it can only

be tested by measuring many copies of the sys-

tem under consideration, it does not hold for

non-replicable systems. For these systems, we

give a procedure to predict the future statistics

of measurement outcomes through Repeated

Measurements (RM). This is done by extend-

ing the validity of quantum mechanics to those

systems admitting no replicas; we prove that

if the statistics of the results acquired by per-

forming RM on such systems is suﬃciently

similar to that obtained by the Born rule, the

latter can be used eﬀectively. We apply our

framework to a repeatedly measured Unruh-

DeWitt detector interacting with a massless

scalar quantum ﬁeld, which is an example of a

system (detector) interacting with an uncon-

trollable environment (ﬁeld) for which using

RM is necessary. Analysing what an observer

learns from the RM outcomes, we ﬁnd a regime

where history-dependent RM probabilities are

close to the Born ones. Consequently, the lat-

ter can be used for all practical purposes. Fi-

nally, we numerically study inertial and accel-

erated detectors, showing that an observer can

see the Unruh eﬀect via RM.

1 Introduction

In quantum mechanics, the Born rule has an intrinsic

frequentist meaning [1,2]. Given a quantum system

in the pure state |ψ⟩and POVM described by a set of

eﬀects {ˆ

Em}, the Born rule tells that the probability

of measuring the system and ﬁnding the outcome m

Nicola Pranzini: nicola.pranzini@helsinki.ﬁ

pm=⟨ψ|ˆ

Em|ψ⟩.(1)

This means that assuming an observer has access to

inﬁnite identical copies of that system in the same

state, the fraction of times they would obtain the out-

come mis pm. Similarly, if we take a large-but-ﬁnite

number of copies Nof the same experiment, the rel-

ative frequency of the outcome mapproaches pmas

Nincreases, and one can describe the system’s state

by the collection of the results obtained from these

ﬁnitely many experiments [3].

In most cases, one can test the Born rule by

taking a suﬃciently large number of copies of the

system, preparing them identically, and performing

many identical measurements, making the frequen-

tist interpretation eﬀective. Once the rule is tested,

one can apply it without the need to repeat measure-

ments. However, if for any reason one cannot repli-

cate a system, and hence test the Born rule, there

is no a priori motivation to take it as valid. At the

same time, the precision to which QM has been tested

over the last century makes it possible to conjecture

that its mathematical framework describes a broader

class of systems for which the above large-number ap-

proach has no right to exist, i.e. systems that admit

no replicas. In particular, we assume all systems are

fundamentally described by the postulates of quan-

tum mechanics and suppose some systems cannot be

replicated; we call these systems non-replicable. No-

tice that the idea of QM being the fundamental theory

of reality can be found in most quantum-related dis-

ciplines: while we do not necessarily share this view,

we take it as a working assumption throughout the

paper.

Relevant to our work, examples of such systems are

those typically studied by Quantum Cosmology (QC)

and Quantum Field Theory (QFT). In QC, the system

under consideration is the whole universe [4,5]: as

it is clear, testing the Born rule by making multiple

Accepted in Quantum 2024-08-29, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.13347v4 [quant-ph] 30 Sep 2024

copies is impossible in this case. Similarly, in QFT one

often works under the assumption that experiments

can be replicated by starting from some initial state

(e.g. the vacuum) and that the ﬁeld’s state can be

reset to some target state. However, such a degree

of control on QFT degrees of freedom is, in practice,

out of reach, at least due to causality reasons. Hence,

there is no reason to deem using QM’s postulates in

these disciplines well-deﬁned.

This paper discusses the possibility of extending

QM to non-replicable systems. By extension we do

not mean that the Born rule is changed in its for-

mal expression, but rather that we change its domain

of validity, i.e. we extended the set of systems for

which it can be used. The logic is the following. First,

we assume all systems behave following the standard

postulates of quantum mechanics. Then, we notice

that this assumption is not enough to enable the us-

age of the Born rule in all settings. In other words,

assuming a system is quantum is not enough for us-

ing the Born rule. Next, we notice that the Born

rule trivially holds for all quantum systems admitting

replicas (those for which QM was formulated) and

search for a class of systems not admitting replicas

for which the Born rule holds. We ﬁnd that some

such systems exist, hence providing an extension of

the domain of validity of the Born rule: the outcomes

of measurements performed on these systems resem-

ble those performed on replicable ones. In particular,

we study what statistics can emerge from measuring

non-replicable systems many times and deﬁne con-

ditions under which observers can still meaningfully

talk about the Born rule. This is done by looking at

what an observer performing Repeated Measurements

(RM) on a non-replicable system could infer from a

string of measurement outcomes.

As an example, we work out the details of RM per-

formed on an Unruh-DeWitt (UDW) detector (e.g. a

two-level system) interacting with a massless scalar

ﬁeld. This paradigmatic system provides an oper-

ational way to prove that the notion of particle is

observer-dependent, which is a milestone result of

QFT in non-inertial frames [6,7]. In particular, the

detector model can be used to study the Unruh ef-

fect, which states that the quantum state observed

to be empty of particles by inertial observers (the

Minkowski vacuum) appears to be a thermal state for

a class of uniformly accelerated observers (Rindler ob-

servers). A detector in uniform acceleration will excite

and de-excite as if it would be absorbing and emitting

particles in interaction with a thermal heat bath. An

additional question is what happens when the state

of the detector is measured. The measurement col-

lapses the state of the detector, while the state of the

composite detector-quantum ﬁeld system factorizes to

a product state. For an accelerated detector, estab-

lishing a thermal distribution for the state of the de-

tector would require many measurements. However,

the probabilistic interpretation assumes an i.i.d set-

ting, which becomes unrealistic by requiring either

many identical copies of the detector and the quan-

tum ﬁeld, or the ability to reset the ﬁeld state af-

ter the measurement. As an alternative, we resort to

our repeated measurements protocol, and investigate

whether a thermal distribution for the detector state

can be reliably established.

Summarising, the scope of this paper is two-fold.

First, we propose the RM framework as a way of

making predictions for a single run of an experiment

performed on a non-replicable system, for which the

standard frequentist interpretation of QM is not ap-

plicable. Second, we illustrate this approach for an

UDW detector interacting many times with a quan-

tum ﬁeld; by doing so, we investigate the popula-

tions obtained by repeated measurements performed

between the many interactions and see if and how

much they diﬀer from those obtained in the standard

frequentist interpretation.

The paper is organized as follows. In Sec. 2, we

introduce our RM scheme as a procedure for mak-

ing predictions about measurements performed upon

systems interacting with non-replicable environments.

In Sec. 3, we review the theory of Unruh-DeWitt de-

tectors and a recent proposal using them for deﬁning

quantum measurements in QFT. Then, we argue that

an UDW detector interacting with a quantum ﬁeld

can be treated via RM and obtain the probability of

observing any string of results, together with upper

and lower bounds. Next, in Sec. 4, we numerically

study the cases of UDW detectors moving along in-

ertial and accelerated trajectories, and compare our

results with those already present in the literature,

obtained in the ﬁnite time interaction case within

the frequentist interpretation. Finally, in Sec. 5, we

present conclusions and discuss future prospects.

Throughout this paper, we work in four-

dimensional Minkowski spacetime with metric signa-

ture (−,+,+,+). Four-vectors are denoted by upper-

case letters, e.g. X, and their three-vectors spatial

parts by lowercase bold letters, e.g. x. We assume

the units convention for which ℏ=c=kB= 1.

2 Repeated measurements

In this section, we present our scheme for making pre-

dictions about non-replicable systems. Let us ﬁrst re-

call and formalize the standard frequentist approach

to the Born rule. Suppose Alice investigates a system

Sin some state |ψ⟩, and that she can replicate it a

large number of times N. In this way, she performs N

measurements of a selected observable getting Nout-

comes, and builds statistics from these results 1. This

procedure is schematized in Fig. 1a. Let us consider

1Note that, as far as |ψ⟩is known, this procedure is not

forbidden by the no-cloning theorem.

Accepted in Quantum 2024-08-29, click title to verify. Published under CC-BY 4.0. 2

also the case where she can not produce replicas, yet

has full control over the whole system, meaning she

can apply any quantum operation to its state. Then,

she can still test the Born rule by preparing the sys-

tem in the initial target state, measuring it, and re-

setting it to the above state; the outcomes she obtains

by this procedure are identical to those obtained by

using replicas, as it is clear by Fig. 1b. Therefore, hav-

ing full control over the system is enough to test the

Born rule even without being able to produce replicas.

In contrast, we call non-replicable a system which, for

any reason, we cannot prepare many times in the same

state or we have no full control over it.

The relevance of the picture described by Fig. 1b

is that it is easily generalized to describe the case of

non-replicable systems. To do so, we restrict Alice to

measure only a part of of the whole system, called de-

tector and labelled by D, while we call the rest of the

system an environment, and label it with E. Further-

more, we suppose she does not have full control over

the system, meaning that she cannot apply quantum

operations at will on the total state. In particular, we

suppose she can measure and apply reset operators

on D, but all other operations she can use inevitably

couple Dand E. Hence, all the measurement results

she obtains from measuring Dare perturbed by the

state of E, and no Born rule is expected to emerge

from performing Repeated Measurements. This sce-

nario is further explained by Fig. 1c, from which it

is clear that each of Alice’s measurements is testing

the same system in a diﬀerent state, and it is only

by having full control of Ethat she would be able to

observe the Born rule on D.

It is important to note that calling the measured

system detector is not standard in quantum measure-

ment theory, as this name is often reserved for the

apparatus making the tested system decohere. Here

we followed a diﬀerent naming convention to be con-

sistent with the literature regarding Unruh-DeWitt

detectors, which we often reference in the second part

of this work. To avoid confusion, we here stress that,

in this work, measurements are idealized as instanta-

neous projective operators applied over the detector

degree of freedom, and collapse is considered to hap-

pen as in the standard von Neumann prescription,

without any mediator nor decoherence time needed.

2.1 Statistical inference from RM

Following the above discussion, let us consider a sys-

tem Scomposed of two parts, one called the detector

D, and the other the environment E. In particular,

without loss of generality we study the case of Dbe-

ing a non-degenerate two-level system on which we

can perform projective measurements, and take the

total Hamiltonian of Sto be

HS(t) = ˆ

HD+ˆ

HE+λˆ

Hint(t),(2)

where

Hint(t) =

k=1

χk(t)ˆ

H(k)

int (t) (3)

with {χk(t)}being a set of window functions with dis-

joint compact supports (i.e. Supp(χi)∩Supp(χj) =

∅,∀i, j), and λa small parameter modulating the

strength of the interaction. Note that by using the

interaction Hamiltonian (3)we implicitly assume our

ability to switch on and oﬀ the interaction, a task that

might be hard to accomplish in practice. As a result

of this setup, Dand Einteract Ntimes via as many

possibly diﬀerent Hamiltonians. Finally, let us take S

to be initially in the separable state

|ψ0⟩=|0⟩⊗|e0⟩,(4)

where |0⟩is the lowest energy eigenstate of ˆ

HD, and

|e0⟩some environmental state. For the sake of sim-

plicity, we take the detector’s Hamiltonian to be

HD=ω|1⟩⟨1|(5)

with ω > 0, so that |0⟩has zero energy. Working in

the interaction picture [8], the unitary operator de-

scribing the evolution generated by the k-th interac-

tion has the form

Uk=T(exp −iλ ZSupp(χk)

χk(t)ˆ

H(k)

int (t)dt!) .

(6)

After the end of each interaction, we measure Dvia

the PVM operators

(ˆ

M0=|0⟩⟨0|

M1=|1⟩⟨1|,(7)

store the corresponding results in a bit string

BL= (b1, b2, . . . , bL),(8)

where L≤N, and reset the detector’s state to |0⟩

via some unspeciﬁed erasure procedure [9]. Note

that these measurements are always applied between

the interactions, meaning that the measurement giv-

ing the bit string element bkis obtained after ˆ

ended, and before ˆ

Uk+1 starts. Moreover, the fact

that [ˆ

Mi,ˆ

HD]=0means that the times at which the

measurements happen are not relevant.

Starting from a generic separable state |0⟩ ⊗ |f⟩,

the k-th operator of the set (6)evolves the system

to ˆ

Uk|0, f⟩, and hence the outcome-dependent post-

measurement state is

|ψb⟩=ˆ

Mbˆ

Uk|0, f⟩

ppb(k)[f],(9)

where

pb(k)[f] = 



⟨b|ˆ

Uk|0, f⟩



2(10)

Accepted in Quantum 2024-08-29, click title to verify. Published under CC-BY 4.0. 3

(a) Standard approach for testing the Born rule. (b) Alternative procedure for testing the Born rule via resets.

Figure 1: Procedures for getting many measurement outcomes from a system prepared in the state |ψ⟩=ˆ

U|ψ0⟩, or in the

RM setting. In panel (a), the standard procedure of replicating the system Ltimes is presented. In panel (b), an alternative

way of getting the Born rule is considered. After the system is prepared and measured, its state is reset to the initial one, so

that no replicas are needed. In panel (c), the RM scenario is presented; after each measurement ˆ

Mof panel (c) the global

state of the system is separable due to the measurement update postulate.

is the probability of getting bafter the k-th unitary is

applied, given that the initial environmental state was

|f⟩. Therefore, since after each measurement the state

of Dis reset to |0⟩, one can deﬁne the environmental

state-dependent operators

Vb(k)[f] = ⟨b|ˆ

Uk|0⟩

ppb(k)[f],(11)

describing the collection of operations 1) application

of Uk, 2) measurement of Dwith outcome b, and 3) re-

set of Dto |0⟩; all together, these operations combine

to give

|0⟩⊗|f⟩ −→ |0⟩ ⊗ ˆ

Vb(k)[f]|f⟩.(12)

When the RM procedure is repeated Ltimes upon

the initial state (4), giving the bit string (8), the ﬁnal

state is

|ψL⟩=|0⟩ ⊗ L

i=1

Vi|e0⟩!(13)

where the operators ˆ

Viare recursively deﬁned as











V1=ˆ

Vb1(1)[e0]

Vn+1 =ˆ

Vbn+1 (n+ 1) "n

i=1

Vi|e0⟩#(14)

and where the operator products appearing in

Eq.s (13)and (14)are ordered starting from the right

by decreasing i. Therefore, each V-operator applied

on the environment depends on all V-operators ap-

plied before it, making Eq. (13)extremely hard to

evaluate.

In order to make Eq. (13)tractable, let us con-

sider the simpler case of weak interaction and slowly-

evolving environment, i.e. take the unitaries (6)to

Uk=ˆ

U⊗IE+ϵX

Al⊗ˆ

Bl(k)+ϵ2X

Cl⊗ˆ

Dl(k)+O(ϵ3)

(15)

where the set of operators {ˆ

Al,ˆ

Cl}and {ˆ

Bl(k),ˆ

Dl(k)}

act on HDand HErespectively, and ϵis a small pa-

rameter. First, let us brieﬂy analyze the case of van-

ishing ϵ, corresponding to having no interaction be-

tween Dand E, and applying many times the same ˆ

to the detector. Then, measuring D, gathering data,

and resetting it to |0⟩many times, means that all the

measurements are performed on the same state ˆ

U|0⟩,

and the results strictly satisfy the Born rule (1)with

Em=ˆ

Mm, i.e.

pm=



⟨m|ˆ

U|0⟩



2.(16)

In fact, this is equivalent to preparing many copies

of Din ˆ

U|0⟩and measuring them, hence reproducing

Accepted in Quantum 2024-08-29, click title to verify. Published under CC-BY 4.0. 4

the standard prescription for getting the Born rule, as

in the case pictured in Fig.s 1a-1b. Next, by taking

ϵto be small, and substituting the unitaries (15)in

Eq. (10)one gets

pm(k)[f] = pm+ϵQ(1)

m(k)[f] + ϵ2Q(2)

m(k)[f] + O(ϵ3)

(17)

where











Q(1)

m(k)[f] = X

l⟨ˆ

A†

lˆ

Mmˆ

U⟩0⟨ˆ

B†

l(k)⟩f+h.c.

Q(2)

m(k)[f] = X

l,l′⟨ˆ

A†

lˆ

Mmˆ

Al′⟩0⟨ˆ

B†

l(k)ˆ

Bl′(k)⟩f+

+"X

l⟨ˆ

C†

lˆ

Mmˆ

U⟩0⟨ˆ

D†

l(k)⟩f+h.c.#

(18)

where we used the shorthand notation ⟨ˆ

O⟩ψ=

⟨ψ|ˆ

O|ψ⟩. Hence, for the generic state |0, f ⟩the prob-

ability of obtaining the outcome mis given by the

Born rule of the ϵ= 0 case plus corrections at most

of order ϵ; yet, these corrections are as convoluted as

Eq. (13), meaning that Qm(k)[ek]will depend on all

the states the environment explored since the RM pro-

cedure started, hence making any usage of Eq. (17)

very complicated.

2.1.1 Bayes’ updates rules

Inspired from the the vanishing ϵcase, one can try

overlooking the contribution given by the memory-

dependent part of the probabilities (17), and predict-

ing the statistics of future outcomes as if the proba-

bilities were well described by Eq. (16), i.e. as if the

Born rule was valid. Clearly enough, observers fol-

lowing this strategy will deduce wrong probabilities.

However, if the error they obtain can not be seen by

confronting the predictions with the actual outcomes,

one can still rely on the Born rule for all practical pur-

poses (FAPP). To formalize this idea let us introduce

Bayesian updating in this setting [10].

Suppose we are given a set of data Dand have

to decide between a set of hypotheses {Hi}which

ones are correct. Moreover, suppose we have some

prior knowledge about the hypotheses, summarized

as probabilities of them being true, i.e. P(Hi). By

observing the data, one can update these probabili-

ties from their initial value to P(Hi|D), which is the

probability of Hibeing true given the new knowledge

acquired. This is done via the Bayes rule, i.e.

P(Hi|D) = P(D|Hi)P(Hi)

P(D),(19)

where P(D)is a normalization factor given by

P(D) = X

P(D|Hi)P(Hi),(20)

and P(D|Hi)is the probability of obtaining the data

assuming Hi. A general update strategy then follows

by applying Eq. (21)recursively: assuming we ob-

served the data set D, our knowledge about the hy-

potheses is described by P(Hi|D)which become our

new prior, so that once we obtain the new data D′,

all we have to do to update our current belief is to

apply Eq. (21)again as

P(Hi|D′) = P(D′|Hi)P(Hi|D)

P(D′).(21)

If, after enough data D′′ it is

P(Hi|D′′)≫P(Hj|D′′),(22)

for one iand all j̸=i, then we claim Hiis the

right hypothesis. If new data are later gathered in

favour of other hypotheses, one must stop calling Hi

the true one. If, at the same time, all hypotheses

{Hi1,...,HiR}satisfy Eq. (22)for all j̸=i1, . . . , iR,

yet between all these there is no one for which Eq. (22)

holds with respect to all other i1, . . . , iR, then all of

them retain the same degree of trueness, meaning that

we can interpret the data by means of any.

We are now ready to discuss if and how an observer

can use the Born rule in situations where it does not

apply strictly. Suppose Alice measures Ltimes the

detector and gets the string of outcomes BL. To in-

terpret these data, Alice formulates the following set

of hypotheses:

•H1(q): The Born rule holds as given by Eq. (16),

and p1=q.

•H2(q): The Born rule holds approximately, with

the Born part given as above and corrections of

at most of order ϵ, as in Eq. (17).

Finally, let us suppose Alice has no previous knowl-

edge or bias about the experiments, and therefore she

assigns uniform priors to all hypotheses, i.e. h=

P(H1(q)) = P(H2(q)) = 1/2. Thanks to this choice,

the priors are correctly normalized

i=1 Z1

P(Hi(ϱ))dϱ = 1 .(23)

Next, she looks at the data contained in BLand up-

dates the probabilities about the hypotheses via

P(Hi(q)|BL) =

2P(BL|Hi(q))

2R1

0(2Pϱ(BL) + ϵ∆Pϱ(BL)) dϱ ,

(24)

where we deﬁned

(P(BL|H1(q)) = Pq(BL)

P(BL|H2(q)) = Pq(BL) + ϵ∆Pq(BL)(25)

and where

Pq(BL) = qn(1 −q)L−n(26)

with nbeing the number of ones in BL, and where

∆Pq(BL)depends on the speciﬁc Hamiltonian and

initial environmental state considered. When ex-

panded, Eq. (24)gives

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Repeatedmeasurementsonnon-replicablesystemsandtheirconsequencesforUnruh-DeWittdetectorsNicolaPranzini1,2,3,GuillermoGarc´ıa-P´erez2,3,4,5,EskoKeski-Vakkuri1,6,3,andSabrinaManiscalco2,3,4,7,81DepartmentofPhysics,P.O.Box64,FIN-00014UniversityofHelsinki,Finland2QTFCentreofExcellence,DepartmentofPhysics...

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Repeated measurements on non-replicable systems and their consequences for Unruh-DeWitt detectors

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