Delayed-choice quantum erasers and the Einstein-Podolsky-Rosen paradox Dah-Wei Chiou1 2 1Department of Physics National Sun Yat-sen University Kaohsiung 80424 Taiwan_2

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Delayed-choice quantum erasers and the Einstein-Podolsky-Rosen paradox

Dah-Wei Chiou1, 2, ∗

1Department of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

2Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan

Considering the delayed-choice quantum eraser using a Mach-Zehnder interferometer with

a nonsymmetric beam splitter, we explicitly demonstrate that it shares exactly the same for-

mal structure with the Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) experiment. Therefore,

the eﬀect of quantum erasure can be understood in terms of the standard EPR correlation.

Nevertheless, the quantum eraser still raises a conceptual issue beyond the standard EPR

paradox, if counterfactual reasoning is taken into account. Furthermore, the quantum eraser

experiments can be classiﬁed into two major categories: the entanglement quantum eraser

and the Scully-Dr¨uhl-type quantum eraser. These two types are formally equivalent to each

other, but conceptually the latter presents a “mystery” more prominent than the former. In

the Scully-Dr¨uhl-type quantum eraser, the statement that the which-way information can be

inﬂuenced by the delayed-choice measurement is not purely a consequence of counterfactual

reasoning but bears some factual signiﬁcance. Accordingly, it makes good sense to say that

the “record” of the which-way information is “erased” if the potentiality to yield a conclusive

outcome that discriminates the record is eliminated by the delayed-choice measurement. We

also reconsider the quantum eraser in the many-worlds interpretation (MWI), making clear

the conceptual merits and demerits of the MWI.

I. INTRODUCTION

The idea of the delayed-choice quantum eraser was ﬁrst proposed by Scully and Dr¨uhl in 1982 [1].

The quantum eraser is an interferometer experiment in which the which-way information of each

single quanton (i.e., quantum object such as photon) is “marked” and therefore the interference

fringe pattern is not seen, but the which-way information can later be “erased” and correspondingly

the interference pattern can be “recovered”, apparently exhibiting some kind of retrocausality. The

ﬁrst experiment of the delayed-choice quantum eraser was realized by Kim et al. in 1999 [2] in a

double-slit interference experiment. A similar double-slit experiment involving entanglement of

photon polarization was later performed by Walborn et al. in 2002 [3]. More diﬀerent scenarios

∗dwchiou@gmail.com

arXiv:2210.11375v2 [quant-ph] 30 Oct 2022

framing the same concept have been experimentally realized (see [4] for a comprehensive review)

including a recent work performed in a quantum circuit on the IBM Quantum platform [5].

Ever since the idea of quantum erasure was proposed, its interpretation and implication have

been a subject of ﬁerce controversy that continues to today [6–13] with divided opinions ranging

from “a magniﬁcat aﬀront to our conventional notions of space and time” [14] to “an experiment

that has caused no end of confusion” [15]. Particularly, by analogy to the Einstein-Podolsky-Rosen-

Bohm (EPR-Bohm) experiment [16, 17], Kastner argued that the quantum eraser neither erases nor

delays any information, and does not present any mystery beyond the standard EPR correlation

[12]. Later on, by considering a Mach-Zehnder interferometer, which conveys the core idea of the

quantum eraser more elegantly than a double-slit experiment, Qureshi further elaborated on the

analogy between the quantum eraser and the EPR-Bohm experiment and claimed that there is no

retrocausal eﬀect whatsoever [13].

In this paper, by generalizing the Mach-Zehnder interferometer considered in [13] with a non-

symmetric beam splitter (i.e., the transmission and reﬂection coeﬃcients are not equal), we show

that the quantum eraser shares exactly the same formal (i.e., mathematical) structure with the

EPR-Bohm experiment, as the modiﬁed Mach-Zehnder interferometer is exactly analogous to a

Stern-Gerlach apparatus used in the EPR-Bohm experiment. However, although the quantum

eraser is formally equivalent to the EPR-Bohm experiment, the former still raises conceptual is-

sues that cannot be explained out by the analogy to the latter, as opposed to what is claimed in

[12, 13]. Speciﬁcally, if one applies counterfactual reasoning about the which-way information, then

whether a quanton travels along either of the two paths or both of them indeed can be aﬀected

by the delayed-choice measurement. This does not violate causality, because what can be altered

is not the detection outcome but the which-way information, which is counterfactual in nature in

most situations. The quantum eraser does present an additional conceptual “mystery” beyond the

standard EPR puzzle, unless counterfactual reasoning is completely dismissed.

Furthermore, the quantum eraser experiments can be classiﬁed into two major categories: the

entanglement quantum eraser and the Scully-Dr¨uhl-type quantum eraser. The entanglement quan-

tum eraser relies on the entanglement of some internal states between a pair of quantons (referred

to as the signal and idler quantons), such as the experiment performed by Walborn et al. [3],

which involves the entanglement of polarization between a pair of photons. In the entanglement

quantum eraser, the which-way information of the signal quanton is “recorded” in terms of some in-

ternal state of the idler quanton, which can be either read out or erased by diﬀerent delayed-choice

measurements. However, as the which-way information of the signal quanton is inferred from the

internal state of the idler quanton through counterfactual reasoning, one can always maintain that,

without regard to counterfactual inference, the which-way information of the signal quanton is not

recorded in the ﬁrst place at all and not erased later. On the other hand, in the Scully-Dr¨uhl-type

quantum eraser as originally proposed by Scully and Dr¨uhl [1] and performed by Kim et al. [2],

the which-way information of the signal quanton is “recorded” in terms of the states of two objects

that are spatially separated. The which-way information inferred from the measurement upon the

states of the two objects becomes factual if the measurement yields a conclusive outcome that

discriminates the record. Therefore, it makes good sense to say that the two objects serve as the

“recorders” of the which-way information and the record of the which-way information is “erased”

if the potentiality to yield a conclusive outcome is eliminated. In this paper, we investigate the

entanglement quantum eraser and the Scully-Dr¨uhl-type quantum eraser separately in depth. In

terms of the Mach-Zehnder interferometer with a nonsymmetric beam splitter, we make it explicit

that both kinds of quantum erasers are formally equivalent to the EPR-Bohm experiment. Never-

theless, both raise conceptual issues beyond the standard EPR puzzle, and the Scully-Dr¨uhl-type

quantum eraser presents a “mystery” deeper than that of the entanglement quantum eraser.

We also consider the quantum eraser in view of the many-worlds interpretation (MWI) of

quantum mechanics [18–20]. The MWI provides an appealing ontological framework, wherein

all possible experimental outcomes exist simultaneously and thus many paradoxes of quantum

mechanics (including the quantum eraser) are simply resolved as they are no longer matters of

concern [21]. Our analysis aﬃrms that the standard (i.e., Copenhagen) interpretation and the

MWI yield identical experimental predictions beyond a trivial model. However, the investigation

of the quantum eraser also reveals the subtle disharmony between the theoretical formulation of

the MWI and its practical application: in principle the classical notion of being in a deﬁnite state

is completely repudiated, but in practice it still has to resort to (semi)classical reasoning about

deﬁnite states in order to theorize the dynamics of evolution.

This paper is organized as follows. In Sec. II, we consider the modiﬁed Mach-Zehnder inter-

ferometer with a nonsymmetric beam splitter, which draws a close analogy to the Stern-Gerlach

apparatus. In Sec. III, we study the entanglement quantum eraser in terms of the modiﬁed Mach-

Zehnder interferometer and investigate its equivalence to and diﬀerence from the EPR-Bohm ex-

periment. In Sec. IV, the same investigation is carried out for the Scully-Dr¨uhl-type quantum

eraser. In Sec. V, we reconsider the quantum eraser in the MWI. Finally, the conclusions are

summarized in Sec. VI.

Path2

Path1

PBS

mirror

polarization

rotator

D−

D+

FIG. 1. The schematic diagram of a Mach-Zehnder interferometer that separates diﬀerent polarizations into

the two paths.

II. MODIFIED MACH-ZEHNDER INTERFEROMETER

Consider the experimental setup using a Mach-Zehnder interferometer as sketched in Fig. 1,

which couples the photon’s spatial degree of freedom with its degree of polarization. An incident

photon is split by the polarizing beam splitter PBS into two paths, Path1and Path2, with hori-

zontal (↔) and vertical (l) polarizations, respectively. Along Path1, an adjustable phase shift φis

introduced (e.g, by inserting a phase-shift plate). Along Path2, a polarization rotator that rotates

linto ↔is introduced in order to make the two paths interfere with each other. The two paths are

ﬁnally recombined by the beam splitter BS before the photon strikes either of the two detectors,

D+and D−.1The detection probabilities at D+and D−, which are measured as accumulated

counts of repeated experiments of individual photons, are said to exhibit the two-path interference

pattern if they appear as modulated in response to the phase shift φ.

For generality, we consider BS to be a nonsymmetric beam splitter (i.e., the transmission and

reﬂection coeﬃcients are not equal). Supposing BS is lossless, we can describe the trasfer of BS by

a 2 ×2 unitary matrix as



|Path1i

|Path2i



−→





α β

β∗−α∗





|D+i

|D−i



,(2.1)

where |α|2and |β|2are the transmission and reﬂection coeﬃcients, respectively, which satisfy

|α|2+|β|2= 1,(2.2)

1Denoting the two detectors as D+and D−, instead of D1and D2, underscores the analogy to the Stern-Gerlach

apparatus, as will be seen shortly.

|Path1/2irepresents the photon state localized on Path1/2, and |D±irepresents the photon state

that is going to strike D±. Because the arguments of the complex number αand βcan be absorbed

into the phases of |D+iand |D−ias well as the phase shit φ, we only need to consider the case

that both αand βare real. For convenience, we parameterize them by θ∈[0, π] as

α= cos(θ/2), β = sin(θ/2).(2.3)

The modiﬁed Mach-Zehnder interferometer captures the core concepts of the corresponding double-

slit experiment.2

Given a horizontally polarized photon entering the interferometer, we denote its state as |↔i|ψi,

where |ψirepresents its spatial degree of freedom. The polarizing beam splitter PBS transfers |ψi

into |Path1i. The photon then undergoes the phase shift φand BS. The state transfer of |↔i|ψi

before hitting D+or D−is given by

|↔i|ψi−→

PBS |↔i|Path1i(2.4a)

−→

φeiφ|↔i|Path1i(2.4b)

−→

BS |↔ieiφα|D+i+eiφβ|D−i.(2.4c)

Similarly, given a vertically polarized photon, denoted as |li|ψi, it undergoes PBS, the polarization

rotator, and BS consecutively, and the state transfer is given by

|li|ψi −→

PBS |li|Path2i(2.5a)

−→

rotator |↔i|Path2i(2.5b)

−→

BS |↔i(β∗|D+i − α∗|D−i).(2.5c)

Deﬁne two orthonormal basis states of arbitrary (elliptical) polarizations parameterized by ϑ

and ϕin the “spinor” style as

|ˆnϑ,ϕ,+i:= cos(ϑ/2)|↔i +eiϕ sin(ϑ/2)|li ≡ 



cos(ϑ/2)

eiϕ sin(ϑ/2)



,(2.6a)

|ˆnϑ,ϕ,−i := sin(ϑ/2)|↔i − eiϕ cos(ϑ/2)|li ≡ 



sin(ϑ/2)

−eiϕ cos(ϑ/2)



.(2.6b)

2In the double-slit experiment, we can place two polarizers with horizontal and vertical polarizations in front of the

two slits to emulate the function of PBS. A polarization rotator is then placed behind the second slit to make the

paths from the two slits interfere with each other. The diﬀerent positions on the screen correspond to diﬀerent

values of φ. We can also place two optical attenuators with diﬀerent attenuation rates behind the two slits to

emulate diﬀerent values of θ.

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Delayed-choicequantumerasersandtheEinstein-Podolsky-RosenparadoxDah-WeiChiou1,2,1DepartmentofPhysics,NationalSunYat-senUniversity,Kaohsiung80424,Taiwan2CenterforCondensedMatterSciences,NationalTaiwanUniversity,Taipei10617,TaiwanConsideringthedelayed-choicequantumeraserusingaMach-Zehnderinterferomet...

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Delayed-choice quantum erasers and the Einstein-Podolsky-Rosen paradox Dah-Wei Chiou1 2 1Department of Physics National Sun Yat-sen University Kaohsiung 80424 Taiwan_2

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