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arXiv:1409.5098v1 [quant-ph] 17 Sep 2014

Foundations of Physics manuscript No.

(will be inserted by the editor)

John G. Cramer ·Nick Herbert

An Inquiry into the Possibility of

Nonlocal Quantum Communication

Received: date / Accepted: date

Abstract The possibility of nonlocal quantum communication is considered

in the context of several gedankenexperiments. A new quantum paradox is

suggested in which the presence or absence of an interference pattern in a

path-entangled two photon system, controlled by measurement choice, pro-

vides a nonlocal signal. We show that for all of the cases considered, the

intrinsic complementarity between one-particle and two-particle interference

blocks the potential nonlocal signal.

Keywords quantum ·nonlocal ·communication ·interference ·comple-

mentarity ·entanglement

PACS 03.65.Aa ·03.65.Ud ·03.67.Hk

1 Introduction

Quantum mechanics, our standard model of the physical world at the small-

est scales of energy and size, has built-in retrocausal aspects. For exam-

ple, Wheeler’s delayed choice gedankenexperiment[1] describes a scheme in

which the experimenter’s later choice of measurement retroactively deter-

mines whether a light photon that had previously encountered a two-slit

aperture had passed through both slits or through only one slit.

In the present work we describe a new quantum-mechanical paradox in

which the presence or absence of an interference pattern in a path-entangled

two photon system, controlled by measurement choice, would seem to permit

John G. Cramer

Dept. of Physics, Box 351560, Univ. of Washington, Seattle WA 98195-1560

Tel.: +001 206-525-3504

E-mail: jcramer@uw.edu

Nick Herbert

Box 261, Boulder Creek, CA 95006

2

retrocausal signaling from one observer to another. We also present an anal-

ysis of this scheme, showing how the subtleties of the quantum formalism,

in particular the complementarity between one-particle and two-particle in-

terference, block such potential retrocausal signals and preserve macroscopic

causality.

2 Quantum Nonlocality and Entanglement

Quantum mechanics diﬀers from the classical mechanics of Newton that pre-

ceded it in one very important way. Newtonian systems are always local. If

a Newtonian system breaks up, each of its parts receives a deﬁnite and well-

deﬁned energy, momentum, and angular momentum, parceled out at breakup

by the system while respecting the conservation laws. After the component

subsystems are separated, the properties of each subsystem are completely

independent and do not depend on those of the other subsystems.

On the other hand, quantum mechanics is nonlocal, meaning that the

component parts of a quantum system may continue to inﬂuence each other,

even when they are well separated in space and out of speed-of-light contact.

This unexpected characteristic of standard quantum theory was ﬁrst pointed

out by Albert Einstein and his colleagues Boris Podolsky and Nathan Rosen

(EPR) in 1935, in a critical paper[2] in which they held up the discovered

nonlocality as a devastating ﬂaw that, it was claimed, demonstrated that

the standard quantum formalism must be incomplete or wrong. Einstein

called nonlocality “spooky actions at a distance”. Schr¨odinger followed on the

discovery of quantum nonlocality by showing in detail how the components

of a multi-part quantum system must depend on each other, even when they

are well separated[3].

Beginning in 1972 with the work of Stuart Freedman and John Clauser[4],

a series of quantum-optics EPR experiments testing Bell-inequality violations[5]

and other aspects of linked quantum systems were performed. These ex-

perimental results can be taken as demonstrating that, like it or not, both

quantum mechanics and the reality it describes are intrinsically nonlocal.

Einstein’s spooky actions-at-a-distance are really out there in the physical

world, whether we understand and accept them or not.

How and why is quantum mechanics nonlocal? Nonlocality comes from

two seemingly conﬂicting aspects of the quantum formalism: (1) energy, mo-

mentum, and angular momentum, important properties of light and matter,

are conserved in all quantum systems, in the sense that, in the absence of

external forces and torques, their net values must remain unchanged as the

system evolves, while (2) in the wave functions describing quantum systems,

as required by the uncertainty principle, the conserved quantities are often

indeﬁnite and unspeciﬁed and typically can span a large range of possible val-

ues. This non-speciﬁty persists until a measurement is made that “collapses”

the wave function and ﬁxes the measured quantities with speciﬁc values.

These seemingly inconsistent requirements of (1) and (2) raise an important

question: how can the wave functions describing the separated members of a

system of particles, which may be light-years apart, have arbitrary and un-

3

speciﬁed values for the conserved quantities and yet respect the conservation

laws when the wave functions are collapsed?

This paradox is accommodated in the formalism of quantum mechanics

because the quantum wave functions of particles are entangled, the term

coined by Schr¨odinger to mean that even when the wave functions describe

system parts that are spatially separated and out of light-speed contact, the

separate wave functions continue to depend on each other and cannot be

separately speciﬁed. In particular, the conserved quantities in the system’s

parts (even though individually indeﬁnite) must always add up to the values

possessed by the overall quantum system before it separated into parts.

How could this entanglement and preservation of conservation laws pos-

sibly be arranged by Nature? The mathematics of quantum mechanics gives

us no answers to this question, it only insists that the wave functions of sepa-

rated parts of a quantum system do depend on each other. Theorists prone to

abstraction have found it convenient to abandon the three-dimensional uni-

verse and describe such quantum systems as residing in a many-dimensional

Hilbert hyper-space in which the conserved variables form extra dimensions

and in which the interconnections between particle wave functions are rep-

resented as allowed sub-regions of the overall hyper-space. That has led to

elegant mathematics, but it provides little assistance in visualizing what is

really going on in the physical world.

In this paper, for reasons of space and focus, we will not attempt to ac-

count for nonlocality by considering any interpretation of quantum mechan-

ics. We will simply note that the transactional interpretation[6] of quantum

mechanics, introduced by one of the authors in 1986, seems to be unique

among the plethora of interpretations of the quantum formalism in provid-

ing a deﬁnite mechanism that accounts for nonlocality and facilitates visu-

alization of nonlocal processes. Here we will take the existence of quantum

nonlocality and entanglement as established facts and consider their impli-

cations.

3 Quantum Nonlocality and Communication

Given that a measurement on one part of an extended quantum system can

aﬀect the outcomes of measurements performed in other distant parts of

the system, the question that naturally arises is: can this phenomenon be

used for communication between one observer and another? Demonstration

of such nonlocal quantum communication would be a truly “game-changing”

discovery, because it would break all the rules of normal communication. No

energy would pass between the send and receive stations; the acts of sending

and receiving could occur in either order and would depend only on the

instants at which the measurements were made; there would be no deﬁnite

signal-propagation speed, and messages could eﬀectively be sent faster than

light-speed, or “instantaneously” in any chosen reference frame, or even, in

principle, backwards in time.

The average member of the physics community, if he has any opinion

about nonlocal communication at all, believes it to be impossible, in part

because of its superluminal and retrocausal implications. Indeed, over the

4

years a number of authors have presented “proofs” that nonlocal observer-

to-observer communication is impossible within the formalism of standard

quantum mechanics[7]. These theorems assert that in separated measure-

ments involving entangled quantum systems, the quantum correlations will

be preserved but there will be no eﬀect apparent to an observer in one sub-

system if the character of the measurement is changed in the other sub-

system. Thus, it is asserted, nonlocal signaling is impossible.

However, Peacock has pointed out[8] that the “proofs” ruling out non-

local signaling are in some sense tautological, assuming that the measure-

ment process and its associated Hamiltonian are local, thereby building

the ﬁnal no-signal conclusion into the starting assumptions. Standard quan-

tum mechanical Bose-Einstein symmetrization has been raised as a counter-

example, shown to be inconsistent with the initial assumptions of such no-

signal “proofs”. Therefore, it seems reasonable to entertain the possibility of

nonlocal communication and to consider possible gedankenexperiments that

might implement it. One successful superluminal experiment would trump

all theoretical impossibility proofs.

We note that it is also sometimes asserted that nonlocal communication is

not possible because it would conﬂict with special relativity. This assertion is

incorrect. The prohibition of signals with superluminal speeds by Einstein’s

theory of special relativity is related to the fact that a condition of deﬁnite

simultaneity between two separated space-time points is not Lorenz invari-

ant. Assuming that some hypothetical superluminal signal could be used to

establish a ﬁxed simultaneity relation between two such points, e.g., by clock

synchronization, this would imply a preferred inertial frame and would be

inconsistent with Lorenz invariance and special relativity. In other words,

superluminal signaling would be inconsistent with the even-handed treat-

ment of all inertial reference frames that is the basis of special relativity.

However, if a nonlocal signal could be transmitted through measurements

at separated locations performed on two entangled photons, the signal would

be “sent” at the time of the arrival of the photon in one location and “re-

ceived” at the time of arrival of the other photon, both along Lorenz-invariant

lightlike world lines. By varying path lengths to the two locations, these

events could be made to occur in any order and time separation in any ref-

erence frame. Therefore, nonlocal signals (even superluminal and retrocausal

ones) could not be used to establish a ﬁxed simultaneity relation between two

separated space-time points, because the sending and receiving of such sig-

nals do not have ﬁxed time relations. Nonlocal quantum signaling, if it were

to exist, would be completely compatible with special relativity. (However,

it would probably not be compatible with macroscopic causality.)

4 Polarization-Entangled EPR Experiments and Nonlocal

Communication

First, let us examine a fairly simple EPR experiment exhibiting nonlocality.

Following Bell[5], a number of EPR tests[4,9] have exploited the correla-

tions of polarization-entangled systems that arise from angular momentum

conservation. Their results, to accuracies of many standard deviations, are

5

Fig. 1 (color online) An EPR two-photon experiment using linear polarization

entanglement.

consistent with the predictions of standard quantum mechanics and can be

interpreted as falsifying many local hidden variable alternatives to quantum

mechanics.

A modern version of this type of EPR experiment is shown in Fig. 1.

Two observers, Alice and Bob, operate polarimeters measuring the linear

polarization (H or V) of individual photons and record photon detections.

The H-V plane of Alice’s polarimeter can be rotated through an angle θwith

respect to the plane of Bob’s polarimeter, so that the basis of her polarization

measurements can be changed relative to Bob’s. Here the source of photons is

taken to be a Sagnac entangled two-photon source of the type developed by

the Zeilinger Group[10], in which the degree of entanglement can be varied

by rotating a half-wave plate in the system, as characterized by the variable

α, producing a two-particle wave function of the general form:

Ψ(α) = (|HAi |HBi+|VAi |VBi)(cos β+ sin β)/2

+i(|HAi | VBi− | VAi | HBi)(cos β−sin β)/2 where β=α−π/4.

The degree of photon-pair entanglement from this source is adjustable. When

α= 0, the two-photon polarization entanglement is 100% in a pure Bell state

with the wave function Ψ(0) = i(|HAi |VBi− |VAi |HBi)/√2; when α=π/4

the entanglement is 0 in a non-entangled product state with Ψ(π/4) = [(|

HAi − i|VAi)×(|HBi+i|VBi)]/2.

Let us assume that we set α= 0 for 100% entanglement. When θis

zero and the polarimeters are aligned, there will be a perfect anti-correlation

between the polarizations measured by Alice and by Bob. The random po-

larization (H or V) that Alice measures will always be the opposite of that

measured by Bob (HAVBor VAHB). However, when θis increased, the per-

fect HAVBand VAHBanti-correlations are degraded and correlated detec-

tions HAHBand VAVB, previously not present, will begin to appear. Local

theories require that for small θrotations this correlation degradation should

increase linearly with θ, while quantum mechanics predicts that it should in-

crease as θ2, i.e., quadratically[11]. This is the basis of Bell’s Inequalities[5],

counting ratio inequalities that are observed for linear behavior in θbut are

dramatically violated for the quadratic behavior characteristic of quantum

mechanics.

6

Fig. 2 (color online) Joint detection probabilities vs. θfor the four detector com-

binations with: α= 0 (red/solid, 100% entangled),α=π/8 (green/dashed, 71%

entangled), and α=π/4 (blue/dot-dashed, 0% entangled)

The quantum mechanical analysis of this system is fairly simple because,

assuming that the entangled photons have a single spatial mode, their trans-

port through the system can be described by considering only the phase shifts

and polarization selections that the system elements create in the waves. We

have used the formalism of Horne, Shimony and Zeilinger[12] to perform such

an analysis and calculate the joint wave functions for simultaneous detections

at both detectors[15]. These are:

ΨHH (α, θ) = [−sin(α) cos(θ) + icos(α) sin(θ)]/√2 (1)

ΨHV (α, θ) = [−cos(α) cos(θ) + isin(α) sin(θ)]/√2 (2)

ΨV H (α, θ) = [cos(α) cos(θ)−icos(α) sin(θ)]/√2 (3)

ΨV V (α, θ) = [sin(α) cos(θ)−icos(α) sin(θ)]/√2.(4)

The corresponding joint detection probabilities are:

PHH (α, θ) = [1 −cos(2α) cos(2θ)]/4 (5)

PHV (α, θ) = [1 + cos(2α) cos(2θ)]/4 (6)

PV H (α, θ) = [1 + cos(2α) cos(2θ)]/4 (7)

PV V (α, θ) = [1 −cos(2α) cos(2θ)]/4.(8)

Fig. 2 shows plots of these joint detection probabilities vs. θfor the four de-

tector combinations with: α= 0 (100% entangled), α=π/8 (71% entangled),

and α=π/4 (0% entangled) .

7

Now consider the question of whether, at any setting of α, observer Al-

ice by operating the left system and varying θcan send a nonlocal signal

to observer Bob operating the right system. Some overall observer who is

monitoring the coincidence counting rates HAHB,VAVB,HAVB, and VAHB

could reproduce Fig. 2 and would have a clear indication of when θwas var-

ied by Alice, in that the relative rates would change dramatically. However,

observer Bob is isolated at the system on the right and is monitoring only the

two singles counting rates HB≡HAHB+VAHBand VB≡HAVB+VAVB.

Bob would observe the probabilities PBH (α, θ ) = PHH (α, θ)+PV H (α, θ) =

1

2and PBV (α, θ) = PH V (α, θ) + PV V (α, θ) = 1

2, both independent of the val-

ues of αand θ. Thus, Bob would see only counts detected at random in one

or the other of his detectors with a 50% chance of each polarization, and

his observed rates would not be aﬀected by the setting of θ. The late Heinz

Pagels, in his book The Cosmic Code[13], examined in great detail the way in

which the intrinsic randomness of quantum mechanics blocks any potential

nonlocal signal in this type of polarization-based EPR experiment.

We emphasize the point that linear polarization is an interference eﬀect

of the photon’s intrinsic circularly-polarized spin angular momentum S= 1,

Sz=±1 helicity eigenstates. As we will see below, the interference com-

plementarity observed here is an example of the intrinsic complementarity

between two-photon interference and one-photon interference[14] that blocks

nonlocal communication. When two-photon interference is present in an en-

tangled system, it blocks the observation of any single photon interference

that depends on θ.

5 The Ghost Interference Experiment with momentum-entangled

photons

Although the entanglement of linear polarization is a very convenient medium

for EPR experiments and Bell-inequality tests, in many ways the alternative

oﬀered by path-entangled EPR experiments provides a richer venue. Per-

haps the earliest example of a path-entangled EPR experiment is the 1995

“ghost interference” experiment of the Shih Group at University of Maryland

Baltimore County[16]. Their experiment is illustrated in Figure 3.

Here a nonlinear BBO (β-BaB2O4) crystal pumped by a 351 nm argon-

ion laser produces co-linear pairs of momentum-entangled 702 nm photons,

one (extraordinary or “e”) polarized vertically and the other (ordinary or

“o”) polarized horizontally. A polarizing beam splitter (BS) directs the two

entangled photons to separate paths. The experimenters demonstrated that

when the pair of photons is examined in coincidence, passing the e-photon

through a double slit system before detection at D1produced either (1) a

“comb” interference distribution or (2) a “bump” diﬀraction distribution in

the position X2of the o-photon detected at D2, depending on whether (1)

both slits were open so the o-photon could take both paths through the

slits or (2) one of the slits was blocked, so that which-way information was

obtained about the o-photon.

Thus, on can make the interference pattern of the o-photon observed at

D2appear or disappear, depending on what is done to the e-photon. If the e-

8

Fig. 3 (color online) The “ghost interference” experiment of the Shih

Group/UMBC. If both slits are open for the e-photon, the o-photon produces a

2-slit interference pattern. If only one slit is open for the e-photon, the o-photon

produces a broad diﬀraction pattern.

photon is made to exhibit particle-like behavior by passing through only one

slit, the o-photon also exhibits particle-like behavior. If the e-photon is made

to exhibit wave-like behavior by passing through both slits and interfering,

the o-photon also exhibits the wave-like behavior of an interference pattern.

This suggests a paradox: that in a system with a momentum-entangled pho-

ton pair, a nonlocal signal might be sent from one observer to another by

controlling the presence or absence of an interference pattern. To send such a

signal, however, one would have to be able to see the interference in singles,

without a coincidence with detection of the other member of the photon pair.

The Shih Group, however, reported that no interference pattern was observed

in singles in their experiment.

6 The Dopfer Experiment with momentum-entangled photons

Another path-entangled EPR experiment was the 1999 PhD thesis of Dr.

Birgit Dopfer at the University of Innsbruck[17], performed under the direc-

tion of Prof. Anton Zeilinger. The Dopfer experiment is illustrated in Fig.

4.

9

Fig. 4 (color online) The Dopfer experiment of the Zeilinger Group/Innsbruck. If

detector D2is a distance ffrom the lens, detector D1observes a 2-slit interference

pattern. If detector D2is a distance 2ffrom the lens, detector D1observes a broad

diﬀraction pattern.

Here a nonlinear LiIO3crystal pumped by a 351 nm laser produces

momentum-entangled pairs of 702 nm photons and selects pairs that emerge

from the crystal at angles of 28.2oto the right and left of the pump axis.

The lower photon in the diagram passes through 2-slit system S1and is de-

tected by single-photon detector D1. The upper photon passes through a lens

of focal length fand is detected by single-photon detector D2. The system

geometry is arranged so that the distance from S1to the crystal plus the

distance from the upper lens to the crystal add to a total distance of 2f.

Beyond the upper lens, detector D2can be positioned either (Case 1) at a

distance of 2ffrom the lens or (Case 2) at a distance of ffrom the lens. It is

observed that for Case 2 an interference pattern is observed at D2, while for

Case 1 there is no interference pattern, but only a broad aperture-diﬀraction

distribution.

Dopfer demonstrated that for Case 1, the position distribution measured

by detector D2showed two sharp spikes, which were interpreted as “ghost”

images of the slits at S1. The slit-lens-detector geometry was such as to pro-

duce a 1:1 image, and momentum entanglement caused a right-going photon

in the lower system to be mirrored by a left-going photon in the upper system.

Thus, in Case 1 detector D2in eﬀect was measuring which path the lower

photon took through the slit system S1and forcing particle-like behavior in

both photons that suppressed the interference pattern.

In Case 2 the distributions measured by detectors D1and D2were both

two-slit interference patterns. Detector D2was placed in the “circle of confu-

sion” region of the lens where no image was formed and virtual rays from both

slits would overlap, resulting in interference. Thus, in Case 2 both photons

of the entangled pair exhibited wave-like behavior and formed interference

patterns.

Therefore, one can make the interference pattern at detector D1appear

or disappear, depending on the location of detector D2. Again, this suggests

that in a system with a momentum-entangled photon pair, a nonlocal signal

10

Fig. 5 (color online) A Mach-Zehnder interferometer.

might be sent from one observer to another by controlling the presence or

absence of an interference pattern.

Examination of the ghost-interference and Dopfer experiments raises a

very interesting question: Can the coincidence requirement be removed? The

answer is subtle. In principle, the two entangled photons are connected by

nonlocality whether they are detected in coincidence or not. The coincidence

may perhaps be removable. However, in both experiments the authors re-

ported that no two-slit interference distribution was observed when the co-

incidence requirement was removed.

These considerations lead to a new quantum mechanical paradox: it ap-

pears possible that nonlocal observer-to-observer signals can be transmitted

by controlling the presence or absence of an interference pattern by forcing

wave-like or particle-like behavior on an entangled photon pair.

7 Two-Slit Problems and Improved Experiments

From the point of view of moving to a path-entanglement situation in which

the coincidence requirement could be relaxed, the problem with both of the

experiments discussed above is that their use of a two-slit system blocks and

absorbs most of the photons from the nonlinear crystal that illuminate the slit

system. Further, the down-conversion process is intrinsically very ineﬃcient

(∼1 photon pair per 108pump photons). An additional complication is that

most detectors capable of detecting individual photons are intrinsically noisy

and somewhat ineﬃcient. For these reasons, there is a large advantage in

using all of the available entangled-photon pairs in any contemplated path-

interference test of nonlocal communication.

The Mach-Zehnder interferometer[18], as illustrated in Fig. 5, provides an

interesting alternative to a two-slit interference system. Here, a light beam is

sent on two paths and recombined. At each reﬂection at a splitter or mirror,

11

Fig. 6 (color online) A path-entangled dual interferometer experiment.

the waves receive a 90ophase shift. For the waves emerging horizontally, the

waves that took Path H and Path V were both reﬂected twice, so they should

have the same phase and should reinforce. However, for the waves emerging

vertically, the waves on the Path V were reﬂected three times while the waves

on Path H were reﬂected only once, so they should emerge 180oout of phase

and should interfere destructively and cancel. Thus, if the path lengths and

split fractions are precisely equal, the waves should emerge only on the path

that is parallel to the original beam. In the Mach-Zehnder interferometer

interference is achieved using all the photons entering the device.

Fig. 6 shows a path-entangled experimental test using Mach-Zehnder in-

terferometers. This type of system was originally developed by the Zeilinger

Group at the Institute for Quantum Optics and Quantum Information, Vienna[19].

Here, the interferometers are a variant of the basic Mach-Zender design that

uses an initial polarizing beam splitter (P B SA,B) that directs the vertical

(v) and horizontal (h) linear polarizations to diﬀerent paths and then con-

verts horizontal to vertical polarization on the upper path with a half-wave

plate (HWA,B ). This has the eﬀect of converting polarization entanglement

from the source to path entanglement and placing waves on both paths in

the same polarization state, so that they can interfere. Again observers, Alice

and Bob operate the interferometers and count and record individual photon

detections. A phase shift element (φA,B ) allows the observers to alter the

phase of waves on the upper path.

As in the EPR example, the source of photons is taken to be the Sagnac

entangled two-photon source developed by the Zeilinger Group[10], in which

the degree of entanglement depends on the value of αin this setup. Following

the path separation the extended two-particle wave function has the general

form:

Ψ(α) = (|a1i |b1i+|a2i |b2i)(cos β+ sin β)/2

+i(|a1i | b2i− | a2i | b1i)(cos β−sin β)/2 where β=α−π/4. When

α=π/2, the two-photon wave function is a fully path-entangled Bell state

of the form Ψ(π/2) = (|a1i | b1i+|a2i | b2i)/√2, and when α=π/4 the

path entanglement is 0 and the wave function is a product state of the form

Ψ(π/4) = (|a1i − i|a2i)×(|b1i+i|b2i)/2.

12

Alice’s last beam-splitter (BSA) is removable. When BSAis in place, the

two paths are remixed, the left-going photons exhibit the wave-like behavior

of being on both paths, and two-path overlap and Mach-Zehnder interfer-

ence will be present. When BSAis removed, path detection occurs, the left-

going photons exhibit the particle-like behavior of being on a path ending

at detector DA0or at detector DA1, so that Alice’s measurements provide

which-way information about both photons. Bob’s last beam splitter (BSB)

remains in place. and, in the absence of decoherence eﬀects, should measure

Mach-Zehnder interference.

This experiment is thus the equivalent of the ghost-interference experi-

ment and the Dopfer experiment described above, in that it embodies entan-

gled paths and two-path interference. However, it improves on those exper-

iments by using all of the available entangled photons and by employing a

source that has an adjustable entanglement.

It has been argued[20, 21] that this situation presents a nonlocal signaling

paradox, in that Alice. by choosing whether BSAis in or out, can cause the

Mach-Zehnder interference eﬀect to be present or absent in Bob’s detectors.

In particular, with BSAout we expect particle-like behavior, and Bob should

observe equal counting rates in DB1and DB0. With BSAin we expect wave-

like behavior, and Bob, for the proper choice of φB, should observe all counts

in DB1and no counts in DB0due to Mach-Zehnder interference. It was

further argued[21] that possibly the nonlocal signal might be suppressed by

the complementarity of entanglement and coherence[22], but by arranging for

71% entanglement and 71% coherence (i.e., α=π/8 for the Sagnac source),

a nonlocal signal might be permitted.

As in the EPR example discussed above, the quantum mechanical analysis

of this system is fairly simple because, assuming that the entangled photons

have a single spatial mode, their transport through the system can be de-

scribed by considering the phase shifts that the system elements create in

the waves. To test the validity of the above arguments, we have used the

formalism of Horne, Shimony and Zeilinger[12] in Mathematica 9 to ana-

lyze the dual-interferometer conﬁguration[23] and to calculate the joint wave

functions for detections of the entangled photon pairs in varous combinations.

For BSAin, these are:

ΦA1B1(α, φA, φB) = [icos(α)(eiφA−eiφB)

+ sin(α)(1 + ei(φA+φB))]/(2√2) (9)

ΦA1B0(α, φA, φB) = [−cos(α)(eiφA+eiφB)

+isin(α)(1 −ei(φA+φB))]/(2√2) (10)

ΦA0B1(α, φA, φB) = [cos(α)(eiφA+eiφB)

+isin(α)(1 −ei(φA+φB)]/(2√2) (11)

ΦA0B0(α, φA, φB) = [icos(α)(eiφA−eiφB)

−sin(α)(1 + ei(φA+φB)]/(2√2).(12)

13

Fig. 7 (color online) Bob’s non-coincident singles detector probabilities PB1(α, φB)

and PB0(α, φB) (Eqns. 17 and 18) for α= 0 (red/solid, 100% entangled), α=π/8

(green/dash, 71% entangled), and α=π/4 (blue/dot-dash, 0% entangled).

The corresponding joint detection probabilities are:

PA1B1(α, φA, φB) = {1−sin(φA)[sin(2α) + sin(φB)]

−cos(2α) cos(φA) cos(φB) + sin(2α) sin(φB)}/4 (13)

PA1B0(α, φA, φB) = {1−sin(2α)[(sin(φA) + sin(φB)]

+ cos(2α) cos(φA) cos(φB) + sin(2α) sin(φB)}/4 (14)

PA0B1(α, φA, φB) = {1 + sin(2α)[(sin(φA) + sin(φB)]

+ cos(2α) cos(φA) cos(φB) + sin(2α) sin(φB)}/4 (15)

PA0B0(α, φA, φB) = {1−sin(φB)[sin(2α) + sin(φA)]

−cos(2α) cos(φA) cos(φB) + sin(2α) sin(φA)}/4.(16)

The non-coincident singles detector probabilities for Bob’s detectors are

obtained by summing over Alice’s detectors, which he does not observe. Thus

PB1(α, φB)≡PA1B1(α, φA, φB) + PA0B1(α, φA, φB)

= [1 + sin(2α) sin(φB)]/2 (17)

PB0(α, φB)≡PA1B0(α, φA, φB) + PA0B0(α, φA, φB)

= [1 −sin(2α) sin(φB)]/2.(18)

Note that these singles probabilities have no dependences on Alice’s phase

φA.

Fig. 7 shows plots of Bob’s non-coincident singles detector probabilities

PB1(α, φB) and PB0(α, φB) for the cases of α= 0 (100% entangled), α=π/8

(71% entangled), and α=π/4 (not entangled).

We see here a demonstration of the complementarity of entanglement

and coherence[22], in that the probabilities for fully entangled system are

constant independent of φBbecause the absence of coherence suppresses

the Mach-Zehnder interference, while the unentangled system shows strong

Mach-Zehnder interference. The α=π/8 case also shows fairly strong Mach-

Zehnder interference and raises the intriguing possibility that a nonlocal sig-

nal might survive.

14

Therefore, the question raised by the possibility of nonlocal signaling

is: What happens to Bob’s detection probabilities when Alice’s beam split-

ter BSAis removed? To answer this question, we re-analyze the dual in-

terferometer experiment of Fig. 6 with BSAin the “out” position. These

calculations[25] give the joint wave functions for simultaneous detections of

detector pairs:

ΨA1B1(α, φA, φB) = [sin(α)−ieiφBcos(α)/2 (19)

ΨA1B0(α, φA, φB) = [isin(α)−eiφBcos(α)]/2 (20)

ΨA0B1(α, φA, φB) = [eiφA(cos(α)−ieiφBsin(α)]/2 (21)

ΨA0B0(α, φA, φB) = [ieiφA(cos(α) + ieiφBsin(α)]/2.(22)

The corresponding joint detection probabilities are:

PA1B1(α, φA, φB) = [1 + sin(2α) sin(φB)]/4 (23)

PA1B0(α, φA, φB) = [1 −sin(2α) sin(φB)]/4 (24)

PA0B1(α, φA, φB) = [1 + sin(2α) sin(φB)]/4 (25)

PA0B0(α, φA, φB) = [1 −sin(2α) sin(φB)]/4.(26)

The non-coincident singles detector probabilities for Bob’s detectors are iden-

tical to the singles detector probabilities of Eqns. 17 and 18 obtained when

BSAwas in place.

The conclusion is that no nonlocal signal can be sent by inserting and

removing BSA. We have also found that even when the left-going photons

from the source are intercepted before entering Alice’s interferometer with

a black absorber, Bob will observe the same singles counting rates given by

Eqns. 17 and 18. As in the EPR case, the intrinsic complementarity of two-

photon and one-photon interference/citeJa93 has erased the nonlocal signal.

8 The Wedge Modiﬁcation and Nonlocal Communication

A possible reason that all of the above attempts at nonlocal communication

have failed is that the left-going photons are directed to both of Alice’s de-

tectors, the two having complementary interference proﬁles, so that when

these proﬁles are added the potential nonlocal signal is erased. Suppose that

instead we direct all the photons on both paths to a single detector, where

they will have only one interference proﬁle. Could this change permit nonlo-

cal signaling? To investigate this question we have analyzed the experiment

shown in Fig. 8.

Here, we have replaced Alice’s last beam splitter and detectors with a

45owedge mirrorWAthat directs the left-going photons on paths a1and a2

to a single detector DA. We assume that the angles of Alice’s mirrors are

tweaked slightly so that the two beams have a maximum overlap at DAand

that WAis positioned so that it reﬂects all of the two beams except their

extreme tails (∼10σ). Also, a removable beam stop has been placed in the

path of the left-going photons near the source. As stated above, when the

left-going photons from the source are intercepted by such a beam stop, the

15

Fig. 8 (color online) Wedge modiﬁcation of the path-entangled dual interferometer

experiment.

non-coincident singles probabilities for Bob’s detectors will be given by Eqns.

17 and 18. We wish to investigate the question of whether Bob will observe

any change in the counting rates of his detectors that depends on whether

the beam stop is in or out.

Naively it might appear that the new conﬁguration would produce a large

change in Bob’s counting rates, because Alice could choose a phase φAfor

which the left-going waves arriving at DAwill interfere destructively and

vanish or will interfere constructively and produce a maximum. Arguments

along these lines have been advanced by Anwar Sheikh[24] to justify a clever

(but ﬂawed) one-photon FTL communication scheme. However, such expec-

tations cannot be true, because they would violate quantum unitarity and

the requirement that any left-going photon must be detected somewhere with

100% probability. Unitarity (or equivalently, energy conservation) requires

that any wave-mixing device that produces destructive interference some-

where must produce a precisely equal amount of constructive interference

somewhere else. The 45owedge is no exception.

The ﬂaw in such cancellation arguments is that in the previous examples

we have always dealt with conﬁgurations in which only a single spatial mode

of the photon is present. In that case, superposition can be used without

considering wave trajectories, since the wave front for any given path arrives

at a detector with a constant overall phase. In the present conﬁguration, the

spatial proﬁles of the waves on Alice’s two paths are truncated at the apex

of the wedge mirror and also must propagate in slightly diﬀerent directions

in order to overlap at the detector. Therefore, the phase of arriving waves is

not constant and will depend on the location on the detector face. Therefore,

simple position-independent superposition cannot be used.

Instead, in order to calculate the diﬀerential probability of detection at a

speciﬁc location on the face of detector DA, one must propagate the waves

from the wedge to the detector by doing a path integral of Huygens wavelets

originating across the eﬀective aperture of the wedge. To get the overall

detection probability, one must then integrate over locations on the detector

face.

The analysis of the wedge system is therefore much more challenging that

those of the previous examples. While analytic expressions can be obtained

16

Fig. 9 (color online) Magnitudes of the wave functions Ψa1(red/solid) and Ψa2

(blue/dotted) as functions of position xon the face of detector DA.

for the diﬀerential probability of two-particle detection with one of Bob’s

detectors and at some speciﬁc lateral position on DA, the integration of

that diﬀerential probability, a highly oscillatory function, over the face of

DAcannot be done analytically. Thus the analysis cannot produce equations

predicting Bob’s single counts that can be directly compared with Eqns. 17

and 18 for the signal test. Instead one must subtract the results of numerical

integration from evaluations of Eqns. 17 and 18 using the same values for α,

φA, and φBused in the numerical integration, and observe how close to zero

is the calculated diﬀerence (which represents the potential nonlocal signal).

We have performed this analysis[26] of the experiment shown in Fig. 8,

tweaking the mirror angles for maximum overlap of the waves on the two

paths to detector DA. The calculation gives large analytical expressions for

joint detection probability as a function of position on detector DA, but these

must be integrated numerically to obtain the position-independent probabil-

ities. Here Fig. 9 shows the overlap of the magnitudes of the wave functions

for paths a1and a2vs. position. The wave functions have a basic Gaussian

proﬁle with oscillations arising from the truncation of one Gaussian tail by

WA.

Fig. 10 shows the corresponding probabilities for α= 0 (e.g., fully entan-

gled) of coincident photon pairs at Alice’s detector DAand at Bob’s detectors

DB1and DB0. The probabilities are highly oscillatory because of the inter-

ference of the two waves and the phase walk of the wave functions with angle,

analogous to two-slit interference.

To test the possibility of a nonlocal signal, we must integrate these proba-

bilities over the extend to the detector face and calculate diﬀerence functions

from these results and similar evaluations of Eqns. 17 and 18. We can expect

17

Fig. 10 (color online) Probabilities of coincident detections at DAand DB1

(red/solid) and at DAand DB0(blue/dotted) with α= 0, φA= 0, and φB= 0.

Fig. 11 (color online) Diﬀerence between numerical singles probabilities and eval-

uations of Eqns. 17 and 18. Here the regions labeled “A” reach minima of 5.7×10−7,

the regions labeled “B” reach maxima of 6.08 ×10−6, and the regions labeled “C”

reach maxima of 5.51 ×10−6. Small blotches indicate regions in which numerical

integration has produced errors.

errors in numerical integration due to the oscillation shown in Fig. 10. The

diﬀerence functions as 2-D contour plots in φBvs. αare shown in Fig. 11.

Thus, the diﬀerences between the probabilities predicted by of Eqns. 17

and 18 and the numerically-integrated probabilities of Fig. 10 are on the

order of a few parts per million. This is equivalent to saying that they are

the same, and that no nonlocal signal is possible using the wedge-modiﬁed

conﬁguration of Fig. 8.

18

9 Conclusions

We have investigated the possibility of nonlocal quantum signaling by ana-

lyzing several path-entangled systems. The conjecture[21] that nonlocal sig-

naling might be possible by adjusting the entanglement to 71% to permit

coherence has proved to be incorrect. Instead, we ﬁnd that in all cases inves-

tigated the intrinsic complementarity between two-photon interference and

one-photon interference[14] blocks any potential nonlocal signal.

Our conclusion is that no nonlocal signal can be transmitted from Alice

to Bob by varying Alice’s conﬁguration in any of the ways discussed here.

Nature appears to be well protected from the possibility of nonlocal signaling.

Acknowledgements This work was supported in part by the U. S. Department

of Energy Oﬃce of Scientiﬁc Research. We are grateful to Prof. Anton Zeilinger,

Dr. Radek Lapkiewicz, Prof. Gerald Miller, Prof. Yahuna Shih, and Prof. James

Woodward for valuable comments, suggestions, and criticisms during the course of

this work.

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