Counterfactuality of \counterfactual" communication

Counterfactuality of \counterfactual" communication L. Vaidman Raymond and Beverly Sackler School of Physics and Astronomy Tel-Aviv University, Tel-Aviv 69978, Israel The counterfactuality of the recently proposed protocols for direct quantum communication is analyzed. It is argued that the protocols can be counterfactual only for one value of the transmitted bit. The protocols achieve a reduced probability of detection of the particle in the transmission channel by increasing the number of paths in the channel. However, this probability is not below the probability of detection of a particle actually passing through such a multi-path channel which was found to be surprisingly small. I. INTRODUCTION Penrose [1] coined the term \counterfactuals" for de- scribing quantum interaction-free measurements (IFM) [2]. Counterfactuals are things that might have happened, although they did not in fact hap- pen. Penrose 1994 He argued that in the IFM, an object is found because it might have absorbed a photon, although actually it did not. The idea of the IFM has been applied to \counter- factual computation" [3], a setup in which one particular outcome of a computation becomes known in spite of the fact that the computer did not run the algorithm. Noh [4] created counterfactual cryptography, a method for se- cret key distribution using events in which the particle was not present in the transmission channel. Noh used a random choice of orthogonal input states (like in [5]) in contrast with the non-orthogonal states of BB84 crypto- graphic protocol [6]. It was argued [7], that a modication of the counterfac- tual computation proposed above which includes quan- tum Zeno eect can achieve counterfactuality for all out- comes of the computation. Recently, this idea has been used for performing \counterfactual communication" [8], which supposedly allowed to send information from Bob to Alice without transferring any particle between them. The transmission happens in a counterfactual way: the mere possibility of transmitting the particle allows trans- mitting the value of the bit. I nd all these results very paradoxical: they contra- dict physical intuition of causality: information is usually transmitted continuously in space. I argued [9] that to re- solve the paradoxical feature of the IFM one has to adopt the many-worlds interpretation (MWI) of quantum me- chanics [10, 11] in which the particle touches the object in a parallel world restoring causality at least within the complete physical universe which includes all the worlds. However, I believe that a protocol which can transmit both values of a bit without any particle present in the transmission channel is impossible irrespective of the in- terpretation of quantum mechanics one adopts. I have expressed this opinion already [12, 13], but more pro- tocols were suggested [14] and the controversy remains open [15{17]. The clarication of these conceptual issues is particularly important due to the recent increasing in- terest in the applications of counterfactual protocols [18{ 26]. Here I discuss this issue in more detail and try to resolve the controversy. The plan of the paper is as follows. In Section II I introduce the general setup of quantum communication protocols. In Sections III and IV I describe two recent protocols which are claimed to be counterfactual. In Sec- tion V I analyze various possible denitions of counterfac- tuality and argue that the best criteria is the magnitude of the trace left in the transmission channel. In Sections VI I calculate the trace left by a single particle present in the channel, i.e. the trace of a non-counterfactual com- munication protocol. In Sections VII-IX I show that the trace in the protocols claimed to be \counterfactual" is not less than the trace in a non-counterfactual protocol. In Section X I compare security of \counterfactual" pro- tocols of bit 0 with counterfactual protocols of bit 1. I summarize the results in Section XI. II. COMMUNICATION WITH QUANTUM PARTICLES There is a surprisingly low bound on the number of bits which can be sent using 1 qubit: The Holevo bound of 1, when the qubit is not entangled [27], and 2, when entanglement is allowed [28]. This is when we transmit one particle with an internal structure of a qubit such as a polarization state of a photon. In this paper I analyze protocols in which the particle does not have an internal structure: the information is encoded in the presence or absence of the particle. Let Alice and Bob be on the two separate sides of a region, see Fig. 1. Bob has a mirror on his side which causes particles sent by Alice to be bounced back to her. For a bit value of 1, Bob places a shutter which absorb Alice's particles, while for 0, the shutter is absent. Quantum mechanics, via IFM, allows, at least some- times, to transmit the bit 1 in a counterfactual man- ner, i.e., without having any particle in the transmission channel, see Fig. 2. Alice arranges a Mach-Zehnder in- arXiv:1410.2723v2 [quant-ph] 3 Apr 2015 2 FIG. 1: Simple communication with a quantum particle. Alice sends a particle to Bob and knows the bit chosen by Bob through observation if the particle comes back to her or not. terferometer (MZI) tuned to destructive interference in one of the ports, say D1, such that one arm of the in- terferometer crosses the place where Bob's shutter might be. Detection of the particle in the dark port of the in- terferometer tells us with certainty that bit is equal to 1 (the shutter is present). The simplest argument that in this case the particle was not present in the transmission channel is: \If the particle were in the transmission channel, it could not be detected by Alice". In my view, this argument cannot be used for claims about quantum particles [29]. I, instead, suggest to rely on the fact that the particle does not leave any trace in the transmission channel. Note that counterfactual transmission of just one bit value can be achieved using a classical particle [30]. Alice and Bob agree in advance that at a particular time, for a particular value of a bit, Bob sends the particle to Alice, while for the other bit value, he sends nothing. Note, however, that this classical protocol cannot achieve the task of the quantum IFM. In the IFM, Alice learns about the shutter in Bob's place without prior agreement with Bob and without Bob knowing that she acquired this information. In the IFM shown in Fig. 2, Alice does not obtain a denite information about the classical bit 0. Without the shutter, the click in the bright port happens with certainty, but it might happen (with probability 25%) with the shutter too. This protocol is also not the most FIG. 2: A single bit value communication with IFM. The interferometer is tuned in such a way that detector D1 never clicks if the paths are free. Alice knows that Bob chose bit 1 (blocked the path) when she observes the click in D1. ecient method for communication of the bit 1. When the particle is detected by a bright port (probability 25%) we get no decisive information, and in half of the cases the particle is lost (then we get the information that the bit is 1 but not in a counterfactual manner). The quantum method can be modied to be a re- liable transmission of both values of the bit. To this end, instead of the shutter, Bob inserts a half-wave plate (HWP), see Fig. 3. This transforms the dark port to bright port and vice versa. However, half of the wave al- ways passes through the communication channel, so one cannot argue that this is a counterfactual communica- tion. Consider next what happens when we combine the quantum Zeno eect with the IFM [31]. It allows to per- form a counterfactual transmission of bit 1 with proba- bility arbitrary close to 1. The device consists of a chain of N interferometers with identical beam splitters with small transmittance T1 = sin2 , see Fig. 4a. Each one of the beam splitters in the chain performs the following unitary evolution of the state of the particle: jLi ! cos jLi + sin jRi; jRi ! 􀀀sin jLi + cos jRi: (1) A straightforward calculation shows that n beam splitters perform the following evolution of the wave packets of the particle entering the chain: jLi ! cos n jLi + sin n jRi; 3 FIG. 3: Communication with MZI and HWP. The interferometer is tuned to destructive interference towards D1. Bob communicates the bit to Alice by changing the destructive interference to detector D2 by inserting the HWP on the right path of the particle. jRi ! 􀀀sin njLi + cos n jRi: (2) For the particular choice of the transmittance parameter, = 2N , after passing all N 􀀀1 interferometers, the wave packet of the particle moves from one side to the other: jLi ! jRi, jRi ! 􀀀jLi. The Zeno eect takes place when the right arms of the interferometers are blocked, Fig. 4b. The state re- mains jLi with probability close to 1 when N is large, cos2N 2N ' 1 􀀀 2 4N . In summary, for bit 0 Bob does nothing and Alice gets the click with certainty at the right port in the detector D2. For bit 1 Bob blocks the interferometers and Alice gets the click with a very high probability in the detector D1. III. \DIRECT COUNTERFACTUAL QUANTUM COMMUNICATION" In this section I describe the recent protocol by Salih et al. [8] which followed the idea of counterfactual com- putation [7, 12]. Let us rst consider a MZI nested in another MZI, see Fig. 5. The inner interferometer is tuned for destructive interference toward the second beam splitter of the external interferometer, see Fig. 5a. The external interferometer is tuned for destructive in- terference towards D2 when the lower path of the inner interferometer is blocked, see Fig. 5b. This conguration provides (sometimes) denite information about value 0 of the bit, namely the absence of the shutter. Indeed, click in D2 is possible only if the shutter is not present. One can naively argue that Alice obtains this informa- tion in a counterfactual way, since the particle cannot pass through Bob's site and reach D2. However, as de- tailed in Section V, the presence of a weak trace inside the inner interferometer contradicts it. FIG. 4: Ecient communication using IFM and quantum Zeno eect. a). The chain of interferometers wih highly re-

ective beam splitters is tuned such that the particle has destructive

interference towards D1. b). Bob blocks the interferometers

and then, due to Zeno eect, the particle reaches

detector D1 with probability close to 1.

4

FIG. 5: Communication with nested MZIs. a). The inner

interferometer is tuned such that the particle cannot path

through the right arm of the external interferometer. b).

There is a destructive interference towards D1 when the right

path of the inner interferometer is blocked.

Salih et al. [8] further argued that a scheme with

numerous nested interferometers leads to a protocol for

transmitting both values of the bit in a counterfactual

way with an eciency which is arbitrarily close to 100%.

In the protocol, M 􀀀 1 chains of the N 􀀀 1 interferom-

eters described in Fig. 4 are parts of another chain of

interferometers with M beam splitters having transmit-

tance, T1 = sin2

2M . To simplify the analysis, I modify

Salih et al. protocol making it slightly less ecient, but

still good according to their line of argument. I replace

the mirrors of the external chain of interferometers by

highly re

ective beam splitters. Transmitted waves are

lost, but the modication balances the losses in the inner

chains when shutters are introduced, such that the states

of particles which are not lost are still described by the

same equation (2).

The setup is described in Fig. 6. The external chain of

interferometers has M beam splitters with transmittance

T2 = sin2

2M and the transmittance of the side beam

splitters serving as mirrors is T3 = 1 􀀀 cos2N

2N .

All right mirrors of the internal chains are in Bob's

territory. He knows that Alice sends a particle at a par-

ticular time at the top of the external chain in the state

jLi. For communicating bit 1, Bob blocks all inner in-

terferometers, see Fig. 6a. Then, after m beam splitters

of the external interferometer, the normalized quantum

state is

j (1)

m i = cos(m􀀀1)N

2N

cos

m

2M

jLi + sin

m

2M

jRi

+::: ;

(3)

and after all M beam splitters the state is

j (1)

M i = cos(M􀀀1)N

2N

jRi + ::: : (4)

In both equations ..." signify states orthogonal to the

term which is shown. If 1 M N, then the norm of

the leading term in (4) is close to 1:

cos2(M􀀀1)N

2N

' 1 􀀀

2M

4N

(5)

Thus, in the limit of large N, Bob's choice of bit 1 leads

to a click of Alice's detector D2. Since the state j (1)

M i

is orthogonal to the state jLi at the output port of the

interferometer, there is zero probability for a click of D1.

The particle can be lost in Alice's or Bob's territories,

and then none of the detectors click, but the probability

of such a case vanishes for large N.

If Bob wants to communicate the bit 0, instead, he

does nothing, Fig. 6b. Then, every wave packet enter-

ing any of the inner chains of the interferometers follows

evolution (2) inside this chain and does not come back

to the external interferometer. At the output of the in-

terferometer, the normalized quantum state is

j (0)

M i = cos(M􀀀1)N

2N

cosM

2M

jLi + ::: (6)

Under the condition 1 M N, the norm of the lead-

ing term in (6) is also close to 1:

cos2(M􀀀1)N

2N

cos2M

2M

' 1 􀀀

2

4

M

N

+

1

M

(7)

Then, the detection of the particle in the other port by

D1 tells Alice that Bob sent bit 1. Note, that the prob-

ability for a failure might become large if the condition

5

FIG. 6: \Direct counterfactual quantum communication" a). For bit 1 Bob blocks all inner interferometers. In this case

detector D2 clicks with probability close to 1, while D1 cannot click. b). For bit 0 Bob leaves all inner interferometers

undisturbed. For large N, D1 clicks with probability close to 1, while the probability for a click in D2 goes to zero.

6

1 M N is not fullled. The particle can be lost

or detected by D2. However, if the condition holds, the

probability for a failure is vanishingly small. The prob-

ability for loosing the particle and getting no result is of

order 2M

4N for bit 1 and 2

4

􀀀M

N + 1

M

for bit 0. The click

of D2 tells Alice with certainty that the bit is 1, and the

click of D1 tells that the bit is 0 with only a very small

probability for an error: about 2

4M2 . This is a good direct

communication protocol.

IV. \DIRECT QUANTUM COMMUNICATION

WITH ALMOST INVISIBLE PHOTONS"

As in the simple example in Sec. II (Fig. 3), using

HWPs instead of absorbing shutters leads to a communi-

cation protocol which is theoretically free of errors. Li et

al. [14] suggest such a protocol and argue that it has \ar-

bitrarily small probability of the existence of the particle

in the transmission channel".

The conguration is similar to the experiment of Salih

et al. [8]: a chain of M 􀀀 1 interferometers with inner

chains of N 􀀀 1 interferometers (but now M;N have to

be even numbers). Without absorbers, the evolution is

unitary and the Zeno eect is not used in this protocol.

There is no need to modify the protocol by replacing

mirrors with beam splitters, because there are no losses

to compensate.

The new protocol is dieret also in the transmittance

of the beam splitters in the inner chains. The parame-

ter is bigger by a factor of 2: =

N . As a result,

the chain (without the HWPs) works as two consecutive

inner chains of the protocol discussed in the previous sec-

tion. The rst half of the chain moves the wave packet

to the right side and the second brings it back to the left.

From (2) we obtain the transformation of the wave packet

in the inner chain of the interferometers jLi ! 􀀀jLi, see

Fig. 7a.

When the HWPs are inserted in every interferometer of

the inner chain, see Fig. 7b, they cause a phase change

of every state jRi and, consequently, every second beam

splitter reverses the operation of the previous one:

jLi ! cos jLi + sin jRi ! cos jLi 􀀀 sin jRi ! jLi:

(8)

Since every chain has an even number of beam splitters,

the transformation of the wave packet in the inner chain

is jLi ! jLi.

The setup for sending bit 0 is described on Fig. 8a.

Bob leaves the inner interferometers untouched. Then,

each inner chain of the interferometers changes the phase

of the quantum state of the particle: jLi ! 􀀀jLi. A

state jLi of the inner interferometer is a state jRi of the

external interferometer. Thus, the operation of the rst

external interferometer is

jLi ! cos jLi+sin jRi ! cos jLi􀀀sin jRi ! 􀀀jLi:

(9)

FIG. 7: The chain of interferometers with highly re

ective

beam splitters manipulated by the HWPs. a). The wave

packet of the particle moves from left to right and then to

left again but obtains the phase . b). HWPs \undo" the

transformation on every second interferometer and the wave

packet ends up in the original state on the left without additional

phase.

Since the number of beam splitters in the external chain

is even, at the end of the process the particle is on the

left side and it is detected by detector D1 with certainty.

If Bob wants to communicate the bit 1, instead, he in-

serts HWPs in all the interferometers of the inner chains

Fig. 8b. Now, after every two beam splitters of the in-

ner chain, the wave packet comes back to the left side

without acquiring additional phase. Thus, every inner

chain works as a mirror and the external chain of the

7

FIG. 8: \Direct quantum communication with almost invisible photons" a). The chains of the interferometers are tuned such

that D1 clicks with probability 1. b). If Bob inserts HWPs in all inner interferometers, then D2 clicks with probability 1.

8

interferometers moves the particle from left to right, to

be detected with certainty by detector D2. Alice knows

with certainty the choice of Bob by observing which de-

tector clicks. This is an ideal direct communication pro-

tocol: theoretically, when there are no losses, there is

zero probability for an error.

V. CRITERIA FOR COUNTERFACTUALITY

The question I want to answer in this paper is: Can the

protocols of Sections III and IV be considered as counter-

factual communication protocols? Let us consider the fol-

lowing three statements which try to capture the meaning

of counterfactuality.

1) The probability of nding the particle in the trans-

mission channel is zero or can be made arbitrarily small.

2) The particle did not pass through the transmission

channel.

3) The particle was not present in the transmission

channel (or the probability of its existence in the trans-

mission channel can be made arbitrarily small).

In the papers on counterfactual communication [8, 14]

all these statements were considered to be interchange-

able, i.e. all are true and each one of them represents

counterfactuality. I argue that the situation is more sub-

tle and clarication is needed.

Statement (1). A non-demolition measurement of the

presence of the particle in the transmission channel dis-

turbs completely the operation of the communication

protocols which are considered. When such a measure-

ment is present, Bob does not transmit information to

Alice by his actions. So, the truth or falsehood of this

statement is not a decisive indication of the counterfac-

tuality of the protocols.

However, since there is a separate controversy about

the validity of this statement for the two protocols un-

der discussion, it should be claried too. The papers on

counterfactual communication claim that this is a cor-

rect statement while I [13] claim that in these protocols

the probability of nding the particle in the transmission

channel is 1.

The source of this controversy is our dierent assump-

tions. The communication protocols involve preparation

and detection of the particle. I consider the probability

of nding the particle in the transmission channel under

the condition of the same nal detection as in the pro-

tocol without intermediate measurement. In this case,

the probability of nding the particle is exactly 1, since

had it it not been found, the result of the nal detec-

tion could not be that of the undisturbed protocol. On

the other hand, without the condition on the result of

the nal detection, the probability to detect the particle

in the transmission channel is vanishingly small. These

are two correct statements about the probability of nd-

ing the particle in the intermediate measurement: the

probability is 1 when both the proper preparation and

the proper nal detection are done, and it is vanishingly

small when only the preparation of the particle is as-

sumed. None of these statements shed much light on the

issue of counterfactuality of the protocols without inter-

mediate non-demolition measurements.

Statement (2). In contrast to such a claim for a clas-

sical particle, the meaning of this statement for a quan-

tum particle is not well dened. For a classical particle,

the operational meaning of (2) is (1), but as discussed

above, statement (1) is not helpful in the quantum case.

In a double slit experiment with a screen, there is no

good answer through which slit the particle passed and

through which it did not pass. However, if the detector

which nds the particle is placed after one of the slits, and

the wave packet passing through the other slit does not

reach the detector, then it is frequently declared, follow-

ing Wheeler [32], that the particle did not pass through

the second slit. (Note that this contradicts the textbook

picture, attributed to von Neumann, according to which

the wave passes through both slits and then collapses

to the location of the detector.) If we adopt Wheeler's

denition, then statement (2) is correct for the protocol

of Section III. The wave packets of the particle passing

through the transmission channel do not reach the detec-

tor which detects the particle in this protocol. I, however,

argued that we should not adopt Wheeler's denition for

discussing the past of a quantum particle [29].

The concept of a quantum particle passing through a

channel has no meaning in standard quantum mechanics,

since particles do not have trajectories. It is rigorously

dened in the framework of Bohmian mechanics. But due

to \surrealistic trajectories" [33, 34] this interpretation

is not useful for analyzing issues of communication. The

fact that a Bohmian trajectory does not pass through the

transmission channel does not tell us that Eve, which has

an access to this channel, cannot get some information

about this communication.

To summarize: statement (2) is not dened in standard

quantum mechanics. One has to add something else for

assigning meaning to (2). Adopting Bohmian trajecto-

ries or Wheeler's postulate makes (2) a correct statement,

but I argue that this does not help the analyses of com-

munication protocols.

Statement (3). Apart from Bohmian mechanics, quan-

tum mechanics does not provide a rigorous meaning also

for statement (3). It seems to me that without a clear on-

tological denition the way to proceed is to introduce an

operational meaning. We cannot rely on an operational

denition based on statement (1), since strong, even non-

demolition, measurements change the setup we want to

analyze. So, my proposal is to look at the weak trace the

particle leaves.

All particles interact locally with the environment. If

the particle is present in a particular place, it leaves some

trace there, and it does not leave any trace where it was

not present. Therefore, we can run the protocol we want

to analyze and then look at the trace left in the environ-

ment. If in a particular region there is no trace, we will

say that the particle was not there. Although very un-

9

FIG. 9: A single particle in a single localized wave packet

passes from Alice to Bob in the transmission channel. Some

trace is invariably left in the channel. We model it as a shift

of a local degree of freedom of the channel (the pointer) described

by (11).

likely, it is possible that the particle changed the local en-

vironment via some local interactions, but then changed

it back to its original state. I will use the variant of state-

ment (3) which is immune to this criticism: If a particle

left a trace in a particular place, one cannot claim that

the particle was not there.

Since we are all along analyzing interference experi-

ments, the trace left by the particles has to be small, as

otherwise the interference is destroyed. When the trace is

small one may argue that it can be neglected. I, however,

claim that it can be neglected only if it is small compare

to the trace which a single particle with the same cou-

pling being at the same place would leave. Hence the

remaining important task in this paper is the compar-

ison between the trace left in the transmission channel

in the \counterfactual" communication protocols [8, 14]

and the trace left by a single particle passing through

the channel. In the next section I will analyze the mini-

mal trace left by a single particle being in a transmission

channel and the above comparison will be made in the

following sections.

VI. THE TRACE LEFT BY A SINGLE

PARTICLE

In the framework of standard quantum mechanics

there is no rigorous way to decide if statement (3) holds,

that is: was the particle present or was it not present in

the transmission channel. In a two-slit experiment it is

not clear whether the particle was in a particular slit. If

a particle starts on one side of a plate with two slits and

is found later on the other side, we do not know if it was

in the two slits together, or in one of them, but we rmly

believe that it cannot be that it was not present in both.

Consider rst a single-path transmission channel with

a single particle in the form of a single localized wave

packet. The wave packet passes from Alice to Bob, see

Fig. 9. Let us model the interaction of the particle with

the transmission channel as von Neumann measurement

with a Gaussian probe. The initial wave function of the

pointer is

hxj0i =

1 p

p

e􀀀 x2

22 : (10)

When a particle is present in the transmission channel,

the interaction shifts x by . Thus, the state of the mea-

suring device after the interaction is

ji =

p

1 􀀀 2j0i + j?i; (11)

where j?i is orthogonal to the initial pointer state and

=

q

1 􀀀 e􀀀 2

2 .

How to quantify the strength of the trace? One op-

tion is to consider the probability, 2, of detecting the

particle in the channel in an idealized experiment. An-

other option is just to use the shift of the pointer via the

parameter

.

For a strong trace, the probability criterion does not

represent the trace well: it remains almost 1 for

=

10 and also for

= 1000. In practice, however, it is

plausible that in a realistic experiment, when in addition

to the quantum uncertainty of the pointer there is an

uncertainty of the grid on which we read the pointer,

only very large

can be observed.

For a small value of

, the probability of detection is

proportional to the square of this parameter. The trans-

mission of two particles doubles the shift, but increases

the probability of detection by a factor of 4. The linear

response seems to be a better representation of the trace.

Consider now a single particle passing through the

transmission channel which consists of N identical paths

as above. The quantum state of the particle is an

equal weight superposition of wave packets in all paths,

j ini = p1

N

PN

i=1 jii, where jii signies the wave packet

of the particle inside path i in the transmission channel,

see Fig. 10. After the particle passes the transmission

channel, the state of the particle and the pointer repre-

senting the transmission channel becomes

1

p

N

XN

i=1

Y

j6=i

j0ij(

p

1 􀀀 2j0ii + j?ii)jii: (12)

The probability to detect the particle in the transmission

channel is the probability to nd one of the states j?ii.

It is 2 as in the case of the single-path channel. The sum

of the expectation values of the shifts of the xis is also

the same,

P

hxii = .

It is important to consider the post-selection measure-

ment of the state of the particle made by Bob. Let Bob

select the undisturbed state j fini = p1

N

PN

i=1 jii. This

corresponds to detection of the particle by Bob's detector

in Fig. 11. For a good, low noise channel, the probabil-

ity to nd this state is very close to 1. The state of the

10

FIG. 10: A single particle in a superposition of several localized

wave packets passes from Alice to Bob. We assume that

the beam splitters are arranged in such a way that all packets

have equal amplitudes and beam splitters on Bob's side are

tuned to interfere constructively toward the detector.

transmission channel then becomes

N

p

1 􀀀 2

QN

j=1 j0ij +

PN

i=1 j?ii

Q

j6=i j0ij p

N2(1 􀀀 2) + N2

(13)
At this stage, the probability to nd one of the states
j?ii is reduced dramatically. This is because the fail-
ure of post-selection by Bob implies that the probability
to nd one of the states j?ii is 1. For small , the prob-
ability to detect the particle in the transmission channel
after a succesful post-selection is approximately 2
N .
It is interesting that the post-selection of the particle
state does not change the expectation value of the sum
of the pointer variables h
P
xii = . One way to see this
is to note that h
P
xii = is proportional to the weak
value [35] of the sum of the projections on all parts of
the channel which equals 1 because the initial state is
the eigen state of the sum of the projections with the
eigenvalue 1 [36]. For describing the magnitude of the
trace, the directions of the shifts are not important. So
the relevant parameter is
P
i jhxiij. We have found a
lower bound,
P
i jhxiij .
VII. THE WEAK TRACE
I analyse next the trace left in the transmission channel
in communication protocols discussed above. All proto-
cols are based on interference, therefore, when they work
properly, only a small trace can be left. I use the same
model: in every path of the transmission channel, the
presence of the particle shifts the Gaussian pointer, see
(11). I assume that the coupling is weak: , and
consequently, 1. For simplicity, I consider the trace
created by particles moving from Alice to Bob and dis-
regard the trace created on the way from Bob to Alice.
Then, the example described in Fig. 1, is identical to that
of Fig. 9. and the trace in the communication channel is
the shift of the pointer and the probability to discover
the presence of the particle by observing the trace, is 2.
In the communication of the bit 0 using IFM, Fig. 2,
and in the protocol with HWPs, Fig. 3 the trace is of
the same order of magnitude.
In the IFM communication of the bit 1, see Fig. 2,
after the interaction with the transmission channel, the
state of the particle and the pointer is
1
p
2
h
0ijLi + (
p
1 􀀀 2j0i + j?i)jabsorbredi
i
:
(14)
If the particle is absorbed by Bob, the trace in the channel
is exactly as in the single-path channel of Section VI:
shift by and the probability to nd the trace is 2. If
the particle is detected by Alice, there will be no trace
in the transmission channel. The wave packet \tagged"
by an orthogonal state of the channel, j?i cannot reach
Alice.
Now I consider the IFM experiment with the chain
of the interferometers, Fig. 4, starting with the com-
munication of bit 0, when Bob leaves the interferometers
undisturbed. The exact expressions are complicated, but
for weak coupling, only the rst order contibution in is
signicant. Neglecting the coupling to the channel, we
obtain from (2) the state of the particle in the nth inter-
ferometer
cos
n
2N
jLi + sin
n
2N
jRi: (15)
The wave packet jRi, \tagged" at the nth interferometer
by the state j?in in the transmission channel, interferes
only with itself and reaches detectors in the state
cos
(N 􀀀 n)
2N
jRi 􀀀 sin
(N 􀀀 n)
2N
jLi: (16)
In this experiment, the particle ends up in detector D2
(state jRi) with a probability close to 1. The state of the
transmission channel then is
N
0
@
NY􀀀1
n=1
j0in +
NX􀀀1
n=1
sin2 n
2N
Y
j6=n
j?in
1
A; (17)
with normalization jN j close to 1. Therefore, the proba-
bility to detect the particle in the transmission channel,
i.e., to nd at least one of the states j?in,
2
NX􀀀1
n=1
sin4 n
2N
2N
2

Z
2
0
sin4 xdx =
32N
8
; (18)
is much larger than the minimal probability to nd a sin-
gle particle present in this multiple-path channel, which
11
can be as low as 2
N . Thus, the case of the bit 0 is def-
initely not a \counterfactual" communication. This can
also be seen by calculating the pointers shifts. These
shifts are proportional to the expectation value of the
projection on the paths of the transmission channel. The
shift in path n is sin2 n
2N and the sum of all shifts, N
2 ,
is much larger than , the standard for the presence of a
single particle in the channel.
The situation is dierent for communication of bit 1,
when Bob blocks the paths of the interferometers, Fig.
4b. Due to Zeno eect, the probability of absorbtion by
Bob is negligible. Detector D1 clicks with probability
close to 1 telling Alice that the bit value is 1. In this case
there is no trace in the communication channel. It is,
therefore, a counterfactual communication for bit value
1.
Let us turn now to the case of nested interferometers,
Fig. 5. The case which is particularly interesting is de-
scribed in Fig. 5a. Bob does not put the shutter in,
and detector D1 clicks. Alice knows that the bit is 0,
and naively, it seems to be a counterfactual communica-
tion since the particle \could not pass through the trans-
mission channel". Indeed, the wave packet entering the
nested interferometer does not reach detector D1.
The interferometer was dened only by demanding de-
structive interferences in particular situations. To make
quantitative predictions, we have to specify the beam
splitters of the interferometer. In a possible implementa-
tion of the interferometer [40], the rst two beam splitters
transform the localized wave packet entering the interfer-
ometer into a superposition:
j ini !
1
p
3
(jAi + jBi + jCi); (19)
and the other two beam splitters transform each of the
states as:
jAi !
1
p
3
j1i 􀀀
1
p
6
j2i +
1
p
2
j3i;
jBi ! 􀀀
1
p
3
j1i +
1
p
6
j2i +
1
p
2
j3i; (20)
jCi !
1
p
3
j1i +
r
2
3
j2i;
where state jii signies a wave packet entering detector
Di. It is is easy to see that these rules ensure the required
destructive interferences.
After the interaction, the joint state of the particle and
the channel is
1
p
3
[jAi(
p
1 􀀀 2j0i + j?i) + (jBi + jCi)j0i]: (21)
From (20) we see that the detection of the particle
in detector D1 post-selects the state p1
3
(jAi 􀀀 jBi +
jCi): Therefore, the state of the channel after the post-
selection is
p
1 􀀀 2j0i + j?i: (22)
This is exactly the same state of the channel as in the
case that a single particle passed through it, see (11).
Thus, contrary to the naive expectation, the scheme with
nested interferometers does not provide counterfactual
communication [12].
VIII. THE WEAK TRACE IN \DIRECT
COUNTERFACTUAL QUANTUM
COMMUNICATION"
Now we are ready to analyze the trace left in the \Di-
rect counterfactual quantum communication". The case
of bit 1, Fig. 6a, is simple. The trace in the communi-
cation channel is correlated with the nal location of the
particle. If it is absorbed by Alice, which happens with a
probability close to 1 and corresponds to the proper oper-
ation of the protocol, the trace is zero. The wave packets
\tagged" by orthogonal states of the channel, j?im;n,
cannot reach Alice. The trace is present only if the par-
ticle is absorbed by one of the Bob's shutters which hap-
pens with vanishing probability. This is a counterfactual
communication protocol for bit 1.
The more interesting case is that of bit 0, when Bob
leaves the interferometer undisturbed, Fig. 6b. We
assume that the interaction with the channel is small,
1
M , 1
N , so only the rst order in should be
considered.
The amplitude of the wave packet of the particle in
the nth path of mth chain of the inner interferometers
(m; n), is
cos(m􀀀1)N
2N
cos(m􀀀1)
2M
sin

2M
sin
n
2N: (23) A particle present in the path (m; n) changes the state of the corresponding pointer according to (11): j0im;njm; ni ! p 1 􀀀 2j0im;n + j?im;n jm; ni: (24) The wave packet jm; ni \tagged" by the orthogonal state j?im;n interferes only with itself and leaves the inner chain in the state, see (16): jm; ni ! sin n 2N jRi 􀀀 cos n 2N jLi: (25) State jRi is lost and the state jLi of the last inner interfer- ometer of the chain m enters from the right the remaining M 􀀀m􀀀 1 large interferometers. The transformation of this state (which is named jRi in the following equation) in the remaining interferometers is: jRi ! sin 2M cos(M􀀀m􀀀1)N 2N cos(M􀀀m􀀀2) 2M

cos

2M

jLi + sin

2M

jRi

+ ::: ; (26)

where ..." signies the wave packets which do not reach

the detectors.

12

In the protocol, the particle is found with probability

close to 1 by detector D1. After the detection of the

particle, the amplitude of the term j?im;n correspond-

ing to detection of the particle in the path (m; n) can be

found by collecting the factors in (23-26). It is

2

cos(M􀀀2)N

2N

sin2

2M

sin

n

N

cos(M􀀀2)

2M

(27)

As explained in Section III, the protocol works properly

if 1 << M << N, so that

cos(M􀀀2)N

2N

1; cos(M􀀀2)

2M

1; (28)

and the probability that one of the orthogonal states

j?im;n will be found in the transmission channel is, ap-

proximately,

X

m;n

2

4

sin4

2M

sin2 n

N

24N

27M3 : (29)

The number of paths in the channel is MN. We

have seen that a single particle present in such a channel

can be found with probability as low as 2

MN which is

smaller than the probability of detection of the particle

in the protocol by a factor of approximately N2

M2 . Since

the protocol works well only when N M, the trace in

the protocol is larger than the trace of a single particle

passing through the channel.

Another criterion of counterfactuality is the sum of

displacements of the pointers in all paths of the channel,

the standard for which is . It can be found by calculat-

ing the absolute values of weak values of all projections:

P

m;n j(Pm;n)wj with pre- and post-selection specied

by the protocol. The scalar product in the denominator

of the weak value is close to 1 since the probability of

the post-selection is close to 1. The amplitude of the for-

ward evolving state at path (m; n) is given by (23) and,

similarly, the amplitude of the backward evolving state

is

cos(M􀀀m􀀀1)N

2N

cos(M􀀀m􀀀1)

2M

sin

2M

sin

(N 􀀀 n)

2N

:

(30)

Thus, the weak value of the projection on the path (m; n)

can be approximated as

(Pm;n)w

2

8M2 sin

n

N

; (31)

and the sum of all the pointer shifts is

X

m;n

j(Pm;n)wj

2N

16M

(32)

Since N M, the trace in the protocol is much larger

than the standard of the trace of a single particle present

in such multiple-path channel.

IX. THE WEAK TRACE IN \DIRECT

QUANTUM COMMUNICATION WITH ALMOST

INVISIBLE PHOTONS"

Let us turn to the \Direct quantum communication

with almost invisible photons" protocol [14]. When Bob

transmits 0, i.e. does nothing, Fig. 8a, the amplitude in

the path (m; n) is

sin

2M

sin

n

N

for m odd;

0 for m even: (33)

The wave packet jm; ni, \tagged" by the orthogonal state

j?im;n, interferes only with itself and it leaves the inner

chain in the state

jm; ni ! sin

n

N

jRi 􀀀 cos

n

N

jLi: (34)

The state jRi is lost and the state jLi of the last inner

interferometer of the chain m enters from the right the

remaining M 􀀀 m 􀀀 1 large interferometers. Using the

fact that \tagging" takes place only for odd m, and that

the number of the large interferometers is odd (M, the

number of beam splitters is even), and that after every

second beam splitter the wave function repeats itself, we

can conclude that the wave packet leaves the last beam

splitter of the last inner chain with the same amplitude.

The wave packet entering the last beam splitter of the

large interferometers transforms into

cos

2M

jRi 􀀀 sin

2M

jLi: (35)

In the protocol, the particle is detected with probabil-

ity 1 by detector D1 which detects the state jLi. Thus,

after detection of the particle, the amplitude of the term

j?im;n corresponding to the detection of the particle in

the path (m; n) can be found by collecting factors from

(33-35) and a factor due to the interaction:

2

sin

2n

N

sin2

2M

(36)

This expression holds only for odd m, the amplitude in

the paths with even m vanishes. Summing the proba-

bilities of nding the record the particle leaves in all the

paths, i.e. summing on odd ms up to M 􀀀1 and on inte-

gers n up to N 􀀀1, we obtain the probability of detection

of the particle:

X

m;n

2

4

sin4

2M

sin2 2n

N

24N

28M3 : (37)

A single particle passing through this transmission

channel can be found with probability of order 2

MN . De-

pending on the ratio N

M , it can be smaller or larger than

(37), so we cannot yet decide on the counterfactuality of

the protocol.

Compare now the sum of the pointer shifts in the chan-

nel. It can be found by calculating the absolute values

13

of weak values of all projections:

P

m;n j(Pm;n)wj with

the pre- and post-selection specied by the protocol. In

this case, the pre- and post-selected states are identical,

so the weak values are expectation values:

(Pm;n)w =

sin2

2M sin2 n

N for m odd;

0 for m even:

(38)

and

X

m;n

j(Pm;n)wj

2N

16M

(39)

The sum of the pointer shifts when one particle in the

transmission channel is , so the ratio N

M tells us when

the sum of displacements in the protocol (39) is smaller

or larger than that of a single particle present in the

channel. We can see that the criterion of the pointer

shifts is in agreement with the criterion of the probability

of detection.

Let us now repeat the analysis for the case of bit 1,

when Bob puts HWPs in every inner interferometer. Now

the amplitude in the path (m; n) is

sin

m

2M

sin

N

for n odd;

0 for n even: (40)

The wave packet jm; ni, can be \tagged" by the or-

thogonal state j?im;n only for odd n. Thus, due to the

presence of the HWPs, the wave packet comes back un-

changed every second beam splitter. It leaves the chain

of the inner interferometers in the state

jm; ni ! cos

N

jRi 􀀀 sin

N

jLi: (41)

State jRi is lost and the state jLi of the last inner in-

terferometer of the chain m enters from the right the

remaining M 􀀀m􀀀1 large interferometers. In the chain

of the large interferometers it performs usual evolution

(2) and ends up in the state

sin

m

2M

jRi 􀀀 cos

m

2m

jLi: (42)

In the protocol, the particle is detected with probabil-

ity 1 by detector D2 which detects the state jRi. Thus,

after detection of the particle, the amplitude of the term

j?im;n corresponding to the detection of the particle in

the path (m; n) can be found by collecting factors from

(40-42) and the factor due to the interaction:

2

sin2

N

sin2 m

2M

(43)
This expression holds only for odd n, the amplitude in the
paths with even n, vanishes. Summing the probabilities
of nding the record the particle leaves in all the paths,
we obtain the probability of detection of the particle:
X
m;n
2
4
sin4 m
2M
sin4
N

234M
26N3 : (44)
Again, as in the case of bit 0, the ratio N
M tells us if
it is larger or smaller than the minimal probability of
detection in case of a single particle present in the trans-
mission channel. However, the dependence is opposite: if
for bit 0 the probability of detection in the protocol was
smaller than single-particle standard, for bit 1 it will be
larger, and vice versa.
The pointer shifts criterion of counterfactuality is in
agreement with these results. The sum of pointers shifts
in all paths of the channel is proportional to the sum
of the weak values of all projections:
P
m;n j(Pm;n)wj.
Also in this case, the pre- and post-selected states are
identical and the weak values are expectation values:
(Pm;n)w =

sin2 m
2M sin2
N for n odd;
0 for n even;
(45)
and

X
m;n
j(Pm;n)wj
2M
4N: (46) The ratio N M tells us when the sum of displacements in the protocol (46) is smaller or larger than that of the one particle. If it is smaller for bit 1, it is larger for bit 0, and vice versa. X. SECURITY OF COUNTERFACTUAL PROTOCOLS One of the motivations for \counterfactual" protocols, in which no particles are present in the transmission channel, is that it is secure against Eve who is trying to eavesdrop the communication: she has no particles to look at [37]. The obvious cryptographic weakness of the protocols with shutters which are 100% counterfactual is that Eve can use an active attack detecting the presence of the shutters. Moreover, the shutter can be detected by Eve in a counterfactual way [38], and recently there has been a claim of a very ecient attack of this kind [39]. If Eve uses an active attack, probing Bob's bit by send- ing her particles, the counterfactuality property does not help. So, for the analysis of counterfactuality we should only consider attacks in which Eve tries to observe the particles used in the protocol. Under this condition, the counterfactual protocols with shutters are secure. Consider the following counterfactual key distribution protocol. There are two identical chains of interferom- eters as in Fig. 4. One of the chains is dened as bit 0, and the other as bit 1. On each run of the protocol, Alice randomly sends a single particle through one of the interferometers and Bob randomly chooses one of the in- terferometers and places the shutters in all of its paths. Every time Alice detects the particle in detector D1 of one of the interferometers, she makes a public announce- ment. Detector D1 can click only if Alice and Bob chose the same interferometer, i.e. they chose the same bit. This creates a common key. 14 The multiple shutters Bob placed represent the weak point of the protocol due to the reason above, although we can improve it using detectors instead of shutters and telling Alice to send particles not every time, but only at some random times (using ideas of [5]). Anyway, we made a postulate here that Eve does not use her particles, so she can only use some (weak) nondemolition measure- ments of the particle running in the interferometer. Let us see if Eve can get some information about the key in this way. If Eve detects the particle in one of the interferometers it cannot be the one which generates a correct bit in the key. For correct bit generation event Alice and Bob have to choose the same interferometer. Eve can detect the particle only if it is present, i.e., it has to be chosen by Alice. If Bob chooses this interferometer, then after detection by Eve, the particle has to be absorbed by Bob, so it will not reach Alice. Only if the interferometer is not chosen by Bob, the particle seen by Eve in the nondemolition measurement can reach Alice, and there is a nonzero probability that the detector D1 will click and thus Alice will declare gen- eration of a bit in the key. But this will be an error bit, since Bob has chosen the other interferometer. Eve intro- duces errors, and the only information she gets is about these error bits. Let us turn to the \Direct counterfactual quantum communication" protocol. When Bob places the shut- ters, Fig. 6a, Eve cannot get information about the cor- rect bit because Eve's detection causes the loss of the particle. (It is not surprising, since in this case the pro- tocol is counterfactual.) If the bit is 0, and the interferometer is free, Fig. 6b, Eve's detection will not necessarily lead to the loss of the particle. Detection of the particle by Eve in the last chain of inner interferometers will lead to part of its wave packet to enter the last beam splitter from the right, so most probably it will create an error: D2 clicks, but Bob's bit is 0. However, Eve's detection of the particle in any of the rst M􀀀2 chains of inner interferometers will lead to part of the particle wave packet to enter the last beam splitter from the left side, so most probably D1 will click, corresponding to a successful transmission of the correct bit. Thus, sometimes, Eve gets a reliable information about the transmitted bit. Eve, which eavesdrops only by observing the original particle of the communication protocol, cannot learn any correct bit of real counterfactual protocols with shutters which transmit bit 1, but she does learn some correct bits in \counterfactual" communication protocols of bit 0. This provides another argument why such protocols should not be named counterfactual. XI. CONCLUSIONS The standard quantum formalism does not specify the position of a quantum particle. Thus, it does not provide an unambiguous answer to the question: Is a particular communication protocol counterfactual? I.e.: Was the particle present in the transmission channel? In this pa- per I analyzed an approach to answering this question based on the weak trace the particle leaves in the chan- nel. I compared the trace left in the channel in recently proposed protocols claimed to be counterfactual with the trace in the protocol constructed to transmit a single par- ticle in the same channel. In the analysis, I considered two criteria for comparing the traces. First, the probability of nding a conclusive evidence for the presence of the particle, and second, the expectation value of the sum of total shifts of some vari- ables of the channel. The question of counterfactuality of the protocols is considered in cases the protocols work properly, i.e. when the particle is detected by the right detector. It means that the particle is pre- and post-selected. The protocols were compared to transmission of a single pre- and post- selected particle. In all these cases the probability of post-selection was closed to 1. The analyses using the two criteria of the trace led to the same conclusion. It is possible to communicate only one value of a bit in a counterfactual way. The protocol \Direct counterfactual quantum commu- nication" [8] is fully counterfactual for bit value 1. The trace is identically 0. Nothing changes in the transmis- sion channel and therefore there is zero probability to detect the particle in the transmission channel. Passive eavesdropping provides no information about the trans- mitted bit. However, the protocol is not counterfactual for the bit value 0. It is true that by increasing the number of paths in the channel, the probability of nding a conclusive evi- dence of the presence of the particle reduces, but increas- ing the number of paths also reduces the probability to nd a single quantum particle when it passes the channel. The probability to nd the presence of the particle in the transmission channel in the event of successful operation of the protocol is larger than the probability to detect a particle successfully passing through this channel. Eve, using passive attack, learns some of the transmitted bit. The criterion of the shifts of variables of the channel tells us the same. The expectation value of the sum of the shifts is zero for bit 1, but for bit value 0 it is larger than the sum of the shifts when a single particle passes the channel. The protocol \Direct quantum communication with al- most invisible photons" is not fully counterfactual for any bit value. Some trace is always left in the transmission channel. However, if we are ready to consider a pro- tocol as counterfactual when it leaves a trace which is much smaller than the trace of a single particle passing through this channel, then we can arrange that it will be counterfactual for one of the bit values. By playing with the numbers N and M of the inner and the ex- ternal interferometers respectively, the protocol can be made counterfactual for value 0 or value 1 of the bit. It 15 cannot be made counterfactual for both. The analysis of the trace left by a particle passing through a N-path channel of Section VI showed a sur- prising result: the probability of detection in the channel of the successfully transmitted particle is reduced by the factor of 1 N . It helped me to analyse the counterfactual- ity of the protocols, but it might also open new avenues for useful quantum communication applications. 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