# Multitime Quantum Communication: Interesting But Not Counterfactual

Robert B. Griffiths
Department of Physics
Carnegie Mellon University
Pittsburgh, PA 15213 Electronic address<EMAIL_ADDRESS>
(Version of 24 June 2023)

###### Abstract

A protocol for transmission of information between two parties introduced by
Salih et al., Phys. Rev. Lett. 110 (2013) 170502 (hereafter SLAZ), involves
sending quantum amplitude back and forth through a quantum channel in a series
of steps, rather than simply sending a signal in one direction. The authors
claimed that their protocol was “counterfactual” in the sense that while a
quantum channel is needed to connect the parties, its actual usage becomes
vanishingly small in the asymptotic limit as the number of steps tends to
infinity. Here we show that this claim is incorrect because it uses
probabilistic reasoning that is not valid at intermediate times in the
presence of quantum interference. When ill-defined probabilities are replaced
with a well-defined measure of channel usage here called “Cost”, equal to the
absolute square of the amplitude sent through the channel, the total Cost does
not go to zero in the asymptotic limit of a large number of steps, but is
bounded below by a rigorous inequality. A detailed analysis shows that this
bound is satisfied in the SLAZ protocol. The analysis leading to the bound
uses the fact that the Gram matrix formed by inner products of a collection of
pure quantum states is additive over Hilbert subspaces and invariant under
unitary time transformations. Its off-diagonal elements, which in general are
not positive, play a significant role in the formal argument as well as
providing a somewhat strange way of visualizing the transfer of information.

###### Contents

  1. I Introduction
  2. II  One-Way Protocols
    1. II A  Multiple Channels in Parallel
    2. II B  One Channel Used Multiple Times
  3. III  Two-way Protocols
    1. III A  Gram Matrices
    2. III B  Basic Two-Way Protocol
    3. III C  Sending One Classical Bit
    4. III D  Lower Bound on Costs
  4. IV  The SLAZ Protocol
    1. IV A  Description of the Protocol
    2. IV B  Calculation of Costs and Overlap
    3. IV C Discussion of Costs and Probabilities
  5. V  Conclusion

##  I Introduction

The motivation for this paper is a scheme for the transmission of quantum
informatiom introduced by Salih et. al [1] with the title “Protocol for direct
counterfactual quantum communication”, and referred to hereafter as SLAZ, the
initials of the authors. One ordinarily thinks of the transmission of
information as sending a signal through a channel from sender to receiver.
However the idea in SLAZ is that information can be sent from Bob to Alice if
the quantum particle used to carry the information starts off in Alice’s
domain, and a part of its quantum amplitude is sent to Bob through a quantum
channel. Bob modifies this is some way before sending (or possibly not
sending) it back to Alice, depending on the signal he wants to send. Alice
then employs what Bob has returned to begin a second round of sending
amplitude to Bob, who again modifies it before returning it, and so forth.
This back-and-forth motion can continue for a large number of rounds until the
information that Bob is sending has arrived in Alice’s domain, where she can
carry out a measurement or perhaps perform additional processing. A key
feature of protocols of this type is that all the intermediate steps can be
represented by purely unitary time evolution, with intermediate time
measurements, if any replaced by unitaries—a process of purification.

The use of _amplitude_ rather than _particle_ in the previous paragraph is
intentional, because the state of the photon or other particle is in general a
coherent superposition of parts associated with different spatial locations:
Alice’s domain, Bob’s domain, and the channel connecting them. One generally
thinks of a particle as something with a spatial location, but in quantum
mechanics one cannot simultaneously ascribe particle and wave properties to
the same entity at the same time because of wave-particle duality. In Hilbert-
space quantum mechanics physical properties, such as location in space, are
represented by _projectors_ (Sec. III.5 of [2]), and when a projector
representing a wave, think of $|\psi\rangle\langle\psi|$, does not commute
with a projector specifying a spatial location, ignoring this fact can rapidly
lead to paradoxes. The double-slit paradox is an example: when a coherent wave
passes through the slit system one cannot say through which slit the particle
passed.

The term “counterfactual” in the original SLAZ paper has the following
significance. A quantum channel connecting the communicating parties is
essential: this is not a case of mysterious nonlocal influences of the sort
which are sometimes invoked to explain quantum violations of Bell
inequalities. However, if the number of steps in an SLAZ protocol is
sufficiently large, the magnitude of the amplitude sent through the channel in
each step can be made very small, and vanishes in the limit as the number of
steps tends to infinity.

A similar claim of counterfactuality has been made in much of the rather
substantial literature motivated by the original SLAZ publication, which
contains various modifications and extensions of the original protocol. There
have also been criticisms of these counterfactual claims, and (of course)
replies to criticisms. The Conclusion, Sec. V, of the present paper contains a
few remarks about how its results apply to some of these publications, but a
review, much less a detailed discussion, lies far outside its scope. The
interested reader is referred to the extensive bibliographies found in [3, 4].

The aim of the present paper is to study the use of quantum channels in
protocols of the SLAZ type, in particular the sense in which this usage is or
is not counterfactual. To this end a technical term, _Cost_ , the absolute
square of the amplitude through the channel in a particular step in the
protocol, is used for reasons discussed in Sec. II, as a useful substitute for
”probability”, which in a quantum context is often ill-defined. The example of
multiple channels in parallel, which few would want to claim are
counterfactual, serves as an introduction to how information can be sent
through a single channel in a single direction at multiple times, in a process
in which all of the intermediate steps are represented by unitary maps.

The main mathematical results of this paper are in Sec. III: Gram matrices and
some of their properties are discussed in Sec. III A, while Sec. III B gives
the basic structure of simple two-way multiple time protocols. Section III C
considers simple schemes for transmitting one classical bit, while the
rigorous lower bound that undermines various counterfactual claims is the
topic of Sec. III D.

The original SLAZ protocol is studied in detail in Sec. IV. In particular the
total Cost of transmitting a classical bit $\lambda=0$, in which Bob reflects
the amplitude back to Alice, and for transmitting $\lambda=1$, in which he
absorbs rather than returns it, are evaluated explicitly. It turns out that in
the asymptotic limit the $\lambda=1$ Cost is miniscule, but that for
$\lambda=0$ is enormous, while the product of the two remains finite and
satisfies the rigorous bound in Sec. III D. The mistaken claim that the SLAZ
protocol is counterfactual results from two errors: a concept of channel use
which would be questionable even for a classical stochastic process, and an
improper use of probabilities in a way that violates quantum principles.

The concluding Sec. V has a summary of the main results of this paper, a few
comments on some parts of the literature related to SLAZ, and some suggestions
for future directions of research. This author believes that protocols of the
SLAZ type are quite interesting, deserve further exploration, and might
contribute to useful ways of studying multipartite and multitime transmission
of quantum information, as in quantum networks. And that such studies would
prove more fruitful in the absence of claims of counterfactuality.

##  II  One-Way Protocols

###  II A  Multiple Channels in Parallel

Think of quantum information as the information carried by a photon as it
passes through a quantum channel, such as an optical fiber. The information
could be encoded in its polarization. Rather than using a single channel, one
could imagine sending the photon as a superposition state through a set of $N$
channels in parallel, using a collection of beamsplitters to divide up the
initial amplitude among the different channels, and a corresponding collection
to later recombine them. Let us suppose that the normalized $|\Phi\rangle$
that represents the photon at some intermediate time is a coherent
superposition of amplitudes

$|\Phi\rangle=\sum_{n=1}^{N}c_{n}|\phi_{n}\rangle$ (1)

associated with the individual channels, labeled by $n$. Define the _Cost_
$q_{n}$ associated with the use of channel $n$, and the _total Cost_ $Q$ for
the channel system as:

$q_{n}:=|c_{n}|^{2},\quad Q:=\sum_{n=1}^{N}q_{n}.$ (2)

If the $|\phi_{n}\rangle$ and $|\Phi\rangle$ are normalized, $Q$ is equal to
1, so one might identify $q_{n}$ with the _probability_ that the photon is in
channel $n$. But what does that mean? In standard (textbook) quantum mechanics
probability refers to the outcome of a measurement, but a measurement carried
out at an intermediate time, when the quantum state is a coherent
superposition over various locations, can alter what occurs later, and hence
it is dangerous to associate such a probability with a situation in which a
measurement does _not_ take place.

Another way of viewing this difficulty is to recall that von Neumann (Sec.
III.5 of [2]) identified quantum _physical properties_ —which in classical
physics are associated with sets of points in the classical phase space—with
_projectors_ , self-adjoint idempotent operators, $P=P^{\dagger}=P^{2}$, on
the quantum Hilbert space. For example, in the case of a spin-half particle
the projectors

$P=(I-\sigma_{z})/2\quad R=(I+\sigma_{x})/2,$ (3)

where I is the identity and $\sigma_{z}$ and $\sigma_{x}$ are Pauli operators,
represent the properties $S_{z}=-\hbar/2$ and $S_{x}=+\hbar/2$, respectively.
In general, if two projectors $P$ and $R$ commute their product $PR=RP$
represents the property $P\ \mbox{\small AND}\ R$. But if they do not commute,
neither $PR$ nor $RP$ is a projector, and so neither represents a quantum
property. In some sense noncommutation is the very essence of quantum
mechanics; it is what distinguishes it from classical physics. The use of
standard (Kolmogorov) probabilities requires a _sample space_ of mutually-
exclusive possibilities, one and only one of which occurs in a particular run
of an experiment. In quantum theory such a sample space is a collection of
mutually orthogonal projectors that sum to the identity, a _projective
decomposition of the identity_. For example, $R$ and $I-R$ in (3) in the case
of spin half; see (7) below for the general definition. In quantum mechanics
there are often many possible sample spaces that one might be interested in,
and carelessly combining incompatible spaces—some projectors in one do not
commute with projectors in the other—inevitably leads to paradoxes rather than
physical understanding.

In the present context the dyad $|\Phi\rangle\langle\Phi|$ is a projector that
does not commute with any of the projectors $|\phi_{n}\rangle\langle\phi_{n}|$
for which $c_{n}$ is nonzero, and thus it is meaningless to assign a
probability to the latter in a situation where the coherent superposition
$|\Phi\rangle$ will later be transformed by the final beamsplitters into the
original state that entered the channel system. For example, in a double-slit
experiment in which the amplitudes from the two slits combine coherently to
produce interference, it makes no sense to talk about the probability that the
photon previously passed through one slit rather than the other. The two costs
are well-defined: they are simply the absolute squares of the two amplitudes.
But attempting to measure which slit the particle passes through in order to
define a probability will destroy the interference pattern.

###  II B  One Channel Used Multiple Times

The possible advantages, if any, of using many channels in parallel can also
be realized by employing a _single_ channel and sending quantum amplitude
through it at a _succession_ of times; this is what makes protocols of the
SLAZ type of some interest. Let us suppose that information is being sent from
Bob to Alice. One can think of the photon at a particular time as being in a
coherent superposition of amplitudes in three different physical locations:
Alice’s domain $A$, Bob’s domain $B$, and the channel $C$ connecting them. The
same symbols can be used for the Hilbert-space projectors associated with
these locations, thus operators which are self-adjoint and idempotent,
$A=A^{\dagger}=A^{2}$, and mutually orthogonal, $AB=BC=AC=0$. They sum to the
identity

$A+C+B=I$ (4)

and hence form a _projective decomposition of the identity_ —see the general
definition in (7) below. A projective decomposition of the identity is the
quantum counterpart of the sample space of mutually exclusive possibilities
essential for using standard (Kolmogorov) probability theory in the case of a
quantum system. Note that $A$, $B$, and $C$ are _subspaces_ of a single
Hilbert space, not _subsystems_ represented by a tensor product. If the
quantum particle possesses other degrees of freedom, these projectors are to
be understood using the usual convention as including the identity operator on
these additional degrees of freedom. Thus for a photon, $A$ means that it is
located in Alice’s domain, whatever may be its polarization.

Bob can send a particular type of information $\lambda$ to Alice by starting
with a normalized _reference state_ $|\psi_{0}\rangle=B|\psi_{0}\rangle$, the
particle is somewhere in his domain $B$, and using a unitary transformation
${\mathcal{B}}^{\lambda}$ acting on the subspace $B+C$ to place it in a state

$|\psi_{1}^{\lambda}\rangle=C|\psi_{1}^{\lambda}\rangle={\mathcal{B}}^{\lambda}|\psi_{0}\rangle,$
(5)

in the channel, at which point it travels through the channel to Alice. As the
channel has no effect except to transmit the particle from one end to the
other, we simplify the discussion (here and later) by using the same symbol
for the ket that arrives at Alice’s end. She then applies a unitary
${\mathcal{A}}$ that does _not_ depend on $\lambda$, for she does not know
what Bob is sending, to empty the channel and arrive at a final state

$|\psi_{2}^{\lambda}\rangle=A|\psi_{2}^{\lambda}\rangle={\mathcal{A}}|\psi_{1}^{\lambda}\rangle,$
(6)

which she can then measure or subject to further processing.

This single-round transmission process can be carried out in a number of
rounds in which during the n’th round Bob employs a unitary
${\mathcal{B}}_{n}^{\lambda}$ acting on the $B+C$ subspace to map an amplitude
$c_{n}|\psi_{0}\rangle$ into $C$, which is initially empty, and which travels
to Alice, who uses a unitary ${\mathcal{A}}_{n}$ acting on $A+C$ to remove it
from the channel, which is then empty and ready for the next round. One way to
visualize this is that Bob has a domain $B$ of high dimension, and at the
outset splits up the initial amplitude $|\psi_{0}\rangle$ into pieces placed
in different subspaces of $B$ with the help of a suitable set of
beamsplitters. At round $n$ the unitary ${\mathcal{B}}^{\lambda}_{n}$
interchanges the appropriate subspace of $B$ with the empty $C$. Alice’s $A$
is also large, and her ${\mathcal{A}}_{n}$ maps whatever Bob has sent into an
empty subspace reserved for this purpose. When the run is completed Alice can
then combine the amplitudes in these different subspaces into a smaller
space—e.g., using beamsplitters—or she can do a similar combination at the end
of each round. Of course Alice’s and Bob’s unitaries cannot be chosen
independently; the two must work together to design the protocol. What is
unknown to Alice is Bob’s choice of $\lambda$ for a particular run; this is
the information that she can extract at the end.

Some multiple-time protocols employ _measurements_ by Alice at intermediate
times. In cases such as the original SLAZ scheme, discussed below in Sec. IV,
it is possible to store the amplitude that could have triggered the measuring
device in an empty subspace in Alice’s domain and put off the measurement
until the protocol is finished. Of course, amplitudes that correspond to
several measurements in succession can be combined, just as in the case of
simultaneous transmission through several channels in parallel, as discussed
earlier.

##  III  Two-way Protocols

###  III A  Gram Matrices

Let $\\{P_{j}\\}$ be a projective decomposition of the Hilbert space identity
$I$:

$I=\sum_{j}P_{j},\quad P_{j}=P_{j}^{\dagger},\quad
P_{j}P_{k}=\delta_{jk}P_{j},$ (7)

and let $\\{|\psi^{\mu}\rangle\\}$, $\mu=0,1,\ldots$, be a collection of kets
on the same Hilbert space. The _Gram matrix_

$G^{\mu\nu}=\langle\psi^{\mu}|\psi^{\nu}\rangle=\sum_{j}G^{\mu\nu}(P_{j})=\sum_{j}\langle\psi^{\mu}|P_{j}|\psi^{\nu}\rangle$
(8)

is _additive_ in that it is a sum over contributions from the different
subspaces. In addition, $G^{\mu\nu}$ is _invariant_ (or _conserved_) under a
unitary operation $U$ that acts on every ket in the collection
$\\{|\psi^{\mu}\rangle\\}$. Also, if this unitary acts on only some of the
subspaces, say $P_{1}$ and $P_{2}$, and is the identity operator on the
others, then while both $G^{\mu\nu}(P_{1})$ and $G^{\mu\nu}(P_{2})$ may
change, their _sum_ $G^{\mu\nu}(P_{1})+G^{\mu\nu}(P_{2})$ remains unchanged.
That Gram matrices are additive and conserved plays an important role in what
follows.

We shall refer to the _diagonal_ elements $G^{\mu\mu}(P_{j})$, which are non-
negative, as _weights_. As these are rather like probabilities, their
additivity and conservation is not surprising. However, that the same is true
of the _nondiagonal_ elements $G^{\mu\nu}(P_{j})$ with $\mu\neq\nu$, hereafter
referred to as _overlaps_ , comes as something of a surprise, especially since
$|\psi^{\mu}\rangle$ and $|\psi^{\nu}\rangle$ may refer to two different runs
of an experiment, one on Friday and one on Monday. Nonetheless, overlaps play
a key role in the following analysis, not only as part of the mathematics but
also in a surprising but useful “intuitive” way of thinking about what is
going on. The absolute value of an overlap corresponds to a notion of
_fidelity_ in quantum information, but in general an overlap is a complex
number, and the fact that it can be negative as well as positive is a key
element in what follows.

###  III B  Basic Two-Way Protocol

In the following discussion the projective decomposition of the identity (7)
that will concern us is $\\{A,C,B\\}$, where $A$ means that the photon or
other quantum particle is in Alice’s domain, $B$ that it is Bob’s domain, and
$C$ in the channel connecting them. At the beginning of a two-way protocol of
the SLAZ type, in which Bob is sending information to Alice, all of the photon
amplitude is in Alice’s domain $A$. She initiates the run by sending some
amplitude to Bob through the channel. He then modifies it and returns some or
all of it to Alice, in a manner that depends on the information $\lambda$ he
wishes to transmit. Alice processes what Bob has returned, and begins the
second round by again sending amplitude to Bob, who again returns it, etc.
This can go on for $N$ rounds, following which Alice makes a measurement to
determine the value of $\lambda$.

In further detail: At the beginning of round $n$, Alice uses a unitary
${\mathcal{A}}_{n1}$ acting on $A+C$ to map some of the amplitude in $A$ into
an empty channel $C$. This amplitude then flows through the channel to Bob,
where he empties the channel into $B$, does some processing, and then maps
some amplitude back into $C$. This flows to Alice, who empties $C$ into $A$
using a unitary ${\mathcal{A}}_{n2}$. We assume that “flow through the
channel” does not change anything, and hence it is convenient not to think of
$C$ as divided into close-to-Alice, close-to-Bob, and in-between subspaces,
but simply imagine that Alice and then Bob and then Alice are acting on a
single $C$ subspace. Alice uses unitaries that act on $A+C$ and are
independent of $\lambda$, while Bob uses unitaries
${\mathcal{B}}_{n}^{\lambda}$, that depend on the information $\lambda$ he
wants to transmit, which act on $C+B$. Both the Alice and Bob unitaries will
in general depend upon the round $n$, but Alice’s do not depend upon
$\lambda$. In addition we impose the restriction that Bob’s actions are
_passive_ in the sense that the magnitude of the amplitude he sends back to
Alice in round $n$ cannot be greater than what he has just received. This last
condition clearly differentiates these two-way protocols from the one-way
protocols of Sec. II B.

The requirement that Alice and Bob only employ _unitary_ operations simplifies
the analysis. It is true that various published protocols of this type,
including the original SLAZ version to be discussed in Sec. IV, employ
nonunitary measurements at intermediate times. In the cases of interest to us
these measurements can be replaced by unitary operations which allow the
measurements to be put off until the end of the run, in a manner indicated in
Sec. II B and employed in the discussion in Sec. IV.

To quantify the channel usage for these protocols we use the notions of _Cost_
, equal to the absolute square of the amplitude for a single use of the
channel, and _Total Cost_ for the sum of the Costs involved in a single
experimental run, as in Sec. II A, see (2). An important issue connected to
claims that these protocols are counterfactual has to do with the difference
between Cost and probability, as will be discussed later for the SLAZ protocol
in Sec. IV—the importance of this has already been noted in Sec. II A. In
particular we will be interested in identifying protocols that minimize the
overall Cost, as in the example discussed next.

###  III C  Sending One Classical Bit

In the simplest SLAZ protocol Bob wants to send a single classical bit,
$\lambda=0$ or $1$, to Alice. At the start all of the amplitude is in $A$ for
both a $\lambda=0$ and a $\lambda=1$ run, so all four of the initial Gram
matrix elements $G^{\mu\nu}_{0}(A)$, where $\mu$ and $\nu$ are the possible
values of $\lambda$, are equal to $1$. The goal is that after $N$ rounds the
Gram matrix will be

$G^{\mu\nu}_{N}(A)=\delta_{\mu\nu}.$ (9)

That is to say, the final result for a $\lambda=0$ run is orthogonal to that
for a $\lambda=1$ run, and Alice, by making a measurement in an appropriate
basis, can determine which bit $\lambda$ Bob sent. Hence the aim of the
protocol is that the _overlaps_ , the diagonal elements $G^{01}(A)$ and
$G^{10}(A)$, relating the two different types of run, should decrease from $1$
to $0$, while the diagonal _weights_ $G^{00}(A)$ and $G^{11}(A)$, remain equal
to $1$.

At this point it is worth noting that if both weights are not maintained—for
example if at the end $G^{00}(A)=1$ while $G^{11}(A)=G^{01}(A)=0$—Alice can
still extract the value of $\lambda$ by measuring whether or not the photon is
in the state $|\psi^{0}\rangle$. Let us call this, for want of a better term,
a _partial_ protocol in contrast to a _full_ protocol that results in (9). A
partial protocol can be used for one-way transmission, and the obvious
advantage is that it costs nothing to transmit $\lambda=1$. A possible
disadvantage is that when Alice’s measurement reveals nothing it could be
because of some failure in the channel or in the measuring device. In the
present discussion we focus on _full_ protocols.

A very simple way to implement such a protocol is that on the very first step
Alice sends the entire amplitude to Bob, with a Cost of $1$ for this use of
the channel. Bob then simply modifies this using the unitary
${\mathcal{B}}^{\lambda}$ and sends it back to Alice, either in one round or
several rounds, with Alice sending nothing back. The Cost for using the
channel in the Bob-to-Alice direction is also $1$, see the discussion in Sec.
II B. Hence a total Cost of $2$ for the protocol as a whole. Notice that since
there is no restriction on ${\mathcal{B}}^{\lambda}$ this rather trivial
protocol can be used to send “quantum” information. From the perspective of
Cost, two-way protocols of the kind under discussion, in which initially all
of the amplitude is on Alice’s side, are interesting because a _classical_
bit, $\lambda=0$ or $1$, can be sent from Bob to Alice at a total Cost of $1$
rather than $2$. And as shown below in Sec. III D, the product of the Costs
for $\lambda=0$ and $1$ cannot be less than $1$.

To discuss the successive steps in protocols that optimize the Cost, we need
an appropriate notation. We will represent kets, thought of as column vectors,
in the way suggested by the following example

$|\psi\rangle=|a;c;b\rangle=|a_{1},a_{2},a_{3};c_{1},c_{2};b_{1},b_{2}\rangle$
(10)

where the dimensions of the $A$, $B$, and $C$ subspaces are $d(A)=3$, $d(B)=2$
and $d(C)=2$, so $|\psi\rangle$ is an element of a 7-dimensional Hilbert
space. Thus $a_{1}$, $a_{2}$, $a_{3}$, are complex numbers forming a
3-component vector $a$; similarly $c$ and $b$ are 2-component vectors. If

$|\hat{\psi}\rangle=|\hat{a};\hat{c};\hat{b}\rangle=|\hat{a}_{1},\hat{a}_{2},\hat{a}_{3};\hat{c}_{1},\hat{c}_{2};\hat{b}_{1},\hat{b}_{2}\rangle,$
(11)

is another vector in the same space, its inner product with $|\psi\rangle$ is
given by

$\langle\hat{\psi}|\psi\rangle=\sum_{j=1}^{3}\hat{a}_{j}^{*}a_{j}+\sum_{k=1}^{2}\hat{c}_{k}^{*}c_{k}+\sum_{l=1}^{2}\hat{b}_{l}^{*}b_{l}$
(12)

Note that we are dealing with a direct _sum_ of subspaces, $A\oplus B\oplus
C$, _not_ a tensor product of subsystems. In much of what follows, $B$ is
empty or can be ignored, so $|a;c\rangle$ will suffice; this and other minor
variants in notation should be self-explanatory.

Let us start with an extremely simple one-round full protocol with $d(A)=2$,
$d(C)=1$. It consists of the following steps:

$\displaystyle|a_{1},a_{2};c\rangle=|1,0;0\rangle\rightarrow|1/\sqrt{2},0;1/\sqrt{2}\rangle$
$\displaystyle\Rightarrow|1/\sqrt{2},0;(-1)^{\lambda}/\sqrt{2}\rangle\rightarrow|1/\sqrt{2},(-1)^{\lambda}/\sqrt{2};0\rangle,$
(13)

where $0$ means this amplitude is equal to zero; do not confuse it with the
label $0$ for one of the two orthogonal states of a qubit. Here $\rightarrow$
indicates the action of a unitary on $A+C$ carried out by Alice, and
$\Rightarrow$ a $\lambda$-dependent unitary on $C$ carried out by Bob. The
action by Bob could involve intermediate steps requiring the $B$ subspace, but
its net effect is only to change the contents of $C$, so there is no need to
include $B$ in the discussion.

In words: At the outset all of the amplitude is in Alice’s $A$, $a_{1}=1$. She
maps half (in the sense of the absolute square) of it into $C$ and sends it to
Bob, who either sends it back unchanged in order to transmit $\lambda=0$, or
with the opposite phase to send $\lambda=1$. Alice then empties the channel
into the $a_{2}$ position, using a unitary on $A+C$ that is independent of
$\lambda$, as it simply requires interchanging two subspaces. A final
measurement by Alice determines which of the two orthogonal states is present
in $A$, and thus which bit Bob was sending.

Next consider what is happening to the Gram matrices $G^{\mu\nu}(A)$ and
$G^{\mu\nu}(C)$ during the successive steps. In particular, the overlap
$G^{01}(A)$ is equal to $1$ at the outset, and the first step reduces it to
$1/2$ by placing $1/2$ in $C$. Bob’s action changes $G^{01}(C)$ from $+1/2$ to
$-1/2$, and this negative contribution to the overlap moves back into $A$ when
Alice empties the channel, leading to the desired $G^{01}(A)=0$. On the other
hand, whereas the weight $G^{00}(A)$ is reduced to $1/2$ during the first
step, Bob’s action does not change the sign of $G^{00}(C)$, so in the final
step Alice moves this weight back to its initial value of $1$, and similarly
for $G^{11}(A)$. Thus the goals of a full protocol have been achieved.

The Costs of using the channel are easily evaluated: $1/2$ for the Alice-to-
Bob step and the same for Bob-to-Alice, for a total Cost of $Q^{\lambda}=1$,
the same for $\lambda=0$ and $1$. These satisfy the rigorous lower bound
worked out below in Sec. III D, so this protocol is optimal if one uses total
Cost as an appropriate measure of channel usage.

This protocol is easily extended to an equally efficient version involving $N$
rounds, $N$ any positive integer. Let

$\epsilon=1/2N,$ (14)

and for the first, $n=1$, round replace (13) with

$|1,0;0\rangle\rightarrow|\sqrt{1-\epsilon},0;\sqrt{\epsilon}\,\rangle\Rightarrow|\sqrt{1-\epsilon},0;(-1)^{\lambda}\sqrt{\epsilon}\,\rangle\rightarrow|\sqrt{1-\epsilon},(-1)^{\lambda}\sqrt{\epsilon};0\rangle,$
(15)

while for round $n+1$,

$\displaystyle|\sqrt{1-n\epsilon},(-1)^{\lambda}\sqrt{n\epsilon};0\rangle\rightarrow|\sqrt{1-(n+1)\epsilon},(-1)^{\lambda}\sqrt{n\epsilon};\sqrt{\epsilon}\rangle$
$\displaystyle\Rightarrow|\sqrt{1-(n+1)\epsilon},(-1)^{\lambda}\sqrt{n\epsilon};(-1)^{\lambda}\sqrt{\epsilon}\,\rangle\rightarrow|\sqrt{1-(n+1)\epsilon},(-1)^{\lambda}\sqrt{(n+1)\epsilon};0\rangle,$
(16)

where it is straightforward to show that there exists a $\lambda$-independent
unitary for the last step. The final result at the end of round $N$ is the
same as in (13), the case in which $N=1$, and again the total Cost is
$Q^{0}=Q^{1}=1$, independent of $\lambda$. One can also let $\epsilon$ depend
on $n$, thus $\epsilon_{n}>0$ for round $n$, subject to the condition

$\sum_{n}\epsilon_{n}=1/2,$ (17)

and the Cost is again equal to $1$.

There are other protocols with larger Costs which may have some practical
advantage. Thus rather than a scalar amplitude, Alice might use photon
polarization, say horizontal $H$, which Bob could return as $H$ to send
$\lambda=0$ or rotate to vertical $V$ to send $\lambda=1$. In this case the
Costs are $Q^{0}=Q^{1}=2$, so twice that for an optimal one-way protocol.
However, there is now no need to maintain a particular phase relation between
what is in Alice’s domain and what is available to Bob during each round. If
polarization is easier to maintain than phase—one leaves that up to the
experts—one could imagine the added Cost being worthwhile if Alice has a large
apparatus capable of generating single photons, while Bob, off on a trip to
spy on Eve, needs only something easily carried in a suitcase.

The protocol used in SLAZ, in which Bob returns the amplitude for $\lambda=0$,
but absorbs it or feeds it to a measuring apparatus for $\lambda=1$, looks
less promising. Because the $\lambda=1$ weight only moves from Alice to Bob it
is difficult to have $G^{11}(A)=1$ at the end of the protocol. In fact SLAZ,
discussed in Sec. IV, employs a clever trick (“Zeno effect”) to get around
this problem, albeit at the cost of a large number of rounds to keep the
probability of failure small, and a large channel usage Cost for one of the
bits.

###  III D  Lower Bound on Costs

The additivity and conservation properties of the Gram matrix $G^{\mu\nu}$
introduced in Sec. III A will now be used to obtain lower bounds on the total
Cost of two-way protocols of the sort exemplified by, but not limited to, the
case of $1$ classical bit discussed above in Sec. III C. Using the
$|a;c\rangle$ notation of (10)—the $b$ entry is not needed in the following
discussion—round $n$ of an $N$ round protocol consists of the following steps
carried out on $A+C$:

$|a^{\mu};0\rangle_{n}\rightarrow|\bar{a}^{\mu};c^{\mu}\rangle_{n}\Rightarrow|\bar{a}^{\mu};\hat{c}^{\mu}\rangle_{n}\rightarrow|a^{\mu};0\rangle_{n+1}.$
(18)

Here $\mu$ labels the bit which Bob is transmitting during this run. Thus
after Alice uses a unitary ${\mathcal{A}}_{n1}$ on $A+C$ to move some
amplitude, $|c^{\mu}\rangle_{n}$ into an initially empty channel. Bob applies
a unitary ${\mathcal{B}}^{\mu}_{n}$ to $C+B$, leading to an amplitude
$|\hat{c}^{\mu}\rangle_{n}$—note the circumflex (hat) added to $c$—in the
channel. If Bob’s action is passive, as assumed in Sec. III C (and in the
later discussion of SLAZ in Sec. IV), one would have

$\|\hat{c}^{\mu}\|_{n}\leq\|c^{\mu}\|_{n},$ (19)

but this conditions is actually not needed to obtain the general results and
inequalities given below, which thus apply equally to one-way multi-time
transmission. As a final step Alice employs a unitary ${\mathcal{A}}_{n2}$ on
$A+C$ to empty the channel by placing its amplitude into $A$. It is important
that Alice’s unitaries ${\mathcal{A}}_{n1}$ and ${\mathcal{A}}_{n2}$, unlike
Bob’s ${\mathcal{B}}^{\mu}_{n}$, _do not depend upon $\mu$_, which can be
different in different runs of the experiment.

The change in the Gram matrix associated with $A$ during round $n$ is given by

$G_{n+1}^{\mu\nu}(A)-G_{n}^{\mu\nu}(A)=\langle
a^{\mu}|a^{\nu}\rangle_{n+1}-\langle
a^{\mu}|a^{\nu}\rangle_{n}=\langle\hat{c}^{\mu}|\hat{c}^{\nu}\rangle_{n}-\langle
c^{\mu}|c^{\nu}\rangle_{n},$ (20)

where $\langle a^{\mu}|a^{\nu}\rangle_{n}$ is the inner product of
$|a^{\mu}\rangle_{n}$ and $|a^{\nu}\rangle_{n}$. The equality follows from the
fact that $G^{\mu\nu}(A+C)$ is invariant under ${\mathcal{A}}_{n1}$ and
${\mathcal{A}}_{n2}$, and additive:
$G^{\mu\nu}(A+C)=G^{\mu\nu}(A)+G^{\mu\nu}(C)$. To discuss the total change
during $N$ rounds, $n=1,2,\ldots N$, it is convenient to define

$|C^{\mu}\rangle:=\\{|c^{\mu}\rangle_{1},|c^{\mu}\rangle_{2},\ldots|c^{\mu}\rangle_{N}\\},\quad|\hat{C}^{\mu}\rangle:=\\{|\hat{c}^{\mu}\rangle_{1},|\hat{c}^{\mu}\rangle_{2},\ldots|\hat{c}^{\mu}\rangle_{N}\\}$
(21)

with inner products

$\langle C^{\mu}|C^{\nu}\rangle=\sum_{n=1}^{N}\langle
c^{\mu}|c^{\nu}\rangle_{n},\quad\langle\hat{C}^{\mu}|\hat{C}^{\nu}\rangle=\sum_{n=1}^{N}\langle\hat{c}^{\mu}|\hat{c}^{\nu}\rangle_{n}.$
(22)

Summing (20) over $N$ rounds yields the following formula

$\Delta
G^{\mu\nu}(A)=G^{\mu\nu}_{N}(A)-G^{\mu\nu}_{0}(A)=\langle\hat{C}^{\mu}|\hat{C}^{\nu}\rangle-\langle
C^{\mu}|C^{\nu}\rangle,$ (23)

for the total change in the $A$ portion of the Gram matrix during the full
protocol. This quantity is bounded by

$|\Delta
G^{\mu\nu}(A)|\leq|\langle\hat{C}^{\mu}|\hat{C}^{\nu}\rangle|+|\langle
C^{\mu}|C^{\nu}\rangle|\leq\|\hat{C}^{\mu}\|\cdot\|\hat{C}^{\nu}\|+\|C^{\mu}\|\cdot\|C^{\nu}\|$
(24)

using the norm $\langle C^{\mu}|C^{\mu}\rangle=\|C^{\mu}\|^{2}$.

Next define the total Cost $K^{\mu}$ for Alice-to-Bob and $\hat{K}^{\mu}$ for
Bob-to-Alice uses of the channel, with $Q^{\mu}$ their sum:

$K^{\mu}=\langle
C^{\mu}|C^{\mu}\rangle=\|C^{\mu}\|^{2},\quad\hat{K}^{\mu}=\langle
C^{\mu}|C^{\mu}\rangle=\|\hat{C}^{\mu}\|^{2},\quad
Q^{\mu}=K^{\mu}+\hat{K}^{\mu}.$ (25)

Combining (24) and (25) gives

$|\Delta
G^{\mu\nu}(A)|\leq\sqrt{K^{\mu}K^{\nu}}+\sqrt{\hat{K}^{\mu}\hat{K}^{\nu}}\leq\sqrt{Q^{\mu}Q^{\nu}}.$
(26)

This yields an upper bound

$\Delta G^{\mu\mu}(A)\leq Q^{\mu}$ (27)

for a non-negative diagonal weight, and for the off-diagonal overlap:

$|\Delta G^{\mu\nu}(A)|\leq\sqrt{Q^{\mu}Q^{\nu}}.$ (28)

In the particular case of the 1-bit two-way protocol, Sec. III C, the aim is
to reduce $G^{01}(A)$ from its initial value of $1$ to $0$ after $N$ rounds.
Setting $\mu=0$ and $\nu=1$ in (28), we see that to achieve this result it is
necessarily the case that the Costs $Q^{0}$ and $Q^{1}$ for sending bits
$\lambda=0$ and $\lambda=1$ must satisfy the condition

$Q^{0}Q^{1}\geq 1.$ (29)

This is satisfied as an equality with $Q^{0}=Q^{1}=1$ for the specific
protocols discussed in Sec. III C, which shows that they are optimal if total
Cost is used as a measure. For more general protocols there is no reason to
expect that the two Costs will be equal, and in that case if, say, the Cost
for $\lambda=1$ is made very small, that for $\lambda=0$ must be very large.
This is in fact the case for the original SLAZ protocol, as discussed below in
Sec. IV, which thus provides an interesting illustration of such a tradeoff.

##  IV  The SLAZ Protocol

###  IV A  Description of the Protocol

The original SLAZ protocol differs from the simpler situation discussed in
Sec. III C in two respects. First, it has a _hierarchical structure_ : there
are a large number $M$ of _outer_ rounds or cycles, each of which consists of
a large number $N$ of _inner_ rounds or cycles, and the protocol will succeed
with high probability provided

$1\ll M\ll N.$ (30)

Second, while Bob sends a bit $\lambda=0$ by reflecting the amplitude sent by
Alice back into the channel, for $\lambda=1$ he simply empties the channel,
which can be described as a unitary operation in which the $C$ amplitude is
placed in Bob’s subspace $B$. In addition, the original SLAZ protocol and some
of its modifications involve measurements at intermediate times, and these
will be replaced in the discussion below by unitary operations in the manner
suggested at the end of Sec. II B.

We use a notation

$|\psi\rangle=|a_{1},a_{2},a_{3},a_{4};c;b\rangle$ (31)

of the form introduced in (10), where the $a_{j}$ are scalar amplitudes in
Alice’s domain $A=A_{1}+A_{2}+A_{3}+A_{4}$, $c$ is the amplitude the channel
$C$, and $b$ is in Bob’s domain $B$. Here capital letters are used to denote
subspaces and the corresponding projectors, while lower case letters indicate
(in general complex) scalar amplitudes. While $A_{4}$ and $B$ are one-
dimensional, one can also make these larger spaces for reasons that will
appear during the discussion. An abbreviated notation is often convenient:
$|a_{2},a_{3}\rangle$ in the case of a unitary acting on $A_{2}+A_{3}$ while
all the other amplitudes remain unchanged.

Central to the discussion are unitary operators that represent a rotation by
an angle $\theta$ on a 2-dimensional space:

$R(\theta)|\alpha,\beta\rangle=|\alpha\cos\theta-\beta\sin\theta,\alpha\sin\theta+\beta\cos\theta\rangle.$
(32)

In particular, $R_{M}$ and $R_{N}$, defined in terms of small angles, play a
central role:

$R_{M}:=R(\theta_{M}),\quad\theta_{M}:=\pi/(2M),\quad\quad
R_{N}:=R(\theta_{N}),\quad\theta_{N}:=\pi/(2N).$ (33)

Note in particular that

$(R_{M})^{M}=(R_{N})^{N}=R(\pi/2);\quad
R(\pi/2)\,|\alpha,\beta\rangle=|-\beta,\alpha\rangle.$ (34)

In view of the fact that $\theta_{N}$ is a small angle, the following
approximations turn out to be useful:

$\displaystyle\cos\theta_{N}\approx\exp[-\theta_{N}^{2}/2]=\exp[-\pi^{2}/(8N^{2})]\approx
1-\pi^{2}/(8N^{2});$
$\displaystyle(\cos\theta_{N})^{N}\approx\exp[-\pi^{2}/(8N)]\approx
1-\pi^{2}/(8N)\approx 1,$ (35)

and similarly if $N$ is replaced by $M$.

These approximations are useful for understanding the overall structure of the
protocol, which is the following. At the beginning of outer round $m$, $1\leq
m\leq M$, $R_{M}$ is applied to $A_{1}+A_{2}$ to yield,

$|a_{1},a_{2}\rangle^{\lambda}=R_{M}|\bar{a}_{1},\bar{a}_{2}\rangle^{\lambda},$
(36)

where $\bar{a}_{1}$ and $\bar{a}_{2}$ are the values of these amplitudes at
the end of the previous outer round. In general they depend upon which bit
$\lambda=0$ or $1$ is being transmitted, whence the superscript label, even
though Alice’s operations do not depend upon $\lambda$. The very first outer
round $m=1$ begins by applying (36) to the starting state (31) with $a_{1}=1$
and all the other amplitudes equal to zero.

The initial step (36) of outer round $m$ is followed by a sequence of $N$
inner rounds, each involving the following steps, here displayed using the
type of notation employed in Sec. III C, but now with reference to the
subspace $A_{2}+A_{3}+C$.

$\displaystyle|a_{2},a_{3};c=0\rangle\rightarrow|a^{\prime}_{2},a^{\prime}_{3};c=0\rangle\rightarrow|a^{\prime}_{2},0;a^{\prime}_{3}\rangle$
$\displaystyle\Rightarrow|a^{\prime}_{2},0;(1-\lambda)a^{\prime}_{3}\rangle\rightarrow|a^{\prime}_{2},(1-\lambda)a^{\prime}_{3};0\rangle,$
(37)

where

$|a^{\prime}_{2},a^{\prime}_{3}\rangle=R_{N}|a_{2},a_{3}\rangle.$ (38)

In words, Alice applies the unitary rotation $R_{N}$, (33), to $A_{2}+A_{3}$,
and then maps $A_{3}$ into the empty channel. Next comes Bob’s action,
indicated by $\Rightarrow$, to either reflect the amplitude $a^{\prime}_{3}$
back into $C$ if he is sending $\lambda=0$, or shift it into his domain $B$,
leaving the channel empty if sending $\lambda=1$. Alice, who does not know the
value of $\lambda$, maps whatever is in the channel back into $A_{3}$ by a
unitary that simply exchanges the contents of $A_{3}$ and $C$, and then begins
the next inner round. The result of $N$ inner rounds in succession is

$|a_{2},a_{3}\rangle\rightarrow\begin{cases}|0,a_{2}\rangle\text{ for
$\lambda=0$, }\\\
|(\cos\theta_{N})^{N}a_{2},0\rangle\approx|a_{2},0\rangle\text{ for
$\lambda=1$. }\end{cases}$ (39)

where the $\lambda=1$ approximation is justified when $N$ is very large, see
(35).

Following the $N$ inner rounds Alice completes this outer round by applying a
unitary to $A_{3}+A_{4}$ that empties the contents of $A_{3}$ into $A_{4}$.
For $\lambda=1$, $a_{3}=0$, (39), so this emptying step is trivial, while for
$\lambda=0$ it is nontrivial, and plays a significant role in understanding
the true Costs of the protocol. In the original SLAZ protocol this emptying
step is replaced by a measurement, but instead of a measurement one can just
as well let the amplitudes accumulate in $A_{4}$, which is the perspective
used here. At the end of the protocol after completing $M$ outer rounds the
final result is

$\displaystyle\lambda=0:|a_{1}=1-r_{1},a_{2}=0,a_{3}=0,a_{4}=r_{4},c=0,b=0\rangle$
$\displaystyle\lambda=1:|a_{1}=s_{1},a_{2}=1-s_{2},a_{3}=0,a_{4}=0,c=0,b=s_{b}\rangle,$
(40)

where the quantities denoted by $r_{j}$ and $s_{k}$ are small corrections, of
order $1/M$ or $M/N$ If these are ignored, all the amplitude is in $A_{1}$ for
$\lambda=0$ or $A_{2}$ for $\lambda=1$, and a simple measurement allows Alice
to determine which bit Bob sent.

###  IV B  Calculation of Costs and Overlap

It is fairly straightforward to work out the Costs for the SLAZ protocol using
approximations justified by $1\ll M\ll N$, and the results are summarized in
Sec. IV C below. We begin with the case $\lambda=1$. If one ignores small
quantities, the nonzero components of $|\psi\rangle_{m}$ at the beginning and
at the end of outer round $m$ are

$a_{1}=\cos(m\theta_{M}),\quad a_{2}=\sin(m\theta_{M}),$ (41)

and since $M\theta_{M}=\pi/2$, at the end of outer round $M$ the result is the
$\lambda=1$ line in (40).

The probability that the photon arrives in $B$ during outer round $m$—the
probability that Bob will detect it if he uses a measuring device—is the sum
of the absolute squares of the amplitudes in the channel $C$ in the $N$ inner
rounds, as this is an incoherent process:

$N(\sin(m\theta_{M}))^{2}(\sin(\theta_{N}))^{2}\approx(\pi^{2}/4)(\sin(m\theta_{M}))^{2}/N.$
(42)

Summing over $m$ gives the total probability

$K^{1}=Q^{1}=(\pi^{2}/8)(M/N).$ (43)

that the photon will end up in Bob’s domain by the end of the protocol, which
is the same as the total Cost for $\lambda=1$.

In the case $\lambda=0$, any amplitude placed by Alice in $C$ is immediately
returned by Bob, and at the end of each outer round is emptied into $a_{4}$,
so that at the end of outer round $m$ the state is

$|\psi\rangle_{m}=|a_{1}=(\cos\theta_{M})^{m},a_{2}=0,a_{3}=0,a_{4},c=0,b=0\rangle.$
(44)

For $m=M$ this is (40) with $r_{1}=\pi^{2}/(8M)$. Thus at the end of the
protocol $a_{2},a_{3},c$ and $b$ are strictly zero. The Cost associated with
inner round $n$—note that the channel is used twice—is

$2[\sin\theta_{M}\cdot\sin(n\pi/2N)]^{2}.$ (45)

Summing over $n$ gives a total of $(\pi^{2}/4)(N/M^{2})$ for each outer round,
and hence for $M$ outer rounds a total Cost of

$Q^{0}=(\pi^{2}/4)(N/M).$ (46)

To compute the total change in overlap $\Delta G^{01}(A)$, note that since for
$\lambda=1$ Bob does not return an amplitude, only the $\langle
C^{\mu}|C^{\nu}\rangle$ term in (23) contributes. The contribution for inner
round $n$ of outer round $m$ is the product of the factors

$[\sin\theta_{M}\sin(n\theta_{N})]\cdot[\sin(m\theta_{M})\sin\theta_{N}]$ (47)

corresponding to $\lambda=0$ and $1$. Summing them yields

$(\sin\theta_{M}\sin\theta_{N})\sum_{m,n}^{M,N}\sin(m\theta_{M})\sin(n\theta_{N})=(\pi^{2}/4MN)(4MN/\pi^{2})=1,$
(48)

and hence

$\Delta G^{01}(A)=-1,$ (49)

as expected.

###  IV C Discussion of Costs and Probabilities

To summarize the results of Sec. IV B: The total Costs $Q^{0}$ and $Q^{1}$ for
$\lambda=0$ and $1$ are:

$Q^{0}=(\pi^{2}/4)(N/M),\quad Q^{1}=(\pi^{2}/8)(M/N),\quad Q^{0}Q^{1}\approx
3.044.\quad Q^{0}/Q^{1}=2N^{2}/M^{2}.$ (50)

Given that $M\ll N$, $Q^{1}$ is miniscule, $Q^{0}$ is enormous, while their
product is of order $1$, and satisfies the rigorous bound (29). The case
$\lambda=1$ is the easiest to understand. Since Bob does not return the
amplitude put into the channel by Alice, the Bob-to-Alice Cost $\hat{K}^{1}$
is zero. The Alice-to-Bob Cost is $|s_{b}|^{2}$ in (40), i.e., the probability
that at the very end the photon is in Bob’s domain. The physical reason for
this is that the process by which the amplitude gets there is _incoherent_ ,
no quantum interference, since no amplitude goes back through the channel. Bob
could either accumulate these amplitudes until the end of the protocol and
then measure to see if the photon is in $B$, or carry out a measurement at the
end of each inner round; in either case the probability of his detecting the
photon is $|s_{b}|^{2}$ in (40). The situation is analogous to the use of
intermediate time measurements in a one-way protocol as discussed at the end
of Sec. II B.

The enormous Cost $Q^{0}$ for $\lambda=0$ comes about because Bob repeatedly
returns the amplitude sent by Alice in a _coherent_ process. While the
amplitude bouncing back and forth through the channel is relatively small, of
order $1/M$, multiplying its absolute square by $2N$, the number of times this
amplitude is is in the channel during each outer round, leads to a Cost of
order $N/M^{2}$ for each outer round, and hence a total of order $N/M$ for the
complete process.

Clearly the large value of $Q^{0}$ means the claim that the protocol is
counterfactual cannot be maintained if Cost is used as a criterion for channel
use, so it is worth discussing how the authors of SLAZ reached a different
conclusion. In essence their reasoning was based on the small value of the
amplitude in $A_{3}$ at the end of an outer round just before it is
transferred to $A_{4}$, as per the discussion in Sec. IV A. The absolute
square of this amplitude is the probability that the corresponding detector
$D_{3}$ in Fig. 2(b) in the SLAZ paper will be triggered. This amplitude was
earlier oscillating back and forth inside the subspace with projector
$S=A_{2}+A_{3}+C$, and hence it is reasonable to assume that if this detector
triggers, the photon was earlier in $S$ during all $N$ inner rounds making up
this particular outer round111This assumption can be justified using
Consistent Histories; see the discussion of measurements in [5, 6]. As this
probability is of order $1/M^{2}$, the probability that one of the $D_{3}$
detectors triggers during the $M$ outer rounds that make up a given run is of
order $1/M$, and hence small.

There are two serious objections to using this small probability to justify
the claim that the protocol is counterfactual: one classical and the other
quantum. Let us start with the former. During a particular outer round the
photon amplitude in a $\lambda=0$ run rattles back and forth inside $S$ a
total of $N$ times, and in particular it is in $C$ a total of $2N$ times.
Consider a stochastic classical protocol for transmitting information in which
most of the time Alice and Bob exchange no information at all. However, with a
small probability $\epsilon$ Alice sends a little white ball into the channel
leading to Bob, who colors it green or red and sends it back to Alice to
convey one bit of information. She records the color, paints the ball white,
and returns it to Bob who again colors it to send a second bit, and so forth,
for a total of $N$ rounds. The average rate of transmitting information is
$N\epsilon$ bits, and one cannot simply throw away the factor of $N$ and claim
that this protocol is in some sense ‘counterfactual’.

The quantum difficulty has to do with what can be inferred from the
probability that the photon was in $S=A_{2}+A_{3}+C$ during the inner rounds
that make up a particular outer round. One may be tempted to use classical
reasoning and assume that the probabilities of being in each of the mutually
exclusive regions, $A_{2}$, $A_{3}$, and $C$, that combine to make up $S$ are
well-defined and sum to the probability of being in $S$. But in the presence
of quantum interference this sort of reasoning is invalid and leads to
paradoxes. See the discussion of parallel channels in Sec. II A.

##  V  Conclusion

The original SLAZ proposal has motivated a large number of papers; see the
extensive bibliographies in [3, 4]. Merely trying to summarize them, much less
provide a detailed review, lies outside the scope of the present paper.
Broadly speaking, this literature consists of modifications, extensions, or
improvements of the original SLAZ scheme; along with criticisms of the claim
that these protocols are counterfactual and replies to such criticisms. It is
hoped that the following rather brief comments will provide some orientation.

Significant extensions of the original SLAZ scheme by the last three members
of the original collaboration include: the use of a phase change rather than
absorption to transmit the $\lambda=1$ bit [7]; a scheme to transmit quantum
states by multiple iterations of the original SLAZ scheme [8]; using many
photons in place of a single photon to transmit a classical bit[4]. These and
others are certainly interesting ideas from the perspective of transmitting
quantum information, and worth further exploration.

On the other hand, in these and all other extensions or modifications of SLAZ
this author has examined, the claim that the protocol is “counterfactual,” in
the sense that the total use of a quantum channel is negligible in the
asymptotic limit, is subject to the same objections discussed in Sec. IV C: An
incorrect use of probabilistic reasoning in a situation where quantum
interference means probabilities cannot be defined, and where even in a
classical situation Cost would be better than probability as a measure of
channel usage. The total Cost remains finite in the asymptotic limit of a very
large number of steps, which means that counterfactual claims should be
dropped. Doing so will aid, not hinder, the serious study of these interesting
quantum schemes for transmitting information.

Shortly after the original SLAZ publication, Vaidman published a Comment[9]
claiming that in the $\lambda=0$ case in which Bob reflects the amplitude
rather than absorbing it, the photon which was later (with high probability)
detected by Alice must at an earlier time have been in the channel $C$. In
their Reply[10] the SLAZ authors pointed out this way of reasoning about
events at an intermediate time in the presence of quantum interference was
invalid, and leads to paradoxes, a position supported by the analysis in Sec.
IV B above. However, they then repeated their original counterfactual claim
which itself is based on a defective understanding of probabilities at an
intermediate time. A later and much more extended criticism of
counterfactuality claims by Vaidman [11] suffers from the same difficulty as
his earlier Comment.

Some years later Aharonov and Vaidman[12] claimed to have found a scheme of
the general SLAZ type which is genuinely counterfactual. However, when
measurements or absorption of a photon at intermediate times are replaced by
unitary processes—mapping amplitude into an empty subspace reserved for this
purpose, as discussed in Sec. IV A—the inequality in Sec. III D applies to
this case and undermines the counterfactual claim. The fundamental difficulty
with such claims is that the Hilbert space projector which identifies the
position of a particle at some intermediate time does not commute with the one
representing the quantum state evolving unitarily in time.

The most significant contributions of the present paper to the analysis of
SLAZ-type protocols is the use of Cost as a measure of channel usage, and the
use of Gram matrices for discussing information transfer at intermediate times
in the presence of quantum interference. In particular, the fact that these
Gram matrices are additive over subspaces and invariant (“conserved”) under
unitary time transformations, plays a key part in the discussions in Sec. III.
A rather surprising feature is the role of off-diagonal elements, “overlaps”,
as a type of information measure which, unlike most such measures, is not in
general positive. That it can be negative plays a very significant part in
understanding its intuitive role in information transfer. That its total
change on Alice’s side must be $-1$ during the course of a successful protocol
is confirmed for the SLAZ protocol in Sec. IV B.

This use of Gram matrices requires that the intermediate time steps be
unitary. In the case of SLAZ, measurements at intermediate times can be
eliminated by mapping photon amplitude into empty subspaces, and this can be
achieved in certain other cases, e.g., the Aharonov and Vaidman protocol[12].
However, it is less clear whether something similar could be done in a case in
which, for example, Alice uses measurements at intermediate times to change
later steps in the protocol in hopes of reducing the total Cost. This author
believes that such an improvement is impossible, because measurements
themselves are quantum processes whose description simply requires a large
enough Hilbert space in Alice’s domain [13]. But this has not yet been
demonstrated.

And what is special about _classical_ information? Sending an arbitrary one-
qubit quantum state from Alice to Bob using the 2-way protocol of Sec. III C
could be done with a Cost of 2, which is to say twice that of simply using a
1-way protocol from Bob to Alice. That this is the minimum seems likely, but
has not been demonstrated. What about a two-way protocol with all the
amplitude starting on Alice’s side, with the aim of a perfect transmission of
each of two specified _nonorthogonal_ states from Bob to Alice—what would be
the minimum total Cost?

An interesting feature of the original SLAZ protocol is the enormous ratio
$2N^{2}/M^{2}$, see (50), of the Costs to transmit $\lambda=0$ and $1$, in
contrast to the relatively simple protocols discussed in Sec. III C for which
the ratio is $1$. Because the success of SLAZ depends upon $N$ being much
larger than $M$, this large ratio presumably has something to do with Bob’s
not sending anything back through the channel when $\lambda=1$. Might there be
some interesting physical principles, in addition to the Zeno effect, hiding
here and waiting to be explored?

In conclusion it is hoped that the thinking and tools employed in this paper
will be useful for studying other problems of quantum information at
intermediate times in situations where the careless use of ill-defined
probabilities generates paradoxes rather than physical understanding. In
particular, information transfer among three or more parties, of current
interest in the study of quantum networks, might benefit from the sort of
analysis used here.

## Acknowledgements

The author thanks Scott Cohen for correspondence and an anonymous referee for
suggestions for improving the presentation. He is also grateful to Carnegie-
Mellon University and its Physics Department for continuing support of his
activities as an emeritus faculty member.

## References

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#  AmbiCoref:
Evaluating Human and Model Sensitivity to Ambiguous Coreference

Yuewei Yuan  Chaitanya Malaviya  Mark Yatskar
University of Pennsylvania
{yuewei, cmalaviy<EMAIL_ADDRESS>

###### Abstract

Given a sentence “Abby told Brittney that she upset Courtney”, one would
struggle to understand who “she” refers to, and ask for clarification.
However, if the word “upset” were replaced with “hugged”, “she” unambiguously
refers to Abby. We study if modern co-reference resolution models are
sensitive to such pronominal ambiguity. To this end, we construct AmbiCoref, a
diagnostic corpus of minimal sentence pairs with ambiguous and unambiguous
referents. Our examples generalize psycholinguistic studies of human
perception of ambiguity around particular arrangements of verbs and their
arguments. Analysis shows that (1) humans are less sure of referents in
ambiguous AmbiCoref examples than unambiguous ones, and (2) most coreference
models show little difference in output between ambiguous and unambiguous
pairs. We release AmbiCoref as a diagnostic corpus for testing whether models
treat ambiguity similarly to humans.111Our dataset and code is available at
https://github.com/LucyYYW/AmbiCoref.

##  1 Introduction

6 |  Type | Ambig. |  Template |  Count
---|---|---|---|---
0.01 1 |  Experiencer Obj (ECO-1) | ✗ |  $[{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}Emily}]_{A}$ told $[Jessica]_{B}$ that $[\emph{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}she}]_{A}$ [saw] [Brian]. |  11336
2 |  Experiencer Obj (ECO-1) | ✓ |  $[Emily]_{A}$ told $[Jessica]_{B}$ that $[\emph{she}]_{?}$ [bored] [Brian]. |  11336
3 |  Experiencer Obj (ECO-2) | ✗ |  $[{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}The\leavevmode\nobreak\ mother}]_{A}$ told $[the\leavevmode\nobreak\ sister]_{B}$ that $[\emph{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}she}]_{A}$ [saw] the client. |  11336
4 |  Experiencer Obj (ECO-2) | ✓ |  $[The\leavevmode\nobreak\ mother]_{A}$ told $[the\leavevmode\nobreak\ sister]_{B}$ that $[\emph{she}]_{?}$ [bored] the client. |  11336
0.01 5 |  Experiencer Sub (ECS-1) | ✗ |  $[{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}The\leavevmode\nobreak\ aunt}]_{A}$ told $[Sarah]_{B}$ that [the daughter] [met with] $[\emph{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}her}]_{A}$. |  4472
6 |  Experiencer Sub (ECS-1) | ✓ |  $[The\leavevmode\nobreak\ aunt]_{A}$ told $[Sarah]_{B}$ that [the daughter] [liked] $[\emph{her}]_{?}$. |  4472
7 |  Experiencer Sub (ECS-2) | ✗ |  $[{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}The\leavevmode\nobreak\ father}]_{A}$ told $[the\leavevmode\nobreak\ son]_{B}$ that the client [met with] $[\emph{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}him}]_{A}$. |  4472
8 |  Experiencer Sub (ECS-2) | ✓ |  $[The\leavevmode\nobreak\ father]_{A}$ told $[the\leavevmode\nobreak\ son]_{B}$ that the client [liked] $[\emph{him}]_{?}$. |  4472
0.01 9 |  Implicit Causality (IC) | ✗ |  $[{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}Abby}]_{A}$ [called] $[Jane]_{B}$ because $[\emph{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}she}]_{A}$ [wanted to apologize]. |  8424
10 |  Implicit Causality (IC) | ✓ |  $[Abby]_{A}$ [called] $[Jane]_{B}$ because $[\emph{she}]_{?}$ [is leaving soon]. |  8424
0.01 11 |  Transfer (TOP) | ✗ |  $[Daniel]_{A}$ [baked] $[{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}the\leavevmode\nobreak\ boy}]_{B}$ [a cake] [after] $[\emph{{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}he}}]_{B}$ [asked for one]. |  8424
12 |  Transfer (TOP) | ✓ |  $[Daniel]_{A}$ [baked] $[the\leavevmode\nobreak\ boy]_{B}$ [a cake] [before] $[\emph{he}]_{?}$ [had lunch]. |  8424
0.01 |  |  |  |
Table 1: Summary of the six template pairs that make up AmbiCoref. Template
slot are indicated in square bracket, and clusters are marked with subscripts
and color. All templates pair an unambiguous sentence with an ambiguous
sentence, where they differ only in the choice of verb phrase.

Ambiguity is a fundamental feature of language (Wasow et al., 2003) that some
linguists believe arises because of a pressure for efficient communication
(Haywood et al., 2005; Piantadosi et al., 2012). Recently, several works have
highlighted the existence of ambiguity in tasks such as question answering
(Min et al., 2020; Guo et al., 2021), frame disambiguation Dumitrache et al.
(2019), anaphora resolution (Poesio and Artstein, 2005) and language modeling
Aina and Linzen (2021). Yet systematic evaluation of how models react to
ambiguity across many types of language processing problems is missing. We
contribute one such study about coreference resolution.

Coreference resolution is crucial to natural language understanding,
especially in long contexts, such as dialog. Ambiguity may arise naturally in
dialog, but existing models do not have well-defined target behavior for such
coreferences. In contrast, when people encounter coreferential ambiguity, they
recognize it, and can ask for clarification. Existing resources, such as
OntoNotes Weischedel et al. (2013), do not provide fine-grained annotations of
such instances to evaluate model behavior. This may result in models not being
calibrated to handle the uncertainty in interpretations of ambiguous
statements. In this work, we ask how sensitive to ambiguity are models trained
on these resources?

To understand how existing coreference models react to ambiguity, we construct
a diagnostic corpus, AmbiCoref. AmbiCoref is composed of minimal pairs with
ambiguous and unambiguous referents, created from four types of templates.
Ambiguity is achieved by reducing context sizes to one sentence, and creating
sentences where participating verbs under-constrain the interpretation of
their arguments. For example, in Table 1, line 2, our first template leverages
ambiguity around verbs expressing subjective experiences.222Such instances
require specific syntactic arrangements: the ambiguous instance in line 2 is
unambiguous if the pronoun is moved to the object position of bored. The
templates are designed by drawing on psycholinguistic studies Springston
(1976); Caramazza et al. (1977); Rohde and Kehler (2014) and a core
contribution of our work is to generalize their observations to create
thousands of instances. We achieve this by identifying VerbNet Schuler (2005)
classes that are likely to contain appropriate verbs, and manually assigning
them to templates. Combined with variability we introduce using noun lists,
AmbiCoref contains over 96 thousand sentences.

We verify that humans perceive instances in AmbiCoref in intended ways by
crowdsourcing judgements (§3). Annotators are asked to find the coreferent for
a pronoun in a sentence, and rate their confidence, to account for the
gradience in ambiguity judgements Schutze (1995). We find that, for
unambiguous instances, humans strongly associate the pronoun with the intended
noun but for ambiguous ones, they show reduced confidence across all
templates, where the majority of participants are either not confident or mark
them as ambiguous. This suggests that humans process ambiguous and unambiguous
sentences in AmbiCoref in qualitatively different ways.

AmbiCoref can be used to evaluate model behavior in the presence of ambiguity.
We analyze five representative English models: three in CoreNLP Manning et al.
(2014), SpanBERT Joshi et al. (2020), and NeuralCoref 4.0 Wolf et al. (2020)
(§4). Our main evaluation involves comparing coreference cluster assignments
of the pronoun, between ambiguous and unambiguous samples. 4 out of the 5
models we analyze show almost no behavioral change. Unlike humans, coreference
models largely do not alter their decisions in the presence of ambiguity. Our
analysis implies models likely need to explicitly account for ambiguity to
achieve human-like behavior in the face of ambiguous input.

##  2 Dataset Construction

Figure 1: Human annotation of ambiguous () and unambiguous () sentences. We
abbreviate human annotations by whether they identified noun A or B and
whether they annotate definitely or likely (marked with ?). For example A?
indicates, noun A, likely. The ground truth for unambiguous instances, from
left to right, corresponds to A, A, A, A, A, B. Annotators read unambiguous
examples as intended, and reduce their confidence on ambiguous examples.

To understand model sensitivity towards coreferential ambiguity, we build
AmbiCoref using four types of templates, shown in Table 1. The templates are
created in minimal pairs, and the only difference between the ambiguous and
unambiguous counterparts lies in the choice of verb phrase. Note that while
ambiguity is a graded phenomenon, we use the the term “ambiguous" for
instances that are more likely to elicit ambiguous human judgements and vice-
versa. Verb phrases are extracted from suitable verb classes in VerbNet
Schuler (2005), identified by manual annotation of VerbNet clusters.333We
consider verbs from verb classes 31: Psych-Verbs (Verbs of Psychological
State), 13: Verbs of Change of Possession, 37: Verbs of Communication as they
conceptually align well with conditions required for ambiguity. Verbs within
clusters were individually evaluated for appropriateness for templates by the
authors. Each template is instantiated with verbs, names, noun-phrases, and
gender-appropriate pronouns, greatly expanding the variation in cases
identified in previous studies.

###  2.1 Template Types

#### Experiencer Constraint for Objects (ECO)

Springston (1976) propose the Experiencer Constraint for complement
constructions which we operationalize in our templates. Verbs that mark their
object as the experiencer of an emotion restrict the assignment of an object
position pronoun to the subject of a declarative communication verb.
Conversely, the assignment is unconstrained when the pronoun is the subject of
an experiencer verb. For example, in row 2 of Table 1, a pronoun in the
subject position of “bored” is ambiguous (but would not be so in the object
position). If the main verb does not impose an experiencer constraint, row 1,
then a pronoun in the subject position is unambiguous. We instantiate two
variants with names (rows 1,2) and general entities (rows 3,4).

#### Experiencer Constraint for Subjects (ECS)

The Experiencer Constraint also suggests that verbs that mark their subjects
as the experiencer of the emotion restrict the assignment of a subject
position pronoun. The assignment of the pronoun is unconstrained when it is in
the object position. For example, in Table 1, row 6, “liked” is ambiguous when
a pronoun is placed in the object position (but not in the subject position).
We instantiate variants with names (rows 5,6) and entities (rows 7,8).

#### Implicit Causality (IC)

Caramazza et al. (1977) hypothesize that implicit causality of a verb can
determine the direction of pronoun assignment. For example, in Table 1 row 9,
the phrase “wanted to apologize” establishes a cause for why “Emily called,”
so the pronoun is constrained to the subject of “call”. Conversely, in row 10,
the phrase “is leaving soon” fails to create such a relationship, leaving the
pronoun ambiguous. For these templates (rows 9,10), we vary the names of the
entities involved, and pair verbs (i.e. called) with constructed phrases that
imply causality (i.e. apologizing), manually.

#### Transfer of Possession (TOP)

Rohde and Kehler (2014) suggests that in transfer-of-possession contexts such
as, “John passed the comic to Bill. _He_ …”, the pronoun is equally likely to
refer back to subject and non-subject. We draw upon this observation, and
create a template around verbs that involve source-goal possession transfers.
We distill the example to one sentence and pair the transfer event with a
reason. For example, in Table 1 row 11, the phrase “asked for one” constrains
the pronoun to be the receiver of “bake”. Conversely, before having lunch
provides no such constraint, because either the receiver or giver could have
“had lunch” before the event. Templates vary the names, verbs, objects,
reasons, and preposition (rows 11,12).

###  2.2 Filling Template Slots

For each template, we construct a list of appropriate verb phrases, reasons
(for IC and TOP templates), and shared list of gendered names and noun-
phrases. Verb phrases were constructed by manually inspecting VerbNet classes.
To control for name bias, we randomly sample names from popular name
lists444https://www.ssa.gov/oact/babynames/decades/ from the last 50 years,
and reuse gendered noun-phrase lists from WinoBias Zhao et al. (2018).
Excluding name and noun-phrase variations, templates have 114, 45, 81, 82
instances for ECO, ECS, IC, and TOP, respectively.

##  3 Human Judgements

The templates used to create AmbiCoref generalize several psycholinguistic
studies using lexical resources. Next, we verify that humans perceive
ambiguity in these examples in the intended ways. We extract a subset of data
for each template and ask Amazon Mechanical Turk workers which person a
pronoun refers to (marked as A or B in Table 1) and assign confidence
(definitely, or likely). Annotators were also allowed to mark the referent as
entirely ambiguous. One sentence was sampled for each template and verb slot,
uniformly at random. We collected 3 annotations per instance.555In ambiguous
cases, annotators do not reliably annotate a particular category, but often
guess with low confidence. As such, we do not only report a majority opinion
per instance, but instead simply report multiple annotations per sentence to
see overall trends. See Appendix A for details on the collection of human
judgements.

Figure 2: Percentage of ambiguous () and unambiguous () instances that fall
into each of our five cases for the SpanBERT-based model across all templates.
All other models show negligible shifts (red and grey distributions are almost
identical). The ground truth for unambiguous instances, from left to right,
corresponds to A, A, A, A, A, B.

Figure 1 summarizes our results. Human judgments for unambiguous templates
favor the intended coreference decision. For unambiguous ECO, ECS, IC, TOP
instances, the intended reading is selected as likely or definitely, 83.2%,
91.9%, and 85.8%, 68.3% of the time, respectively. For ambiguous instances,
annotations display a substantial shift toward ambiguity. As shown in previous
work, humans display substantial disagreement on ambiguous instances Poesio et
al. (2019). This is reflected in many templates, such as TOP, where humans
produce almost uniform responses.

##  4 Model Evaluation

We now examine if we can detect sensitivity to ambiguity in existing
coreference resolution models by evaluating on AmbiCoref. We experiment
666Roughly one week of continuous Colab GPU compute. with five representative
models: NeuralCoref 4.0 model from Hugging Face
777https://github.com/huggingface/neuralcoref, SpanBERT Joshi et al. (2020)
representation within the independent framework for end-to-end coreference
Joshi et al. (2019), and the three models in Stanford CoreNLP Manning et al.
(2014): deterministic Lee et al. (2013), statistical Clark and Manning (2015)
and neural mention ranking Clark and Manning (2016). All models were trained
on the CoNLL 2012 dataset Pradhan et al. (2012).

Here, we evaluate the model’s final predictions, not their distribution over
possible choices. The reason is two-fold: (1) not all models produce a
distribution and (2) initial analysis revealed that the models are
miscalibrated, as in other settings Desai and Durrett (2020); Jiang et al.
(2021), making it unreliable to interpret their output scores directly.

###  4.1 Setup

In this section, we ask, are there differences between how models process
similar unambiguous and ambiguous examples? As our examples are synthetically
generated, we use the unambiguous examples as a form of control. If a model is
unable to link the pronoun with the correct noun on unambiguous examples for
at least 40% of examples, we omit that template during evaluation.

We analyze model behavior by breaking it into cases that cover all possible
cluster assignments for the pronoun in a single sentence. We compute the
percentage of time a model outputs a cluster with:

  * •

case A: the pronoun and noun A

  * •

case B: the pronoun and noun B

  * •

case S: the pronoun as a singleton

  * •

case M: the pronoun, noun A, and noun B

  * •

case O: the pronoun and any other span

Model | Mean EMD % | Templates
---|---|---
SpanBERT | 11.7 | 5
CoreNLP Neural | 3.5 | 5
NeuralCoref 4.0 | 4.0 | 5
CoreNLP Statistical | 1.2 | 3
CoreNLP Deterministic | 0.6 | 5
Table 2: Mean Earth Mover’s Distance between matched ambiguous and unambiguous
case distributions and the number of templates where models get at least 40%
of unambiguous cases correct.

For example, Figure 2 contains SpanBERT’s output distribution over these cases
for each template. For each such distribution where the model’s performance is
above threshold, we compare ambiguous (red bar) and unambiguous (grey bar)
distributions using Earth Mover’s Distance (EMD) Pele and Werman
(2009)888Earth Mover’s distances represent the amount of probability mass
required to match two probability distributions. Hence, they help us compare
distributions for ambiguous and unambiguous instances in a more interpretable
way, than other possible measures like KL divergence.. Table 2 reports the
number of templates above threshold, and their mean EMD.

###  4.2 Results

Overall, most models we evaluated show essentially no change in output
distribution over cases between ambiguous and unambiguous templates, as
evidenced by near zero EMD. Most models are evaluated on five of six
templates, but TOP is often excluded, representing a hard unambiguous case for
most systems in its own right.

Of the models we evaluated, only SpanBERT shows significant deviation in
behavior with ambiguous inputs. Figure 2 breaks down SpanBERT’s performance on
each template. While average EMD is higher than for other models, it still
largely doesn’t change predictions. When decisions change, often the pronoun
is linked with the other noun. For example, in ambiguous cases of ECO-1,
SpanBERT reduces merged outputs, and instead links the pronoun with noun B
more frequently. In ambiguous cases, other models largely link the first noun-
phrase (A) to the pronoun.

##  5 Discussion and Conclusion

Overall, our results suggest that model behavior significantly deviates from
how human treat ambiguous coreference. We lend more evidence that models miss
aspects of how people understand language, especially in discourse Upadhye et
al. (2020). The reason is likely in part that models are trained on resources
which do not account for distributions in judgments. As a result, models do
not have well-defined behavior when ambiguity arises and are poorly
calibrated.

Training models with finer-grained coreference judgments could allow models to
better align with human behavior. Techniques to improve model calibration
could also be effective, allowing models to abstain or seek clarification when
ambiguity arises. We hope that AmbiCoref can serve as a diagnostic set for
future modeling approaches in evaluating their sensitivity to instances of
ambiguity in language.

##  6 Limitations

Our study focuses entirely on coreference in the English language with models
trained in high-resource settings. Furthermore, the cases of ambiguity we
identify are English-specific and the names we insert into templates are
popular American names. It is an open question as to how our results
generalize to low-resource non-American-English settings.

The language we use to evaluate models is templatic. While we make an effort
to account for unnatural data, by only evaluating templates models do well at,
models struggle to completely solve all our unambiguous examples. This
presents a challenge for future model builders. On the other hand, our
templates may not reflect a particular real world distribution that models
will be tested on.

## Acknowledgements

We thank Chris Callison-Burch and the PennNLP group for their helpful comments
on this work.

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##  Appendix A Human Judgement Tests

Figure 3: Annotation interface for the human judgement tests, presented in
section 1.

In all our human judgement tests, we required annotators to be based primarily
in English-speaking countries: the US, UK, Canada or Australia. Further,
annotators needed to have at least 1000 approved HITs and a HIT acceptance
rate of at least 98%. Each HIT contained 10 examples, and we estimated the
completion time for each HIT to be $\sim$5 minutes, so we paid $1.25 per HIT,
for a pay rate of $15 per hour.

For our human judgement tests, we first ran a qualification round to ensure
high-quality annotations. In this round, we asked annotators to complete a
single HIT with 10 examples (5 unambiguous, 5 ambiguous randomly ordered). For
each annotator who completed this round, we compute their accuracy by
measuring how often they responded with the correct referent (or the ambiguous
label), while ignoring their confidence. The top 100 annotators were qualified
to work on the main task.

For our main task, we had 625 sentences labeled in total, with 3 assignments
per sentence. Each annotator was asked to work on not more than 5 HITs, so
that we get a diverse set of judgements. Similar to the qualification round,
we asked each annotator to label the referent (or the ambiguous label) and
their confidence. We group the annotations into 5 options: (Noun A,
definitely), (Noun A, likely), Ambiguous, (Noun B, likely), and (Noun B,
definitely). The human judgement labels for each template type were aggregated
by computing the fraction of annotations in each of the five options. Our
annotation interface for the main task is shown in Figure 3.