Quantum nonlocality without entanglement

TLDR: In this paper, it was shown that there is a finite gap between the mutual information obtainable by a joint measurement on these states and a measurement in which only local actions are permitted.

Abstract: We exhibit an orthogonal set of product states of two three-state particles that nevertheless cannot be reliably distinguished by a pair of separated observers ignorant of which of the states has been presented to them, even if the observers are allowed any sequence of local operations and classical communication between the separate observers. It is proved that there is a finite gap between the mutual information obtainable by a joint measurement on these states and a measurement in which only local actions are permitted. This result implies the existence of separable superoperators that cannot be implemented locally. A set of states are found involving three two-state particles that also appear to be nonmeasurable locally. These and other multipartite states are classified according to the entropy and entanglement costs of preparing and measuring them by local operations.

Figures

TABLE I. Entropies, entanglements, and advice for non-Bell ensembles are upper bounds from protocols, actual values could be less. The entropies of measurement for nine-state and four-Bell en are for entanglement-assisted measurement since these ensembles are otherwise not locally measu nine-state ensemble consists of nine equiprobable statesc1 , . . . ,c9 of Eq. ~3! and Fig. 1. The 2468 and 24 ensembles are equiprobable distributions over$c2 ,c4 ,c6 ,c8% and$c2 ,c4 ,c6%, respectively. The four-Bell ensemble consists of four equiprobable Bell states$F1,F2,C1,C2% and the two-Bell ensemble of two equiprobable Bell states, e.g.,$F1,C1%.

FIG. 2. Sequence of measurements performed to distinguish states of Fig. 1 if the statec4 is excluded. The dashed lines indica the von Neumann measurements, the italic numbers indicate order in which they are performed. Dashed-dotted lines indic measurements in the rotated basis.

FIG. 5. Set of four states, shown in the domino notation, wh can be prepared locally by Alice and Bob in a reversible fashio

FIG. 3. Tree depicting the four stages of measurement indic in Fig. 2. A andB indicate the party performing the measureme B0/1 indicates that the 0 and 1 outcomes are not distinguished. boldfaced numbers at the base of the tree indicate the states th inferred from this chain of measurements.

Full text

Quantum nonlocality without entanglement
Charles H. Bennett,
1
David P. DiVincenzo,
1
Christopher A. Fuchs,
2
Tal Mor,
3
Eric Rains,
4
Peter W. Shor,
4
John A. Smolin,
1
and William K. Wootters
5
1
IBM Research Division, T. J. Watson Research Center, Yorktown Heights, New York 10598
2
Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125
3
De
´
partement d’Informatique et de Recherche Ope
´
rationelle, Succursale Centre-Ville, Montre
´
al, Canada H3C 3J7
4
AT&T Shannon Laboratory, 180 Park Avenue, Building 103, Florham Park, New Jersey 07932
5
Physics Department, Williams College, Williamstown, Massachusetts 01267
~Received 17 June 1998!
We exhibit an orthogonal set of product states of two three-state particles that nevertheless cannot be reliably
distinguished by a pair of separated observers ignorant of which of the states has been presented to them, even
if the observers are allowed any sequence of local operations and classical communication between the
separate observers. It is proved that there is a finite gap between the mutual information obtainable by a joint
measurement on these states and a measurement in which only local actions are permitted. This result implies
the existence of separable superoperators that cannot be implemented locally. A set of states are found involv-
ing three two-state particles that also appear to be nonmeasurable locally. These and other multipartite states
are classified according to the entropy and entanglement costs of preparing and measuring them by local
operations. @S1050-2947~99!00302-9#
PACS number~s!: 03.67.Hk, 03.65.Bz, 03.67.2a, 89.70.1c
I. INTRODUCTION
The most celebrated manifestations of quantum nonlocal-
ity arise from entangled states: states of a compound quan-
tum system that admit no description in terms of states of the
constituent parts. Entangled states, by their experimentally
confirmed violations of Bell-type inequalities, provide strong
evidence for the validity of quantum mechanics and can be
used for different forms of information processing, such as
quantum cryptography @1#, entanglement-assisted communi-
cation @2,3#, and quantum teleportation @4#, and for fast
quantum computations @5,6#, which pass through entangled
states on their way from a classical input to a classical out-
put. A related feature of quantum mechanics, also giving rise
to nonclassical behavior, is the impossibility of cloning @7#
or reliably distinguishing nonorthogonal states. Quantum
systems that for one reason or another behave classically
~e.g., because they are of macroscopic size or are coupled to
a decohering environment! can generally be described in
terms of a set of orthogonal, unentangled states.
In view of this, one might expect that if the states of a
quantum system were limited to a set of orthogonal product
states, the system would behave entirely classically and
would not exhibit any nonlocality. In particular, if a com-
pound quantum system consisting of two parts A and B held
by separated observers ~Alice and Bob! were prepared by
another party in one of several mutually orthogonal, unen-
tangled states
c
1
,
c
2
,...,
c
n
unknown to Alice and Bob,
then it ought to be possible to reliably discover which state
the system was in by locally measuring the separate parts.
Also, it ought to be possible to clone the state of the whole
by separately duplicating the state of each part. We show that
this is not the case by exhibiting sets of orthogonal, unen-
tangled states
$
c
i
%
of two-party and three-party systems such
that the states
$
c
i
%
can be reliably distinguished by a joint
measurement on the entire system, but not by any sequence
of local measurements on the parts, even with the help of
classical communication between the observers holding the
separate parts, and the cloning operation
c
i
→
c
i
^
c
i
cannot
be implemented by any sequence of local operations and
classical communication. Some of the features of this kind of
nonlocality appeared in @8#, which presented a set of or-
thogonal states of a bipartite system that cannot be cloned if
Alice and Bob cannot communicate at all. However, the
states in @8# can be cloned if Alice and Bob use one-way
classical communication.
Many more of the nonlocal properties considered in the
present work were anticipated by the measurement protocol
introduced by Peres and Wootters @9#. Their construction in-
dicates the existence of a nonlocality dual to that manifested
by entangled systems: Entangled states must be prepared
jointly, but exhibit anomalous correlations when measured
separately; the Peres-Wootters states are unentangled and
can be prepared separately, but exhibit anomalous properties
when measured jointly. We note that such anomalies are at
the heart of recent constructions for attaining the highest pos-
sible capacity of a quantum channel for the transmission of
classical data @10–13#.
In the Peres-Wootters scheme, the preparator chooses one
of three linear polarization directions 0°,60°, or 120° and
gives Alice and Bob each one photon polarized in that direc-
tion. Their task is to determine which of the three polariza-
tions they have been given by a sequence of separate mea-
surements on the two photons, assisted by classical
communication between them, but they are not allowed to
perform joint measurements, share entanglement, or ex-
change quantum information.
Of course, because the three two-photon states are nonor-
thogonal, they cannot be cloned or reliably distinguished,
even by a joint measurement. However, Peres and Wootters
performed numerical calculations that provided evidence
~more evidence on an analogous problem was provided by
PHYSICAL REVIEW A FEBRUARY 1999VOLUME 59, NUMBER 2
PRA 59
1050-2947/99/59~2!/1070~22!/$15.00 1070 ©1999 The American Physical Society

the work of Massar and Popescu @14#! indicating that a
single joint measurement on both particles yielded more in-
formation about the states than any sequence of local mea-
surements. Thus unentangled nonorthogonal states appear to
exhibit a kind of quantitative nonlocality in their degree of
distinguishibility. The discovery of quantum teleportation,
incidentally, grew out of an attempt to identify what other
resource, besides actually being in the same place, would
enable Alice and Bob to make an optimal measurement of
the Peres-Wootters states.
Another antecedent of the present work is a series of pa-
pers @15–17# resulting in the conclusion @17# that several
forms of quantum key distribution @18# can be viewed as
involving orthogonal states of a serially presented bipartite
system. These states cannot be reliably distinguished by an
eavesdropper because she must let go of the first half of the
system before she receives the second half. In this example,
the serial time ordering is essential: if, for example, the two
parts were placed in the hands of two separate classically
communicating eavesdroppers rather than being serially pre-
sented to one eavesdropper, the eavesdroppers could easily
cooperate to identify the state and break the cryptosystem.
In this paper we report a form of nonlocality qualitatively
stronger than either of these antecedents. We extensively
analyze an example in which Alice and Bob are each given a
three-state particle and their goal is to distinguish which of
nine product states
c
i
5
u
a
i
&
^
u
b
i
&
, i5 1,...,9, the com-
posite 333 quantum system was prepared in. Unlike the
Peres-Wootters example, these states are orthogonal,sothe
joint state could be identified with perfect reliability by a
collective measurement on both particles. However, the nine
states are not orthogonal as seen by Alice or Bob alone, and
we prove that they cannot be reliably distinguished by any
sequence of local measurements, even permitting an arbi-
trary amount of classical communication between Alice and
Bob. We call such a set of states locally immeasurable and
give other examples, e.g., a set of two mixed states of two
two-state particles @quantum bits ~qubits!# and sets of four or
eight pure states of three qubits, which apparently cannot be
reliably distinguished by any local procedure despite being
orthogonal and unentangled.
In what sense is a locally immeasurable set of states
‘‘nonlocal?’’ Surely not in the usual sense of exhibiting phe-
nomena inexplicable by any local hidden variable ~LHV!
model. Because the
c
i
are all product states, it suffices to
take the local states
a
i
and
b
i
, on Alice’s and Bob’s side
respectively, as the local hidden variables. The standard laws
of quantum mechanics ~e.g., Malus’s law!, applied separately
to Alice’s and Bob’s subsystems, can then explain any local
measurement statistics that may be observed. However, an
essential feature of classical mechanics, not usually men-
tioned in LHV discussions, is the fact that variables corre-
sponding to real physical properties are not hidden, but in
principle measurable. In other words, classical mechanical
systems admit a description in terms of local unhidden vari-
ables. The locally immeasurable sets of quantum states we
describe here are nonlocal in the sense that, if we believe
quantum mechanics, there is no local unhidden variable
model of their behavior. Thus a measurement of the whole
can reveal more information about the system’s state than
any sequence of classically coordinated measurements of the
parts.
The inverse of local measurement is local preparation, the
mapping from a classically provided index i to the desig-
nated state
c
i
, by local operations and classical communica-
tion. If the states
c
i
are unentangled, local preparation is
always possible, but for any locally immeasurable set of
states this preparation process is necessarily irreversible in
the thermodynamic sense, i.e., possible only when accompa-
nied by a flow of entropy into the environment. Of course if
quantum communication or global operations were allowed
during preparation, the preparation could be done reversibly,
provided the states being prepared are orthogonal.
By eliminating certain states from a locally immeasurable
set @such as
$
c
1
,...,
c
9
%
in Eq. ~3! below#, we obtain what
appears to be a weaker kind of nonlocality, in which the
remaining subset of states is both locally preparable and lo-
cally measurable, but in neither case ~as far as we have been
able to discover! by a thermodynamically reversible process.
Curiously, in these situations, the entropy of preparation ~by
the best protocols we have been able to find! exceeds the
entropy of measurement.
Besides entropies of preparation and measurement we
have explored other quantitative measures of nonlocality for
unentangled states. One obvious measure is the amount of
quantum communication that would be needed to render an
otherwise local measurement process reliable. Another is the
mutual information deficit when one attempts to distinguish
the states by the best local protocol. Finally, one can quantify
the amount of advice, from a third party who knows i, that
would be sufficient to guide Alice and Bob through an oth-
erwise local measurement procedure.
The results of this paper also have a bearing on, and were
directly motivated by, a question that arose recently in the
context of a different problem in quantum information pro-
cessing. This is the problem of entanglement purification,in
which Alice and Bob have a large collection of identical
bipartite mixed states that are partially entangled. Their ob-
ject is to perform a sequence of operations locally, i.e., by
doing quantum operations on their halves of the states and
communicating classically, and end up with a smaller num-
ber of pure, maximally entangled states. Recently, bounds on
the efficiency of this process have been studied by Rains @19#
and Vedral and Plenio @20#; other constraints on entangle-
ment purification by separable superoperators have recently
been studied by Horodecki et al. @21#.
In their work, they represent the sequence of operations
using the theory of superoperators, which can describe any
combination of unitary operations, interactions with an ancil-
lary quantum system or with the environment, quantum mea-
surement, classical communication, and subsequent quantum
operations conditioned on measurement results. In the
operator-sum representation of superoperators developed by
Kraus and others, the general final state S(
r
) of the density
operator of the system is written as a function of the initial
state
r
as
S
~
r
!
5
(
k
S
k
r
S
k
†
. ~1!
The S
k
operators appearing in this equation will be referred
PRA 59 1071QUANTUM NONLOCALITY WITHOUT ENTANGLEMENT

to as ‘‘operation elements.’’ A trace-decreasing superopera-
tor satisfies the condition 0<
(
k
S
k
†
S
k
,1 and is appropriate
for describing the effect of arbitrary quantum measurements
on the system ~see @22#, Sec. III!, while a trace-preserving
superoperator specified by
(
k
S
k
†
S
k
51 describes a general
time evolution of the density operator if a measurement is
not made or its outcomes are ignored @23#. Reference @24#
has a useful general review of the superoperator formalism.
To impose the constraint that Alice and Bob act only lo-
cally, Rains and Vedral and Plenio restricted their attention
to separable superoperators, in which the operation elements
have a direct product form involving an Alice operation and
a Bob operation:
S
k
5A
k
^ B
k
. ~2!
We will show in Sec. II B ~see also @22#, Sec. IXC! that all
operations that Alice and Bob can perform during entangle-
ment purification bilocally, in which they can perform local
quantum operations and communicate classically, can be
written in this separable form. This was enough for the deri-
vation of valid upper bounds on the efficiency of entangle-
ment purification. However, the natural question that this led
to is the converse, that is, can all separable superoperators be
implemented by bilocal operations?
The answer to this question is definitely no, as a result of
the examples that we analyze in this paper. Quantum mea-
surements are a subset of the superoperators, and measure-
ments involving only product states are separable superop-
erators. Thus our proof that some unentangled states cannot
be distinguished locally shows that some separable superop-
erators cannot be implemented by only separate operations
by Alice and Bob with classical communication between
them. This indicates that any further investigations of en-
tanglement purification protocols involving separable super-
operators will have to be performed with some caution.
This paper is organized as follows. Section II presents the
333 example and sketches the proof that these states cannot
be distinguished by local measurements. Appendix B gives
many of the important details of this proof and Appendix A
supplies a crucial technical detail that all superoperators can
be decomposed into a sequence of very weak measurements.
Section III shows how the measurement can be done locally
if some states are excluded and presents the best measure-
ment strategy we have found for distinguishing ~imperfectly!
all nine states. Section IV shows how the measurement can
be done for the 333 example if entanglement is supplied.
Section V analyzes the thermodynamics of local state mea-
surement, studying the heat generated in measurement and in
state preparation; Appendix C gives some details. Section VI
analyzes a three-party 23232 example involving eight pure
states. Section VII gives other compact examples ~four pure
states in a 23232 system, two mixed states in a 232 sys-
tem! and poses some questions for the future ~Appendix D
gives details of a specific problem considered there!.
II. A SEPARABLE MEASUREMENT
THAT IS NOT BILOCAL
A. Ensemble of states in a 333 Hilbert space
We will consider the following complete, orthonormal set
of product states
c
i
5
u
a
i
&
^
u
b
i
&
. They exist in a nine-
dimensional Hilbert space, with Alice and Bob each possess-
ing three dimensions. We will use the notation
u
0
&
,
u
1
&
, and
u
2
&
for the bases of Alice’s and Bob’s Hilbert spaces. The
orthonormal set has the form u
a
&^u
b
&:
c
1
5
u
1
&
^
u
1
&
,
c
2
5
u
0
&
^
u
011
&
,
c
3
5
u
0
&
^
u
021
&
,
c
4
5
u
2
&
^
u
112
&
,
c
5
5
u
2
&
^
u
122
&
, ~3!
c
6
5
u
112
&
^
u
0
&
,
c
7
5
u
122
&
^
u
0
&
,
c
8
5
u
011
&
^
u
2
&
,
c
9
5
u
021
&
^
u
2
&
.
Here
u
061
&
stands for (1/
A
2)(
u
0
&
6
u
1
&
), etc. Figure 1
shows a suggestive graphical way to depict the nine states of
Eq. ~3! in the 33 3 Hilbert space of Alice and Bob. The four
dominoes represent the four pairs of states that involve su-
perpositions of the basis states. State
c
1
is clearly special in
that it involves no such superposition.
B. Measurement
We will show that the separable superoperator S(
r
)
5
(
i
S
i
r
S
i
†
consisting of the projection operators
S
i
5
u
i
&
A
u
i
&
B
^
c
i
u
~4!
cannot be performed by local operations of Alice and Bob,
even allowing any amount of classical communication be-
tween them. In Eq. ~4! the output Hilbert space is different
from the input; it is a space in which both Alice and Bob
separately have a complete and identical record of the out-
come of the measurement. See Sec. VII for a discussion of
FIG. 1. Graphical depiction of the nine orthogonal states of Eq.
~3! as a set of dominoes.
1072 PRA 59
CHARLES H. BENNETT et al.

why we use the particular form of Eq. ~4! for the operator;
note that the input state need not be present at the output in
Eq. ~4!.
Since this superoperator corresponds to a standard von
Neumann measurement, we can equally well consider the
problem in the form of the following game. Alice and Bob
are presented with one of the nine orthonormal product states
~for the time being, with equal prior probabilities, let us say!.
This is not important; it is only important that the prior prob-
abilities of states
c
2
through
c
9
be nonzero!. Their job is to
agree on a measurement protocol with which they can deter-
mine, with vanishingly small error, which of the nine states it
is, adhering to a bilocal protocol.
Let us characterize bilocal protocols a little more explic-
itly. Our discussion will apply both to bilocal measurements
and to bilocal superoperators ~in which the measurement out-
comes may be traced out!. By prior agreement one of the
parties, let us say Alice, initiates the sequence of operations.
The most general operation that she can perform locally is
specified by the set of operation elements
A
r1
^ I. ~5!
We will immediately specialize to the case where each value
r1 labels a distinct ‘‘round 1’’ measurement outcome that
she will report to Bob, since no protocol in which she with-
held any of this information from Bob could have greater
power. She cannot act on Bob’s state, so her operators are
always the identity I on his Hilbert space. A
r1
can also in-
clude any unitary operation that Alice may perform before or
after the measurement. Note also that the operator A
r1
may
not be a square matrix; the final Hilbert space dimension
may be smaller ~but this would never be useful! or larger
~because of the introduction of an ancilla! than the original.
After the record r1 is reported to Bob, he does his own
operation
I
^ B
r2
~
r1
!
. ~6!
The only change from round 1 is that Bob’s operations can
be explicit functions of the measurements reported in that
round. Now the process is repeated. The overall set of op-
eration elements specifying the net operation after n rounds
is given by multiplying out a sequence of these operations
S
m
5A
m
^ B
m
, ~7!
A
m
5A
rn
„r1,r2,...,r
~
n21
!
…••• A
r3
~
r1,r2
!
A
r1
, ~8!
B
m
5B
r
~
n2 1
!
„r1,r2,...,r
~
n22
!
…•••
3B
r4
~
r1,r2,r3
!
B
r2
~
r1
!
. ~9!
Here the label m can be thought of as a concatenation of all
the data collected through the n rounds of measurement:
m5r1:r2:r3:•••:rn. ~10!
Equations ~7!–~9! demonstrate the fact that all bilocal opera-
tions are also separable operations. It is the converse state-
ment that we are about to disprove for the operator corre-
sponding to the nine-state measurement ~4!.
We can get some intuitive idea of why it will be hard for
Alice and Bob to perform Eq. ~4! by local operations by
noting the result if Alice and Bob perform simple, local von
Neumann measurements in any of their rounds. These mea-
surements can be represented on the ‘‘tic-tac-toe’’ board of
Fig. 1 as simple horizontal or vertical subdivisions of the
board. The fact that any such subdivision cuts apart one of
the dominoes shows very graphically that after such an op-
eration the distinguishability of the states is spoiled. This
spoiling occurs in any local bases and is more formally just a
reflection of the fact that the ensemble of states as seen by
Alice alone, or by Bob alone, is nonorthogonal.
However, it is not sufficient to show the impossibility of
performing Eq. ~4! using a succession of local von Neumann
measurements, as Alice and Bob have available to them an
infinite set of weak measurement strategies @25#. Much more
careful reasoning is required to rule out any such strategy. In
the remainder of this section we present the details of this
proof, which also results in a computation of an upper bound
on the amount of information Alice can Bob can obtain when
attempting to perform the nine-state measurement bilocally.
C. Summary of the proof
We assume that Alice and Bob have settled on a bilocal
protocol with which they will attempt to complete the mea-
surement as well as possible. We identify the moment in the
execution of this measurement when Alice and Bob have
accumulated a specific amount of partial information. We
will have to show that it is always possible to identify this
moment either in Alice and Bob’s protocol or in an equiva-
lent protocol that can always be derived from theirs. We then
show, based on the specific structure of the nine states, that
at this moment the nine possible input states must have be-
come nonorthogonal by a finite amount. We then present an
information-theoretic analysis of the mutual information ob-
tainable in the complete measurement and show, using an
accessible-information bound, that the mutual information
obtainable by Alice and Bob bilocally is less, by a finite
amount, than the information obtained from a completely
nonlocal measurement. Now we present the steps of this
proof in detail.
D. Information accumulation and the modified continuous
protocol
If the measurement has proceeded to a point where mea-
surement record m has been obtained, an inference can be
made using Bayes’s theorem of the probability p(
c
i
u
m) that
the input state was
c
i
:
p
~
c
i
u
m
!
5
p
~
m
u
c
i
!
p
~
c
i
!
(
j
p
~
m
u
c
j
!
p
~
c
j
!
. ~11!
We take all prior probabilities p(
c
i
) to be equal to
1
9
, so they
will drop out of this equation. The measurement probabilities
p(m
u
c
i
) are given by the standard formula
p
~
m
u
c
i
!
5Tr
~
S
m
u
c
i
&^
c
i
u
S
m
†
!
5
^
c
i
u
S
m
†
S
m
u
c
i
&
. ~12!
PRA 59 1073QUANTUM NONLOCALITY WITHOUT ENTANGLEMENT

Here S
m
is the operation element of Eq. ~7!; the quantum
state in Alice’s and Bob’s possession has been transformed
to
f
i,m
[S
m
u
c
i
&
. ~13!
We imagine monitoring these prior probabilities every
time a new round is added to the measurement record in Eq.
~10!. We will divide the entire measurement into two stages
I and II; ‘‘stage I’’ of the measurement is declared to be
complete when p(
c
i
u
m), for some i, equals a particular
value ~the choice of this value is discussed in detail in Sec.
II E!. ‘‘Stage II’’ is defined as the entire operation from the
end of stage I to the completion of the protocol.
There is a problem with this, however: The measurement
record changes by discrete amounts on each round and it is
quite possible for these probabilities to jump discontinuously
when a new datum is appended to this measurement record
of Eq. ~10!. Thus it is likely that the probabilities p(
c
i
u
m)
will never attain any particular value, but will jump past it at
some particular round. The probabilities would evolve con-
tinuously only if Alice and Bob agree on a protocol involv-
ing only weak measurements, for which all the A
rk
and B
rk
of Eqs. ~8! and ~9! are approximately proportional to the
identity operator. However, in an attempt to thwart the proof
about to be given, Alice and Bob may agree on a protocol
that has both weak measurements and strong measurements
@for which the operators of Eqs. ~8! and ~9! are not approxi-
mately proportional to the identity#.
However, such a strategy will never be helpful for Alice
and Bob because for any bilocal measurement protocol that
they formulate involving any combination of weak and
strong measurements, a modified measurement protocol ex-
ists that involves only weak measurements for which the
amount of information extracted by the overall measurement
is exactly the same. For this modified protocol an appropriate
completion point for stage I of the measurement can always
be identified. Thus we can prove, by the steps described be-
low, that the modified protocol cannot be completed success-
fully by bilocal operations, and we give a bound on the at-
tainable mutual information of such a measurement.
However, since the modified protocol is constructed to have
the same measurement fidelity as the original one, this
proves that any protocol, involving any combination of weak
and strong measurements, also cannot attain perfect measure-
ment fidelity.
The modified protocol is created in a very simple way: It
proceeds through exactly the same steps as the original pro-
tocol, except that at the point where the result of a strong
measurement is about to be reported to the other party by
transmission through the classical channel, the strong mea-
surement record, treated as a quantum-mechanical object, is
itself subjected to a long sequence of very weak measure-
ments. The outcomes of these weak measurements are re-
ported, one at a time, to the other party and appended to the
measurement record in Eq. ~10!.
The precise construction of this weak-measurement se-
quence is described in Appendix A. The weak measurements
are designed so that in their entirety they give almost perfect
information about the outcome of the strong measurement
~the strong measurement outcome itself can be reported at
the end of this sequence as a confirmation!. So the recipient
of this stream of reports from the outcomes of the weak
measurements need only wait until they are done to know the
actual ~strong! measurement outcome in order to proceed
with the next step of the original protocol. However, except
in cases with vanishingly small probability, the information
contained in the accumulating measurement record grows
continuously.
To conclude this discussion of the modified measurement
protocol, we can show how Alice and Bob can be duped into
being unwitting participants in the modified protocol, and
also give an illuminating if colloquial view of how the ‘‘con-
tinuumization’’ of the measurement can take place. What is
required is a modification of the makeup of the classical
channel between Alice and Bob. We imagine that when Al-
ice transmits the results of a measurement, thinking that it is
going directly into the classical channel to Bob, it is actually
intercepted by another party (Alice
8
), who performs the nec-
essary sequence of weak measurements. Here is a way that
Alice
8
can implement this operation: She examines the bit
transmitted by Alice. If the bit is a 0, she selects a slightly
head-biased coin, flips it many times, each time transmitting
the outcome into the classical channel. If the bit is a 1, she
does the same thing with a slightly tail-biased coin. At the
other end of the channel there is another intercepting agent
(Bob
8
) who, after studying a long enough string of coin flips
sent by Alice
8
, can with high confidence deduce the coin
bias and report the result to Bob. Alice and Bob are oblivious
to this whole intervening process; nevertheless, as measured
by the data actually passing through the channel, the modi-
fied protocol with nearly continuous evolution of the avail-
able information has been achieved.
E. State of affairs after stage I of the measurement
Having established that no matter what Alice’s and Bob’s
measurement protocol, we can view the probabilities as
evolving continuously in time and we can declare that stage
I of the measurement is complete when
max
i
p
~
c
i
u
m
I
!
5
1
9
1
e
, ~14!
that is, after the probabilities have evolved by a small but
finite amount away from their initial value of
1
9
. It should be
noted that since some measurement outcomes might be much
more informative than others, the time of completion of
stage I is not fixed; it will in general require a greater number
of rounds for one measurement record m
I
than for another.
The
e
in Eq. ~14! should be some definite, small, but
noninfinitesimal number. Moreover, we will require that all
posterior probabilities p(
c
i
u
m
I
) be nonzero. For this any
value smaller than
1
72
will be acceptable since
min
i
p
~
c
i
u
m
I
!
>
1
9
28
e
. ~15!
We now rewrite Bayes’s theorem from Eq. ~11!:
1074 PRA 59CHARLES H. BENNETT et al.

Frequently asked questions

Q1. What have the authors stated for future works in "Quantum nonlocality without entanglement" ?

It would appear that further work on the tiling problems could produce other interesting examples for numbers of parties between 3 and 10. 

Q2. What contributions have the authors mentioned in the paper "Quantum nonlocality without entanglement" ?

Bennett, DiVincenzo, Christopher A. Fuchs, Tal Mor, Eric Rains, Peter W. Shor, John A. Smolin, and William K. Wootters this paper have published a paper entitled `` 17 June 1998: A Review of the Recent Developments in Computational Artificial Intelligence ''.