What is erased in the quantum erasure?
B. J. Hiley and R. E. Callaghan.
∗
In this paper we re-examine a series of gedanken welcher Weg (WW)
experiments introduced by Scully, Englert and Walther that contain the es-
sential ideas underlying the quantum eraser. For this purpose we use the
Bohm model which gives a sharp picture of the behaviour of the atoms
involved in these experiments. This model supports the thesis that interfer-
ence disappears in such WW experiments, even though the centre of mass
wave function remains coherent throughout the experiment. It also shows
exactly what it means to say “that the interference can be restored by ma-
nipulating the WW detectors
long after
the atoms have passed.” It does
not support Wheeler’s notion that “the past is undefined and undefinable
without the observation [in the present]”
KEY WORDS:
Quantum erasure, Welcher Weg experiments, Bohm
trajectories.
1
INTRODUCTION
We have recently returned to examine the criticism that Bohm-type trajec-
tories have such ‘bizarre’ properties that they must be dismissed as phys-
ically unreasonable and should be regarded as ‘surreal’
(1)
. In Callaghan
and Hiley
(2)
and Hiley
(3)
we show that, if we use the Bohm interpretation
(BI) as defined in Bohm and Hiley
(4
,
5)
, we do not reproduce the strange be-
haviour predicted by Aharonov and Vaidman
(6)
, Englert, Scully, S¨
ussman
and Walther
(7)
and Scully
(1)
. In fact the Bohm trajectories do
not
actu-
ally exhibit the unreasonable behaviour predicted by the above authors, but
produce exactly the same behaviour as those obtained by Scully
(1)
using
standard quantum mechanics (SQM) as they must.
In view of these results we have decided to re-examine what new light
the BI could throw on the phenomenon discussed under the heading of the
∗
Theoretical Physics Research Unit, Birkbeck College, University of London, Malet
Street, London WC1E 7HX, England. E-mail contact: b.hiley@bbk.ac.uk
1
‘quantum erasure’. We will confine our discussion to the particular case
of the two-slit interference experiment introduced by Scully, Englert and
Walther, (SEW)
(8)
(See figure 1 for the experimental set up. This figure
is taken from their paper.)
The reason for restricting our discussion to
this particular experiment is that it uses atoms whose time development is
described by the Schr¨
odinger equation. It is from this equation that the
particle Bohm model used in this paper is derived
1
The device shown in Figure 1 allows for the possibility of later erasing
information as to which slit each atom passed through. In such an experi-
ment the essential question raised by SEW is whether erasing this welcher
Weg (WW) information will restore the interference after the atoms have
passed through the slit system and indeed what this ‘restoration’ actually
means.
Figure 1: The ‘which way’ [WW] experiment of Scully, Englert and Walther
[SEW]. The uniform distribution, as opposed to fringes, appearing when the
shutters are in place is shown in (a). The fringes (antifringes) appearing
when the cavity detector fires (does not fire) are shown in (b)
The original impetus for studying such experiments appear to stem from
Wheeler’s analysis of the delayed choice experiment. Wheeler
(11)
argues that
these experiments leave us no option but to embrace the notion that ‘the past
is undefined and undefinable without the observation [in the present]’. The
1
While many quantum erasure experiments are done using photons, we will not discuss
these here because they require quantum field theory. Even in the Bohm model
(5
,
9)
it is
not possible to construct photon trajectories. For a detailed discussion of this point see
Holland
(10)
2
word ‘undefinable’ seems to suggest that a very radical position has been
adopted, too radical for some.
However the proposal becomes plausible
only if we insist there is no knowable reality underpinning the quantum
formalism. We will show in this paper that the BI does not support this
notion. It provides a consistent model which insists that what has passed is
past and cannot be altered by some measurement that is performed in the
future. We will show here that the BI provides a consistent and physically
reasonable account of these WW experiments without the need to propose
any radical new notions. An important consequence of all this is that the
BI does not support the notion of a ‘quantum eraser’ if it is meant by this
that the past dynamics is changed when we can no longer have access to the
original WW information. In fact we will show that past dynamics cannot
be erased by any future action.
In this paper we will assume by now that the particle Bohm model [BI]
is well known. Those unfamiliar with this model are referred to the books
of Bohm and Hiley
(5)
and Holland
(12)
where references to more detailed
discussions can be found. We will simply adopt its framework and examine
how it treats the WW experiments introduced in the excellent paper of
SEW
(8)
.
To bring out these details we want to concentrate solely on the two-slit
experiments that are clearly laid out in their paper. This experiment (see
figure 1) consists of a beam of atoms, all in the same excited state, incident
on a pair of slits. Two maser cavities are placed in front of the two-slit system
so that it is possible to determine through which slit each atom passes. This
is achieved by assuming that when an excited atom enters a cavity it gives
up its internal energy to the cavity with 100% efficiency without changing
the phase of the centre of mass wave function of the atom. Furthermore
we assume the photodetector within the cavity is also 100% efficient. This
ideal situation is assumed so that we can bring out the principles involved in
the experiment in as simple a way as possible. The reasonableness of these
assumptions have been excellently discussed in SEW so we will not dwell
further on these issues here.
Thus, as each atom passes through the cavity, it gives up all its internal
energy to the cavity and then continues onto the screen. As this process
does not disturb the centre of mass momentum, the emerging beam remains
coherent. In spite of this no fringes appear. This is explained in SQM by the
orthogonality of the cavity states involved (see equation (1)). The vanishing
of the interference is sometimes explained by arguing that future measure-
ment on the cavities will identify through which slit each atom passed and
this knowledge will violate the uncertainty principle.
3
The question that SEW raise is as follows. Suppose one does not measure
the energy of a cavity to find out which way an atom went but rather simply
joined the two cavities by removing the shutters shown in figure 1. This
allows the cavity fields to combine removing any possibility of finding out
which way any particular atom went. Will this process allow the interference
fringes to reappear? In other words will erasing the ‘which way’ information
allow interference to reappear?
To ensure that such an erasure has taken place, a detector is placed in
the cavity behind the pair of shutters. When the shutters are removed the
detector will be called into action (See figure 1 for details). The interaction
Hamiltonian describing the field/detector interaction couples the detector
only to the symmetric combination of the fields.
It does not couple to
the antisymmetric combination. The presence of the cavity detector thus
ensures the WW information is destroyed. Let us therefore compare the
interference pattern before and after the shutters are removed.
In terms of the Copenhagen interpretation, we can say nothing about
what goes on until a measurement is actually made. Thus there appears
to be a different outcome depending on which measurement we choose to
take no matter when we chose to do the measurement, hence the notion of
delayed choice. If one subscribes to the Wheeler analysis then since the past
is only revealed in the present action, the present experiment suggests that
we have somehow miraculously created interference ‘long after the atoms
have passed’ as SEW put it.
No such ambiguity shows up if we use the BI. If the atoms have reached
the screen and been detected at the screen then clearly delayed measurement
of the content of the cavity will have no effect. However, what happens to the
subsequent behaviour of the atoms if the screen is removed and the atoms
are not detected? We will show that, if the atoms have passed through the
slits, the subsequent removal of the shutters allowing the cavity detector to
function produces no effect on the atoms that have passed through the slit
system. The influence of the two-slit/cavity set-up is over, it is complete.
What any subsequent cavity measurement establishes is what
would have
been
the situation had the measurement been carried out before the par-
ticles enter the two slits, that is, the cavity measurement is instantaneous
and not delayed. What the so-called eraser does is to identify positions on
the screen where atoms would have arrived had the cavity detector fired
instantaneously. Of course this is the conclusion one would also arrive at in
SQM provided we examined carefully the time evolution of the total wave
function using the Schr¨
odinger equation. The advantage of the BI is that
this behaviour can be made much more transparent. We will now go through
4
the mathematical details more fully to substantiate our claims.
2
WELCHER WEG EXPERIMENT BEFORE
MAKING A MEASUREMENT OF THE CON-
TENT OF THE CAVITY
Consider a coherent beam of atoms in an internal energy state
|
a
i
incident
on the cavity-slit system. On passing through one of the cavities, the atom
gives up its internal energy leaving it in the internal energy state
|
b
i
. If the
centre of mass wave function is
ψ
(
r
), then the total wave function at the
screen will be
Ψ(
r
) =
1
√
2
[
ψ
1
(
r
)
|
1
1
0
2
i
+
ψ
2
(
r
)
|
0
1
1
2
i
]
|
b
i
(1)
Since the cavity radiation states
|
1
1
0
2
i
and
|
0
1
1
2
i
are orthogonal, the prob-
ability
|
Ψ
|
2
contains no interference terms and, as we have already pointed
out, no fringes are predicted.
In terms of the BI we write the wave function as
Ψ(
r
,
Φ) =
1
√
2
ψ
1
(
r
)Φ(
φ
1
(
r
0
)
, η
0
(
r
0
)) +
ψ
2
(
r
)Φ(
φ
0
(
r
0
)
, η
1
(
r
0
))
(2)
Here Φ(
φ
i
(
r
0
)
, η
j
(
r
0
)) is the wave functional of the cavity fields
φ
i
(
r
0
), and
η
j
(
r
0
).
φ
i
is the field in the top cavity while
η
j
is the field in the bottom
cavity in figure 1. The suffixes
i
and
j
(= 0,1) refer to the number of photons
excited in each cavity. The easiest way to exhibit the interference terms is to
examine the quantum potential. In the case we are considering, the quantum
potential takes the form
Q
=
−
1
2
Z
d
3
r
0
1
m
∇
2
r
R
R
+
δ
2
R
(
δ
Φ)
2
R
(3)
where we re-write Ψ(
r
,
Φ) as
R
(
r
,
Φ) exp[
iS
(
r
,
Φ)]. Here
δ/δφ
in the func-
tional derivative and we have put ¯
h
=1.
We then find
|
R
(
r
,
Φ)
|
2
=
R
1
(
r
)
R
(Φ(
φ
1
(
r
0
)
, η
0
(
r
0
)))
2
+
R
2
(
r
)
R
(Φ(
φ
0
(
r
0
)
, η
1
(
r
0
)))
2
+
2
R
1
(
r
)
R
2
(
r
)
R
(Φ(
φ
1
(
r
0
)
, η
0
(
r
0
)))
R
(Φ(
φ
0
(
r
0
)
, η
1
(
r
0
))) cos ∆
S.
Now the contribution of the cavity fields (second factor of each term in
the sum) to the quantum potential (3) is negligible outside the cavities so
5
that the main contribution comes from the first term. In fact a detailed
calculation of the quantum potential is not necessary, all we need to do is
carefully examine the interference term.
At first sight the last term in the expression for
|
R
(
r
,
Φ)
|
2
suggests that
interference is present. However we will now show that in this case it al-
ways vanishes, contributing nothing to the quantum potential. To see this
remember that the quantum potential must be evaluated over the
actual
position of the atom and all of the variables describing the properties of the
cavities concerned. Thus, if the atom actually goes through the top cavity
then the probability of finding the bottom cavity in an excited state
η
1
(
r
)
(i.e. containing a photon) is zero. Therefore
R
(
φ
0
(
r
0
)
, η
1
(
r
0
)) must be zero,
so the interference term must vanish. In fact we are only left with one term
|
R
(
r
,
Φ)
|
2
=
R
1
(
r
)
R
(Φ(
φ
1
(
r
0
)
, η
0
(
r
0
)))
2
Again if the atom actually went through the lower cavity then the prob-
ability of finding a photon in the upper cavity will be zero. In this case
R
(Φ(
φ
1
(
r
0
)
, η
0
(
r
0
))) will be zero. Again we are left with one term
|
R
(
r
,
Φ)
|
2
=
R
2
(
r
)
R
(Φ(
φ
0
(
r
0
)
, η
1
(
r
0
)))
2
and the interference will therefore once again vanish. Thus there will be
no interference term in the quantum potential for whichever way the atom
actually goes and therefore no fringes will appear. This is in complete agree-
ment with the predictions of SQM as it must be since BI simply uses the
standard quantum formalism.
The distribution on the screen will be the sum of two independent sin-
gle slit distributions. We can confirm this result by calculating individual
trajectories using the so-called guidance condition
p
=
∇
S
(
r
,
Φ). The cor-
responding trajectories are sketched in figure 2.
Notice in this diagram that individual trajectories cross the horizontal
axis of symmetry. That this must be so can be seen from the following ar-
gument. Let us consider one atom incident on the slit system and suppose
it enters at the lower edge of the top slit. It will travel on one of the lower
trajectories emanating from the top slit. Since there is no contribution from
the bottom slit, it will travel along the trajectory crossing the axis of sym-
metry before arriving at the screen. Similarly, consider another later atom
incident on the upper edge of the bottom slit. It will follow the trajectory
that crosses the axis of symmetry from below.
If a bunch of atoms approach the slits together then we can think in
terms of currents as does Erez and Scully
(13)
. This means that we will get
6
Figure 2: Sketch of trajectories expected if the effects of the two slits behave
independently
a current from the top slit crossing the axis of symmetry and there will
also be a current from the bottom slit crossing the axis of symmetry in the
opposite direction. Of course these currents are symmetric about this axis
so that the algebraic sum of the vertical component of the currents is zero
as argued by Erez and Scully, but their conclusion that no atoms cross the
axis of symmetry is clearly incorrect.
3
WHAT HAPPENS WHEN A MEASUREMENT
IS MADE ON THE CAVITIES THAT
DESTROYS THE WELCHER WEG INFOR-
MATION?
Now we must investigate what happens when we remove the shutters be-
tween the two cavities and allow the detector in the cavities to function. The
first point to remember is that the detector in the cavities is described by
an interaction Hamilton that only couples with symmetric combinations of
the two fields. This means that only the symmetric combination will cause
the detector to fire, while the antisymmetric combination will leave the de-
tector unaffected. This process will destroy the WW information. What we
must do now is to analyse how we can apply the quantum formalism to this
situation.
In SQM it is first necessary to find an expression for the wave function
that will allow the field to couple with the detector. To this effect we in-
troduce the symmetric,
|
+
i
, and antisymmetric,
|−i
, states of the radiation
7
fields in the cavity so that
|±i
=
1
√
2
[
|
1
1
0
2
i ± |
0
1
1
2
i
]
(4)
At the same time we introduce the symmetric,
ψ
+
, and antisymmetric
ψ
−
atomic states defined by
ψ
±
(
r
) =
1
√
2
[
ψ
1
(
r
)
±
ψ
2
(
r
)]
(5)
Then it is easy to show that the wave function of the combined system (1)
can be written as
Ψ(
r
,
Φ) =
1
√
2
[
ψ
+
(
r
)
|
+
i
+
ψ
−
(
r
)
|−i
]
|
b
i|
d
i
(6)
where
|
d
i
is the ‘unfired’ state of the detector in the cavities. If we now
introduce the interaction Hamiltonian coupling the cavity field to the cavity
detector, the final wave function becomes
Ψ(
r
,
Φ) =
1
√
2
[
ψ
+
(
r
)
|
0
1
0
2
i|
f
i
+
ψ
−
(
r
)
|−i|
d
i
]
|
b
i
(7)
where
|
f
i
is the ‘fired’ state of the cavity detector. Again because of the
orthogonality of
|
f
i
and
|
d
i
no interference fringes will be seen.
In order to get clear on what is involved in a delayed removal of the
shutters, let us first see how SQM and BI deal with the case when the
shutters are removed
before
the atoms pass through the cavity. We will
later discuss what happens if the removal of the shutters is delayed.
Since in this case the detector either fired or did not, both SQM and
BI claim that the wave function splits into a mixture of two sub-ensembles.
SQM uses the collapse of the wave function to establish this. The BI uses
an argument similar to the one used in section 2.
Let us first use SQM to analyse these two sub-ensembles. First consider
the case where the cavity detector fires. Here the wave function for this
sub-ensemble is
Ψ(
r
,
Φ) =
ψ
+
(
r
)
|
0
1
0
2
i|
f
i|
b
i
(8)
If we expand the term
ψ
+
(
r
), we see that interference terms are present
which is confirmed by writing down the probability of an atom arriving at
the screen
P
f
(
r
) =
1
2
h
|
ψ
1
(
r
)
|
2
+
|
ψ
2
(
r
)
|
2
+
Re
[
ψ
∗
1
(
r
)
ψ
2
(
r
)]
i
(9)
8
This sub-ensemble clearly shows the presence of an interference effect.
If the cavity detector does not fire, then the wave function for this sub-
ensemble is
Ψ(
r
,
Φ) =
ψ
−
(
r
)
|−i|
d
i|
b
i
(10)
Again interference arises as can be seen by expanding the term
ψ
−
(
r
). In
this case the probability is
P
d
(
r
) =
1
2
h
|
ψ
1
(
r
)
|
2
+
|
ψ
2
(
r
)
|
2
−
Re
[
ψ
∗
1
(
r
)
ψ
2
(
r
)]
i
(11)
Here it is the minus sign that produces what SEW
(8)
call the ‘antifringes’
(See figure 1). This difference in sign simply means that the points of max-
imal arrivals on one interference pattern will correspond to the points of
minimum arrivals on the other. When taken together it appears as if there
is no interference present at all.
The BI also offers a very straight forward explanation of these two sub-
ensembles. In the previous section we used the quantum potential to show
why there were no interference terms present, but here we will simply use the
guidance condition,
p
=
∇
S
directly. This is more convenient in this case
because we are only interested in the trajectories and these are calculated
directly from the guidance condition simply by integration.
Figure 3: Fringes produced when WW information is lost and detector fires.
In the case where the cavity detector fires, we clearly must use wave
function (8). However since we are only interested in the general structure
of the trajectories, we need only consider part of the wave function
ψ
+
(
r
) =
R
+
(
r
) exp[
iS
+
(
r
)]. We can then put this in the guidance condition which
9
in this case reads
p
=
∇
r
S
+
(12)
Here
S
+
is given in terms of
ψ
1
and
ψ
2
by the equation
tan
S
+
=
R
1
sin
S
1
+
R
2
sin
S
2
R
1
cos
S
1
+
R
2
cos
S
2
(13)
Integrating equation (12) numerically we find the trajectories shown in figure
3.
In the case where the cavity detector does not fire, we need only consider
the wave function
ψ
−
(
r
) =
R
−
(
r
) exp[
iS
−
(
r
)] and again use the guidance
condition
p
=
∇
r
S
−
(14)
where
tan
S
−
=
R
1
sin
S
1
−
R
2
sin
S
2
R
1
cos
S
1
−
R
2
cos
S
2
(15)
Because of the negative sign appearing in this expression the trajectories
bunch to form ‘antifringes’. These trajectories are shown in figure 4. So
once again we see that the BI merely reinforces the conclusions derived
from SQM.
Figure 4: Antifringes produced when WW information is lost and cavity
detector does not fire.
4
DISCUSSION OF RESULTS
In sections 2 and 3 we have discussed two extreme situations. The first
case, case
A
, involved keeping the cavity shutters in place throughout the
10
experiment and the other, case
B
, involved keeping the shutters open all
the time. In these two cases we see that we obtain very different sets of
trajectories.
In case
A
when the shutters are closed, we find two independent sets of
trajectories. One set arises from atoms that pass through the top slit, the
other from the atoms that pass through the bottom slit. Neither set shows
any interference effects. Notice that in this case atoms do actually cross the
horizontal axis of symmetry, equal numbers crossing in both directions.
In the second case,
B
, an examination of the two sets of trajectories
plotted in figure 3 and 4 show that, in each case, interference is present.
If we superimpose the two sets of trajectories as demonstrated in figure 5,
we see that we get a uniform distribution of atoms arriving at the screen
suggesting no interference present. Nevertheless interference is present but
is hidden by superimposing the fringe and anti-fringe patterns as pointed
out by SEW. Notice that in both figures 3 and 4 no trajectory crosses the
symmetry axis. This shows that very different sets of particle evolution can
give rise to what looks like the same interference-free pattern.
Figure 5: The superposition of trajectories corresponding to superposition
of fringes and antifringes.
Now let us turn to the more interesting case in which we start the ex-
periment with the shutters in place, case
A
, but then remove them at some
later time
t
0
. In order to get a clear account of the behaviour of individual
atoms in this case, let us consider a very weak beam of atoms incident on
the cavities so that at any one time there is only one atom passing through
the apparatus.
Let us first consider the more straight forward situation when each atom
11
is actually detected at the screen before the shutters are removed. Clearly
once an atom has arrived at the screen and has been detected, any subse-
quent removal of the shutters will have no effect on the past behaviour of
this atom. The atom has arrived and been detected and its past behaviour
cannot be changed. This means that any subsequent destruction of WW
information in the cavities will have no effect on how the atom got to the
screen. In other words in this case, although the WW information for each
atom has been ‘erased’, this in no way effects the dynamics because the
dynamics is over as soon as each atom is detected.
We can now turn to the much more interesting case where an atom has
passed through the cavity set up as in case
A
but is still in motion when
the cavity shutters are removed at time
t
0
. What will be the subsequent
behaviour of the atom?
Figure 6: Particle reaches
R
at time
t
0
when shutters are removed. The
trajectory continues unchanged.
Suppose we consider the case where an atom follows a given trajectory
and when the shutters are removed the atom has reached position
R
as
shown in Figure 6. We have already argued in section 2 that the effective
wave function for this particular atom is simply
Ψ(
r
,
Φ
, t
) =
ψ
2
(
r
, t
)Φ(
φ
0
(
r
0
, t
)
, η
1
(
r
0
, t
))
|
b
i
(16)
To understand the effect on the atom of removing the shutters at
t
0
, we must
write this wave function in the form
Ψ(
r
,
Φ
, t
) =
1
√
2
ψ
2
(
r
, t
)[Φ
+
+ Φ
−
]
|
d
i|
b
i
(17)
where
Φ
±
=
1
2
Φ(
φ
0
(
r
0
, t
)
, η
1
(
r
0
, t
))
±
Φ(
φ
1
(
r
0
, t
)
η
0
(
r
0
, t
)
(18)
12
If the cavity detector fires when the shutters are removed, the effective
wave function is
Ψ(
r
,
Φ
, t
) =
ψ
2
(
r
, t
)
|
0
1
0
2
i|
f
i|
b
i
t > t
0
(19)
Using the wave function given in (19), we see that the atom must continue
on its straight line trajectory since its centre of mass wave function
ψ
2
is
unchanged and there is no contribution from
ψ
1
. A corresponding argument
can be made for an atom passing through the top slit.
To sum up then, the BI shows clearly that although the WW information
is erased, the past dynamics of each atom is not changed and, furthermore,
the future dynamics is not changed either. In other words the interference
has not been restored in the sense that the dynamics of the individual atoms
has been changed. All that one can do with this delayed information is to
divide the atom arrival positions on the screen into two sub-ensembles, one
set being identified with the firing of the cavity detector while the other is
identified with its non-firing.
Thus the use of the word ‘eraser’ does not seem to capture the essence of
what is going on here. In one sense the word is misleading because it tends
to imply that somehow the past dynamics is changed and the interference
pattern has been ‘restored’ but nothing of the sort has happened.
Our
“erasing” the WW information has had no effect on the behaviour of the
atoms that have already passed through the cavities and the slits.
5
CONCLUSIONS
By using the Bohm interpretation we have shown that once the atoms have
passed through the slits, any subsequent erasure of the ‘which way’ (WW)
information has no effect either on the past nor future dynamics of the
atoms. Thus we have come to the opposite conclusion reached by Scully et
al.(SEW)
(8)
who claimed that “the interference effects can be restored by
manipulating the WW detectors
long after
the atoms have passed.”
The apparent re-appearence of interference arises because it is possible
to have different underlying dynamics to produce the same final probability
distributions. This was brought out clearly in figures 2 and 5. In figure 2
the atoms cross the horizontal axis of symmetry contrary to figure 5 where
no atoms cross this axis at all, yet the final distribution shows no signs of
interference.
What figure 5 does show is that if we identify the arrival points of atoms
on the screen with the subsequent firing/non-firing of the cavity detector, we
13
can separate out these points into two sub-ensembles. If we take either sub-
ensemble it will look as if interference has somehow ‘re-appeared’. However
the significant point is that the two sub-ensembles could have arisen if the
experiment had been run with the cavity shutters removed from the very
beginning. Thus it is not the erasing of the WW information
per se
that
restores
the interference.
It is easy to miss this point in SQM because strictly there is no way
of analysing what is going on between measurements. We may be able to
obtain more information by representing the atoms by small wave packets,
but even here we can only talk about potential behaviour under exactly
specified experimental conditions. When the experimental conditions are
changed, the potential outcomes are changed, but according to Bohr
(14)
,
there is no way to picture any possible underlying dynamics.
On the other hand the BI presents a clear unambiguous picture of the un-
derlying dynamics. One can build information about the overall experimen-
tal conditions into the quantum potential, and show how these experimental
conditions affect the behaviour of individual atoms. It is the fact that the
quantum potential contains this information about the experimental con-
ditions that ensures that at the level of probabilities there is no difference
between SQM and the BI.
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15