In this report, we summarize the given scenario behind the “Quantum Interrogation" problem, and offer historical background of the development of quantum “interaction-free measurement" theory. We then describe some real-world applications of this procedure, whether already verified experimentally or as potential applications in the future. Finally, we exhibit our calculations that display the feasibility of this quantum effect, and summarize the key takeaways from this problem.

1. Quantum Interrogation:
Interaction-Free Determination of Existence
Benji Kan, Irtiza Iram, Samuel Buckley-Bonanno
Physics 19 Final Project, Harvard University
December 16, 2019
Abstract
In this report, we summarize the given scenario behind the “Quantum Interrogation”
problem, and offer historical background of the development of quantum “interaction-
free measurement” theory. We then describe some real-world applications of this proce-
dure, whether already verified experimentally or as potential applications in the future.
Finally, we exhibit our calculations that display the feasibility of this quantum effect,
and summarize the key takeaways from this problem.

2. 1 Introduction
1.1 Scenario
Suppose we have a perfectly sealed box which may or may not contain a bomb. We are
offered a reward for finding out if the box contains an explosive bomb, but we can only
probe the box by inserting a “trigger” device. This “trigger” has two possible states: a
lit match which would light the bomb, causing an explosion, or an ice cube which has no
effect in any case. Classically, this problem seems impossible: inserting an ice cube gives no
information, while inserting a bomb would confirm the non-existence of a bomb, but also
causes an explosion if there actually were a bomb inside the box. However, quantum theory
offers a clever solution that utilizes nonlocality and wave-particle duality to safely obtain the
reward.
1.2 Background
This problem of quantum interrogation falls under a broader class of quantum measure-
ments known as “interaction-free measurement,” pioneered by Paul G. Kwiat1
. Through an
iterative quantum process, the presence of an object can be detected without directly inter-
acting with it. Similar experiments in this category include the Renninger negative-result
experiment2
, which shows that measurements can occur without detection of particles, and,
most relevantly, the Elitzur-Vaidman Bomb Tester3
, which utilizes quantum superpositions
to experimentally verify the functionality of bombs without needing to detonate them.
1.3 Real-world applications
Quantum interrogation finds its primary application in the field of quantum computing, in
which the unique properties of quantum systems are utilized to vastly accelerate compu-
tations on certain specific problems. The computers work by applying quantum gates to
the state vectors of entangled quantum systems known as “qubits.” The final state vectors
themselves contain the results of the computation.
An effective quantum computer would be capable of factoring prime numbers orders of
magnitude more rapidly than current, classical computers, indicating potential applications
in cryptograpghy, for breaking current encryption protocols and generating new, unbreakable
ones. Quantum computers also have the potential to revolutionize our ability to simulate
molecules, protein folding, AI models, particle collider experiments, and climate models,
among other related uses.
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3. The unique states that quantum computers take advantage of are extremely delicate; for
instance, two dust grains separated by the diameter of a dust grain (about 10−5
m) suspended
in air in a dark room decohere in about 10−31
seconds4
. The longest-lasting qubit created so
far remained coherent for just 39 minutes5
. Thus, a technique such as quantum interrogation,
allowing us to determine what the state vector is without disturbing the system, would be
an invaluable part of any quantum computing system.
2

4. 2 Calculations
(a) Let us consider the trigger to have a state vector given by the following quantum
superposition:
|trigger = cos θ |ice cube + sin θ |match . (1)
Using the Born rule, we can find that the probability that a measurement of the trigger
device will yield the result “ice cube” is
P(ice cube) = | cos θ |2
= cos2
θ . (2)
If a measurement of the trigger device yields that it is an ice cube, then the state vector
will collapse to the “ice cube” state, given by
|triggerf = |ice cube . (3)
Similarly, the probability that a measurement will instead yield the result “match” is
given by the Born rule as
P(match) = | sin θ |2
= sin2
θ (4)
In this case, after such a measurement, the state vector of the trigger device will collapse
to
|triggerf = |match . (5)
The sum of the two probabilities is
P = P(ice cube) + P(match) = cos2
θ + sin2
θ = 1 , (6)
which is expected, as the sum of the probabilities of all possible outcomes should be
1. It is also consistent with the postulate that the state vector describing a quantum
system must have unit norm, which in this case is satisfied by trigger|trigger =
cos2
θ + sin2
θ = 1.
(b) We now consider a physical transformation of our trigger device that changes the
angle θ in its state vector by a small amount (10−N
)◦
, where N 1 is a large, fixed
integer. Such a transformation is known as a quantum rotation.
The term “rotation” here does not indicate a physical rotation of our trigger device.
It is called a “rotation” in the sense that it is analogous to changing the components
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5. of a vector along each basis in two or three dimensional space keeping the norm fixed.
Such a change to a vector in two or three dimensional space would cause a physical
rotation of the vector.
The transformation can be written as:
θ → θ + (10−N
)◦
,
|trigger = cos θ|ice cube +sin θ|match → cos(θ+(10−N
)◦
)|ice cube +sin(θ+(10−N
)◦
)|match .
After performing the quantum rotation, the new probability of getting the result “ice
cube” upon a measurement of the trigger device is (as given by the Born rule):
P (ice cube) = | cos(θ + (10−N
)◦
)|2
= cos2
(θ + (10−N
)◦
) . (7)
After such a measurement yielding the result “ice cube”, the state vector of the trigger
device will collapse to
|triggerf = |ice cube . (8)
The probability of getting the result “match” is now
P (match) = | sin(θ + (10−N
)◦
)|2
= sin2
(θ + (10−N
)◦
) . (9)
Such a measurement will cause the state vector to collapse to
|triggerf = |match . (10)
The sum of the probabilities is again
P = P (ice cube)+P (match) = cos2
(θ+(10−N
)◦
)+sin2
(θ+(10−N
)◦
) = 1 , (11)
as expected.
(c) Let us now consider a scenario in which the trigger device is initially in a state
given by the state vector
|trigger0 = |ice cube =⇒ |trigger0 = 1 |ice cube + 0 |match . (12)
In this state, if we perform a measurement of the trigger device, the probabilities of
getting either of the two results, namely “ice cube” and “match”, are, according to the
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6. Born rule,
P(ice cube) = |1|2
=⇒ P(ice cube) = 1 , (13)
and
P(match) = |0|2
=⇒ P(match) = 0 . (14)
(d) Instead of performing a measurement, we carry out the quantum rotation mentioned
in part (b). Initially the state vector is |trigger = |ice cube , which can be written as
|trigger = 1 |ice cube + 0 |match = cos 0 |ice cube + sin 0 |match (15)
Hence the quantum rotation will be similar to what we did in part (b) with θ now
equal to 0. Thus, the transformation is now
0 → 0 + (10−N
)◦
= (10−N
)◦
. (16)
After the quantum rotation, the new state vector of the trigger device will be
|trigger1 = cos((10−N
)◦
) |ice cube + sin((10−N
)◦
) |match . (17)
(e) We now insert the trigger into the box.
If the box is empty, no measurement of the trigger device takes place, and its state
vector remains unchanged, i.e. it remains
|trigger1 = cos((10−N
)◦
) |ice cube + sin((10−N
)◦
) |match . (18)
If instead the box contains a bomb, then the bomb will carry out a measurement on
the trigger device. The probability that the bomb will find the trigger device to be in
the state “ice cube” (and hence not explode) is then, as given by the Born rule,
P(ice cube) = | cos((10−N
)◦
)|2
= cos2
((10−N
)◦
) , (19)
which is very close to 1, as 10−N
is very nearly 0. This indicates that the bomb is more
likely to find the trigger in the “ice cube” state, and hence it is less likely to explode. If
the bomb carries out a measurement that yields the result “ice cube”, the state vector
of the trigger device will then collapse to
|trigger1 = |ice cube . (20)
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7. (f) We now retrieve the trigger device and, instead of carrying out a measurement, we
perform the aforementioned quantum rotation of its state vector. If the box is empty,
then upon retrieval, the state vector of the trigger device is
|trigger1 = cos((10−N
)◦
) |ice cube + sin((10−N
)◦
) |match . (21)
A quantum rotation of this state vector will then involve the transformation:
(10−N
)◦
→ (10−N
)◦
+ (10−N
)◦
= 2 × (10−N
)◦
.
So the new state vector is given by
|trigger2 = cos(2 × (10−N
)◦
) |ice cube + sin(2 × (10−N
)◦
) |match . (22)
If the box instead contains a bomb and it hasn’t exploded after the first insertion of
the trigger, then upon retrieval of the trigger device, its state vector, as shown in part
(e), is
|trigger1 = |ice cube . (23)
Writing it as |trigger1 = cos 0 |ice cube + sin 0 |match , we can see that, similar to
our calculation in part (d), the quantum rotation is given by the transformation:
0 → (10−N
)◦
.
Hence the new state vector of the trigger device will be
|trigger2 = cos((10−N
)◦
) |ice cube + sin((10−N
)◦
) |match . (24)
(g) Suppose that we repeat the process described in steps (e) and (f) 90 × 10n
times.
If the box is empty, then the final state vector |triggerf of the trigger device would be
|triggerf = cos(90 · 10n
· (10−n
)◦
)|ice cube + sin(90 · 10n
· (10−n
)◦
)|match
This simplifies to the state vector
|triggerf = cos(90◦
)|ice cube + sin(90◦
)|match
|triggerf = (0)|ice cube + (1)|match
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8. |triggerf = |match
(h) We now qualitatively consider the case where the box contains a bomb, but it
never detonates throughout the repetitions. Then, each time the trigger is inserted
into the box, the bomb (which indeed exists) will observe the trigger. Since we assume
that the bomb never detonates, then the state vector of the trigger must collapse to
|trigger = |ice cube after each measurement. This process is repeated, where the
trigger is rotated, placed into the box, collapsed to the ice cube state, and rotated
again. At the end, the state vector of the trigger will simply be one quantum rotation
from the pure ice cube state, so we see that
|triggerf = cos((10−N
)◦
)|ice cube + sin((10−N
)◦
)|match .
However, for large N, 10−N
≈ 0 =⇒ cos((10−N
)◦
) ≈ 1, sin((10−N
)◦
) ≈ 0, so
|triggerf ≈ |ice cube .
Now, we consider the total probability that the box contains a bomb and it actually
detonates at some point during the whole attempted run. Note that this probability
is simply 1 minus the probability that it never detonates, or
P(BOOMsometime) = 1 − P(nothingeverytime).
However, since each measurement is independent, and (barring a detonation) the state
vector of the trigger is “reset” to the pure ice cube state before being quantum rotated
and reinserted. then we can calculate the probability it never explodes by multiplying
the probabilities that the bomb doesn’t explode on a single, given measurement. Then,
P(nothingeverytime) = [P(nothingmeasurement)]90·10N
,
which is the probability the bomb doesn’t detonate on a single measurement taken to
the power of the number of repetitions. We know P(nothingmeasurement) is simply the
square of the coefficient of the ice cube state in the state vector for the trigger, which
is cos2
((10−N
)◦
, so we see that the overall
P(BOOMsometime) = 1 − cos180·10N
((10−N
)◦
) .
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9. Now, we consider how this probability depends on N. Graphing this function on
Desmos, we see that for sufficiently large N, 1 − cos180·10N
((10−N
)◦
) will drop arbitrar-
ily close to 0.
We note that this is because for large enough N, cos((10−N
)◦
) becomes extremely close
to 1 so taking this quantity to higher powers has essentially no effect. In fact, we note
that:
At N = 3 (1,000 repetitions), P(BOOM) = .086, and
By N = 5 (100,000 repetitions), P(BOOM) = .001.
Similarly, we can achieve arbitrarily small probabilities of explosion (and thus corre-
spondingly higher margins of safety) with sufficiently large N.
(i) We can perform a single measurement on the trigger device at the end of the full run
of repetitions, and thus determine whether the box contains a bomb. If the bomb does
contain a bomb, that bomb will have been collapsing the state of the trigger device
to the |ice cube state vector every time the trigger was put into the device with an
arbitrarily high probability, as shown above. Thus, if the trigger collapses to the “ice
cube” state when a measurement is made on it, we will know that the box must have a
bomb inside. If the box does not contain a bomb, then full run of repetitions will have
(quantum) rotated the trigger’s state vector fully to the |match state, as shown in
part (g). In this way, if the trigger collapses to the “match” state when a measurement
is made on it, we will know that the box doesn’t contain a bomb.
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10. 3 Conclusion
The protocol developed here allows one to determine the presence of a classical object without
disrupting its state, using the properties of quantum systems in a superposition of two
possible states. Thus, measurements may take place without interactions, by exploiting the
absence or presence of wavefunction collapse as the deciding factor of an observed system’s
presence. The results of these theoretical calculations have been empirically verified as well,
through various experimental incarnations6,7,8
.
This protocol effectively demonstrates the technique of the Elitzur-Vaidman bomb tester,
and points the way to such protocols as counterfactual quantum computation, in which the
result of a quantum computer’s algorithm may be determined without running the quantum
computer, by a means analogous to how the bomb was measured without ever interacting
with it. Thus, the protocol explored in this report opens the way to a series of tools useful
for quantum computing, elucidating steps for how this technology will be implemented in
the future.
4 Acknowledgements
First and foremost, we would like to thank Dr. Jacob Barandes of the Harvard University
Physics Department for his eternal patience and incredible guidance with this project and
throughout the class. Thank you not only for teaching us theoretical physics, but also for
inspiring us with your indulgence of our curiosity. We would also like to thank our tireless
Teaching Fellow, Sruthi Narayanan, and infallible Course Assistant, Zack Gelles. We never
would have finished our problem sets without your consistent presence and help. Similarly,
we appreciate our peers in the Fall 2019 Physics 19 class as well as the community of physics
enthusiasts at large; it was our pleasure to share this wondrous journey through theoretical
physics with so many excellent friends. Finally, we appreciate the encouraging support of
Harvard Physics for this class and our continual exposure to the incredible world of physics.
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11. 5 References
1. Kwiat, Paul G. “Interaction Free Measurements.” Interaction Free Measurements, Uni-
versity of Illinois Department of Physics, physics.illinois.edu/people/kwiat/interaction-
free-measurements.asp.
2. Renninger, M. (1960). “Messungen ohne St¨orung des Meßobjekts” (Measurement with-
out disturbance of the measured objects). Z. Physik 158: 417–421 doi:10.1007/BF01327019
3. Elitzur, Avshalom C., and Lev Vaidman (1993). “Quantum Mechanical Interaction-
Free Measurements.” Foundations of Physics 23.7: 987–997. Crossref. Web.
4. Ball, Philip, “The Universe Is Always Looking,” Atlantic, 2018.
5. Saeedi, Kamyar et al. (2013). “Room-Temperature Quantum Bit Storage Exceeding
39 Minutes Using Ionized Donors in Silicon-28,” Science, 342.6160: 830-833. doi:
10.1126/science.1239584.
6. P. G. Kwiat; H. Weinfurter; T. Herzog; A. Zeilinger; M. A. Kasevich (1995). “Interaction-
free Measurement”. Phys. Rev. Lett. 74 (24): 4763–4766. doi:10.1103/PhysRevLett.74.4763.
7. Hosten, Onur; Rakher, Matthew T.; Barreiro, Julio T.; Peters, Nicholas A.; Kwiat,
Paul G. (February 23, 2006). “Counterfactual quantum computation through quantum
interrogation”. Nature. 439 (7079): 949–952. doi:10.1038/nature04523.
8. Carsten Robens; Wolfgang Alt; Clive Emary; Dieter Meschede and Andrea Alberti (19
December 2016). “Atomic ‘bomb testing’: the Elitzur–Vaidman experiment violates
the Leggett–Garg inequality”. Applied Physics B. 123 (1): 12.
6 Contributions
The abstract was written by Benji, as was the Scenario and Background (sections 1.1 and
1.2), while Samuel described the Real-world applications (section 1.3). In the Calculations of
this report (section 2), Irtiza performed and wrote up the calculations for parts (a)-(f), while
Samuel finished parts (g) and (i), and Benji completed part (h). Finally, Samuel summarized
the significance of these results in the Conclusion (section 3).
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