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Quantum positioning system
V. Giovannetti, S. Lloyd∗ , and L. Maccone
Massachusetts Institute of Technology, Research Laboratory of
Electronics
∗
Department of Mechanical Engineering MIT 3-160,
Cambridge, MA 02139, USA.
Abstract: We show how quantum entanglement and squeezing can
be employed in measuring a position in space, gaining an accuracy
increase over classical localization procedures. We develop a positioning
protocol based on frequency-entanglement of number-squeezed pulses,
which may be used also for synchronizing the clocks of distant parties,
and show how the accuracy increase scales with the system resources
and how it compares with analogous, but classical, protocols.
c 2004 Optical Society of America
OCIS codes: (030.1670) Coherent optical effects; (270.6570) Squeezed states
Quantum entanglement and squeezing have been exploited in the context of interferometry, frequency measurements, lithography, and algorithms (see for example Refs.
[1]) in order to beat classical limitations. In this paper, we show that entanglement
and squeezing can also be used to increase the accuracy of positioning measurements.
Typically, such measurements can be performed by sending M pulses (each composed
of N photons in average) from the party that is to be localized to M receiving stations
and measuring their times of arrival. We will show that, √by using entangled-squeezed
light, it is possible to obtain an accuracy enhancement of M N over the classical case.
A detailed derivation of the methods described here can be found in [2].
Consider the frequency-entangled state
Z
|Ψi ≡ dω φω |Nω i1 |Nω i2 · · · |Nω iM ,
(1)
where φω is a spectral envelope function, and |Nω ii is the N -photons Fock state (number
squeezed) of frequency ω, directed towards the i-th receiving station. In the case of ideal
detectors, the probability to detect all the photons is given by [3]
* M N
+
Y Y (−)
(+)
p({ti,k }) ∝ :
Ei (ti,k )Ei (ti,k ) : ,
(2)
i=1 k=1
where ti,k is the detection time of the k-th photon at the i-th receiving station. In
R
(−)
Eq. (2), the signal field at the i-th detector position is Ei (t) ≡ dω a†i (ω) eiωt and
†
(+)
(−)
Ei ≡ Ei
, where ai (ω) is the field annihilator of a quantum of frequency ω. The
calculation of this probability with the state |Ψi yields
!2
M X
N
X
(3)
p({ti,k }) ∝ g
ti,k ,
i=1 k=1
where g(t) is the Fourier transform of φω . The uncertainty ∆t in the mean time of
arrival
hti =
M N
1 XX
ti,k
M N i=1
k=1
(4)
can be immediately obtained from Eq. (3), resulting in
∆t =
∆τ
,
MN
(5)
where ∆τ 2 is the variance of the distribution g(t). This result must be compared to
what one would obtain in the classical case where one employs M unentangled pulses
each in a coherent state with N average photons. For a fair comparison with the state
|Ψi, we must consider the situation in which all the photons are described by the same
bandwidth function φω of Eq. (1). In this case, because of the Poissonian statistics, the
times of arrival of each photon are independent. One can thus apply the central limit
theorem, obtaining an accuracy
∆τ
∆t & √
,
MN
(6)
where, by using Eq. (2), it is possible to show that time of arrival uncertainty of the
single photon ∆τ is the same as in Eq. (5). The comparison between (5) and (6) shows
that entanglement and number-squeezing can increase
the accuracy of the mean time
√
of arrival (and hence of positioning) by a factor M N .
Using experimental techniques analogous as the ones described in [4], at least the
simple case of M = 2 detectors and N = 1 photons per pulse can be implemented in
practice.
Assuming that the distance between different parties is known, the same protocol can
also be employed to enhance the accuracy of the synchronization of their clocks. Namely,
they have to exchange entangled and squeezed pulses while measuring the pulses transit
times.
In addition to the difficulty of creating the requisite entanglement, the primary drawback of this scheme is the sensitivity to loss. However, for this same reason, the procedure
exhibits improved security: because the timing information resides in the correlations
between pulses, an eavesdropper who intercepts some but not all of the photons obtains no timing information. Finally, the frequency entanglement allows similar schemes
to be highly robust against pulse broadening due to transit through dispersive media
as experimentally shown in [5]. Since dispersion is one of the main limiting factors in
employing classical narrow pulses, this is a very attractive characteristic for practical
applications.
This work was funded by the ARDA, NRO, and by ARO under a MURI program.
References and links
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Bollinger, et al., Phys. Rev. A 54, R4649 (1996); A. N. Boto, et al., Phys. Rev. Lett. 85, 2733
(2000); L. K. Grover, Phys. Rev. Lett. 79, 325 (1997).
2. V. Giovannetti, S. Lloyd, and L. Maccone, Nature 412, 417 (2001), Eprint quant-ph/0103006.
3. R. J. Glauber, Phys. Rev. 130, 2529 (1963).
4. C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
5. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 68, 2421 (1992).