---
abstract: |
    We examine a situation in which an information-carrying signal is sent from two sources to a common receiver. The radiation travels through free space in the presence of noise. The information resides in a relationship between the two beams. We inquire into whether it is possible, in principle, that the locations of the transmitters can be concealed from a party who receives the radiation and decodes the information.

    Direction finding entails making a set of measurements on a signal and constructing an analytic continuation of the time dependent fields from the results. The fact that this process is generally different in quantum mechanics and in classical electrodynamics is the basis in this investigation.

    We develop a model based upon encoding information into a microscopic, transverse, non-local quantum image (whose dimensions are of the order of a few wavelengths) and using a detector of a type recently proposed by Strekalov et al. The optical system, which uses SPDC (Spontaneous Parametric Down Conversion), functions like a Heisenberg microscope: the transverse length, which encodes the signal information, is conjugate to the transverse momentum of the light. In the model, reading the signal information spoils the directional resolution of the detector, while determining the directions to the sources spoils the information content. Each beam, when examined in isolation, is random and indistinguishable from the background noise.

    We conclude that quantum communications can, in principle, be made secure against direction-finding, even from the party receiving the communication.
---

<span>**Direction Cryptography in Quantum Communications**</span>

<span>**Walter Simmons and Sandip Pakvasa\
**</span>

Introduction
============

Classical cryptography had several weaknesses, including the need for either key distribution, or for the use of a public key system, whose algorithm could be subject to possible decipherment. Other classical deceptive techniques, such as spread spectrum communications, padding the communications with dummy messages, etc., could help achieve better privacy, but it remained for quantum cryptography to remove the limitations on keys and make possible the encipherment of a signal that guaranteed privacy by the laws of quantum mechanics @4b [@Bennett; @1b].

In the case of signals sent through free space, it is often possible for a third party to direction-find the sources of quantum communications (or their concomitant classical signals), even if that eavesdropper cannot decipher the meaning of the signal. Moreover, the signal's intended recipient, who has the necessary cryptographic key and equipment, generally cannot be deceived about the direction of propagation of the incoming radiation.

Quantum mechanics is rich in physical phenomena, such as the superposition of amplitudes from various sources, EPR correlations, the interference of photons of different frequencies @Larchuk, etc., that might be used to engineer a system that could encrypt the direction of propagation of electromagnetic signals. We undertake to investigate these possibilities in this work.

When a single transmitter is involved, the direction to the transmitter can be determined by any party who can read information off of the transmitted signal, (even if the station is buried in noise), by simply measuring the recoil of the receiver. This is true in both classical and quantum mechanics.

We proceed to consider the case of two transmitters, in which the individual signals are pseudo-random, but which convey information to a distant party through a relationship between the two beams, which travel through a noisy environment. If the party receiving the signals cannot separate the signal from the noise, then the recoil of the apparatus points toward a position between the two transmitters and the individual sources cannot be separated. Classically this will protect the sources from direction-finding from an eavesdropper only so long as he or she cannot decipher the signal. Furthermore, it will not conceal the sources from the intended recipient who can identify and record the two components of the signal and determine the directions to the sources, in spite of the noise.

Classically, ideal direction finding consists of measuring the electromagnetic fields throughout a volume of space for a period of time. The measurements are made with arbitrarily high precision. Using analyticity, the full wave front is reconstructed and the positions of the sources are revealed. With enough time and instrumentation, direction finding is always possible for classical fields, even in the presence of noise.

For the reasons just stated, we shall focus our analysis only upon quantum communications in which there are two sources and one receiver. We shall assume the existence of an isotropic background noise.

Direction Finding in Quantum Communications in Free Space
=========================================================

The introduction of quantum mechanics into communications widens the possibilities for cryptography very considerably. The properties of the electromagnetic fields can be divided, at least approximately, into two parts: the properties of the ensemble structure and the properties of the quanta. The properties of the ensemble structure include the density of quanta in space and time (including the times and directions of transmission), and the inclusion of spurious quanta. The quanta, for our purposes, are photons and biphotons.

As viewed from a quantum mechanical perspective, classical direction-finding consists of making ensemble measurements with sufficient statistical precision to enable the reconstruction of the space-time properties of the fields and subsequent location of the sources with desired accuracy. Methods such as spread spectrum communications vary a classical property (e.g. the frequency or time of transmission) but still send ensembles of identical quanta. In other words, the designer breaks up large ensembles into smaller ones.

The essential departure that made quantum cryptography possible is based upon the observation @4b that individual quanta can not only carry information, but have many useful properties such as, fragility under measurement, superposition, and entanglement. Of particular importance here, is the usual property that quantum states can be usefully measured only once.

In the model discussed here, all signal information is encoded into individual quanta (biphotons), which are transmitted as part of a heterogeneous ensemble. The ensemble structure carries no signal information. The biphotons carrying the signal information are transmitted at random intervals and noise quanta are zlee@example.org. The intensity and other single particle statistics are relatively independent of the information content of the biphotons. Thus, with wide latitude, the electromagnetic field can be constructed so that various distributions measured on the receiving end, whether by the intended receiver or by an eavesdropper, can generally be made to look like noise. In a noisy background, the ensemble structure will carry no information useful for direction finding.

Quantum Direction Finding
=========================

In 1989, Ou, Wang, and Mandel @Ou performed what may, in retrospect, have been the first experiment related to quantum ranging. They showed that a positional ambiguity arises when two non-linear crystals are aligned along a common optical path.

Giovannetti, Lloyd, and Maccone @Giovannetti [@I] took up the question of cryptographic ranging in a one-dimensional situation and showed that a quantum mechanical system could ensure range privacy.

This work was stimulated by @Giovannetti [@I], but here, we are interested in direction cryptography in three-dimensional space.

A Specific Problem in Direction Privacy
=======================================

To make our analysis as simple as possible we pose a specific hypothetical problem for which we develop our model.

An agent living somewhere in the Galaxy desires to broadcast a signal to let some other intelligent beings elsewhere, know that she exists. She is cautious, however, because she is concerned that a hostile being, which receives her signal, may direction-find her and make trouble. Incidentally, such caution may offer a simple explanation why no obvious signals from ETs (Extra-Terrestrials), have been yet detected at the earth despite many searches; and thus provide a partial explanation for the Fermi Question(or at least account for the great silence @Tarter). Presumably ETs being smart as well as cautious, would utilise the kind of direction cryptography discussed below, but most likely much more advanced technological versions thereof.

She notes that looking into even a small visual cone in the plane of the Galaxy reveals millions of stars, as well as weak background light and radio noise. She desires to send a signal which, if studied with a direction finding apparatus, will look just like ordinary background noise, but which, if read properly, reveals some information.

She assumes that any agent worth contacting has already done his research, has set up all of his equipment, and will be ready to receive a signal at any time. Moreover, she chooses to broadcast to limited groups of stars at a time.

We endeavor to engineer a communication system to solve this problem.

Preliminary Discussion
======================

Before taking up our model, in the next section, we discuss some aspects of the problem semi-quantitatively here.

Our problem is made easier by setting aside phenomena that do not occur in our hypothetical situation. These include evanescent waves, back reaction on the transmitter, and the need for an informational cryptographic layer (we don't care who understands the message, we only desire to confuse those who do about its spatial origin.)

The assumption that the receiving agent is well prepared, allows us to radiate only a limited amount of total energy and a limited number of quantum states, all of which might be different from one another; therefore, useful high statistics measurements cannot be made by the receiver. The assumption that the radiation is targeted at a limited volume allows us to choose radiation with a small signal strength (or a limited coherence length) away from the optical path, so that wide-angle measurements, made with detectors separated by astronomical distances, will yield no information. Moreover, because of these assumptions, the range to the receiver will be approximately known, allowing convenient conversion of approximate distances into angular measures.

We desire that the signal be readable with an instrument, which measures photon pairs originating from different directions in space, but not with an instrument that detects the photons from the individual sources. This corresponds to a choice of large transverse momentum acceptance, versus small transverse momentum acceptance. Utilizing the Uncertainty Principle, we choose to engineer information into the signals in the form of an image, of microscopic transverse dimensions. The radiation will be transmitted from a pair of radiators separated by an astronomical distance.

As we discuss in a section, below, the transference of images into quantum correlations between optical beams has been studied for several years. Recently, using Spontaneous Parametric Down Conversion, clear images of letters, arrayed transversely to the beam directions, have been encoded in correlations and subsequently recovered from individual photon detection measurements @Group [@B].

For our purposes, we require an image which is microscopic in size. The minimum dimension is of the order of one wavelength, as smaller images do not propagate. To date, there have been no experimental demonstrations of the encoding and decoding of such small images, while there is no problem of principle in doing such experiments. See also the Appendix.

The detection of quantum correlations over ranges of tens of kilometers has been demonstrated @Group [@C]. We shall assume that the detection can be extended to astronomical ranges.

In our hypothetical apparatus, we encode a small transverse image onto a pair of optical beams, prepared by Spontaneous Parametric Down Conversion @Group [@B]. The beam is split and directed toward reflectors located far apart (the distance L, which is of the order of several parsecs), where the two beams are directed at the receiving party, located at a known distance, R. The form of the image is unimportant and may be as simple as some dots separated by a distance, $Y_0$.

Let us call the opening angle, (at the receiver), between the two transmitters, $\Delta \Theta_0$, and call a typical smaller opening angle, such as might be necessary to isolate the radiation from one transmitter from the radiation from the other transmitter, $\Delta \Theta$. With each of these angles, we associate a spread of transverse momentum. $$\Delta P_{y,0} \sim P_Z \Delta \Theta_0 \ \ \mbox{and} \ \ 
\Delta P_{y} \sim P_Z \Delta \Theta$$

Note that the momentum along the direction of propagation, $P_Z$ , can be measured independently of the angles. Thus a measurement of one of the angles (equivalently, restricting the radiation to within that angle) amounts to a measurement of the transverse momentum.

We want to engineer the image so that it can be viewed with quantum mechanical measurements only when no precise measurement or strong restriction is made upon the transverse momentum of either photon. When the radiation from either transmitter is isolated, we want the Uncertainty Principle to prevent the observation (and recording) of the image information in the radiation. We choose $Y_0$ such that, $$Y_0 \gg 
\frac{1}{\Delta P_{y,0}} =
\frac{\lambda}{4 \pi \Delta \Theta_0} \ \
        \mbox{but also,} \ \
Y_0 \ll
\frac{1}{\Delta P_{y}} =
\frac{\lambda}{4 \pi \Delta \Theta}$$

In order that the radiation propagate, we must also have, $$Y_0 \geq  O (\lambda)$$

These conditions are not difficult to achieve if L and R are of astronomical sizes. Therefore, we can engineer the transmitter and image correlations so that the receiver must use a large acceptance angle instrument in order to be able to extract the image. In that case, the recoil momentum measured points back toward a position between the transmitters and gives very little information about their true location. If the receiver measures the transverse momenta of the individual photons with sufficient precision to determine the position of their respective transmitters, then the Uncertainty Principle implies that the information will be washed out and the photons are indistinguishable from background noise.

Non-Local Quantum Image Based Model
===================================

It has been known for some time that the quantum correlations between particle states, originally discussed by EPR, can carry information. In Spontaneous Parametric Down Conversion (SPDC), a biphoton state is created by the interaction of a laser beam with a non-linear crystal @MW. Within a first order perturbation theory treatment, (with the pump beam treated classically), the interaction Hamiltonian has a factor. $$H_I \rightarrow a^+_1 a^+_2 a_p$$ where (1) and (2) refer to outgoing photon states and (p) refers to the laser beam, which is generally treated classically. From this we see that, $$exp (-iH \ _It) |  0 >_1 |0>_2 \ \rightarrow \sum_{n}\ C_n (t) | n >_1 | n >_2$$ and hence, the state consists of two highly entangled multi-photon states with perfectly correlated photon number. The states of greatest interest to date are the biphoton states, consisting of an entangled pair of photons. Within a first order perturbation context these can be written as single photons states which are entangled in momentum. The two modes are traditionally called signal (S) and idler (I). Taking these to be continuous, the Hamiltonian, above, evolves the vacuum into the state, $$|  Biphoton > \ \rightarrow  \int dk_1 \int dk_2 \Phi (k_1, k_2, k_p) \mid k_1, 
\sigma_1 >_I | k_2, \sigma_2 >_S
\label{vectors}$$

A biphoton is an excitation of the electromagnetic field, which need not be represented perfectly in perturbation theory. For example, the biphoton can carry the same k vector as the pump beam. This was noted theoretically by @Lukin [@Physics] and has recently been observed experimentally by @Physics. Strekalov et al. @Strekalov have discussed the point that the biphoton has properties which are not described by perturbation theory.

Just as in EPR states, information encoded in, say, momentum or polarization relationships between the S and I photons, resides in a relationship between those properties of the individual photons. For this reason, this kind of information is often called quantum non-local information.

A special case of Eq. \[vectors\], in which the function $\Phi$ depends only upon the difference of the transverse momenta, could be $$|  Biphoton > \ \rightarrow  \int dq_1 \int dq_2 \Phi (q_1 - q_2)| q_1>_I |q_2>_S
\label{vector}$$ where the $q$ vectors are the transverse momenta of the photons and where other variables have been suppressed.

If we engineer the function to be peaked around zero, then we observe the following properties of the state in Eq. \[vector\]:

a.) A measurement of either single photon state, (S or I), results in a random result for the corresponding q vector.

b.) Following the first measurement, the possible momentum of the other single photon state, (I or S), is sharply peaked around the same value of q.

We know from classical optics @Goodman that when light is passed through an aperture of diameter w and at a range z from the source, (w/z is an angular measure), then the spatial frequency is cut off at $$\frac{w}{z \lambda}
\label{cutoff}$$

It is usually assumed that classical transfer functions for passive optical devices carry over to the quantum mechanical description. Thus, if one of the photons in Eq. \[vectors\], (S or I), is passed through an aperture of sufficiently narrow width, its transverse momentum will be constricted to a value that can pass through the aperture. It follows from the properties listed above, that the amplitude for the other photon (I or S) is similarly reduced.

Our model for communication with direction privacy consists of encoding information on large values of the transverse momentum variables $(q_1$ or $q_2)$ of biphotons, which consist, generally of photons of unequal frequencies. As per property (a), a measurement of an isolated photon yields a noise value. An attempt to restrict the transverse momentum of either photon in order to narrow the direction of propagation into an angle determined by w/z, results in the destruction of the transverse momentum information in both photons.

A measurement of the biphoton as a single quantum state causes the measuring apparatus to recoil in the direction of motion of the biphoton and the individual photon directions of motion are lost.

Obviously, it is essential that it be possible to extract the transverse information from the biphoton. For that reason, we shall briefly discuss the status of transverse, quantum non-local images.

Within the past few years, it has become possible to convey images using biphoton states. Investigations of fringe visibility and the transverse spatial structure of SPDC light were carried out starting around 1993 @Group [@B; @Ribeiro]. Two early experiments are those of Strekalov et al. 1995 @Group [@B] and Pittman et al. 1995 @Group [@B]. Some theoretical discussion is given in @Theory [@A]. Fonseca et al. @Theory [@A] demonstrated that the angular spectrum of the pump beam can be recovered from the fourth order correlations of the signal and idler fields. It has recently been demonstrated theoretically and experimentally @Nogueira that these images have photon anti-bunching in spatial variables transverse to the direction of biphoton propagation, thus extending the concept of temporal anti-bunching and demonstrating that the correlations in the transverse plane have no classical analog.

Our purpose in choosing SPDC is to have a model process. The state of technology is not yet such that we can use SPDC in its present state for direction cryptography. The realm of microscopic images has not been studied experimentally, and our model requires a new kind of biphoton detector such as that proposed by Strekalov et al. @Strekalov. Lukin et al. @Lukin have discussed the theory of the detection of entangled photons in connection with the phenomenon of storage of light in atomic vapor.

Other Direction Finding Approaches
==================================

In this section we consider a specific attack on the cryptography and add some comments about the ensemble properties.

Let us suppose that the receiver attacks the privacy as follows. He focuses an image of the sky onto a plane in his laboratory. Behind this plane, he places a biphoton detector. He then blocks out a small section of the image at the (guessed) position of the image of one of the two sources. When the signal in the detector vanishes, he has obviously covered up the image of the source.

From Eq. \[cutoff\], we see that the image of each source must be broad enough to encompass at least the width of the aperture we considered earlier. The angular width of the source images are greater than the angular separation of the two sources. Thus, this method does not work.

In an attack, in general, the receiver places biphoton detectors throughout a volume of space and records, for each biphoton event at each detector, the arrival time, the total momentum, polarization, and, of course, the transverse displacement data (the microscopic image), which carries the signal information. He next attempts to use this data to derive the source directions.

We emphasize that the signal information is a property of individual biphotons, and the ensemble characteristics can be engineered for maximum deception at the transmitter. We also emphasize that each biphoton can be usefully measured only once, never at several sets of detectors. Correlations between count rates by different detectors generally measure ensemble characteristics.

The introduction of an image distribution into an ensemble of biphotons like those in Eq. \[vectors\] can be expected to bias the single photon statistics. However, the bias can be calculated and offset by introducing single photon noise.

If the microscopic transverse image is static, then the biphotons can arrive in any sequence. The transmitter emits them a few at a time and at random intervals. We have already discussed the idea that the structure of the quantum signal in the transverse direction can be chosen to be random on macroscopic scales. Therefore, neither the time of arrival data, nor the macroscopic spatial distribution data, contains significant direction information.

Conclusions
===========

We have investigated whether quantum communications can be made resistant to direction-finding.

We considered a model in which a heterogeneous ensemble of biphotons carries information, in the form of a microscopic transverse image, to an intended receiver. We conclude that it is possible to engineer the radiation field so that even the receiver, who knows how to read the informational image, cannot determine the precise angular position of sources from which the biphotons were transmitted.

In this paper, we have raised the possibility that quantum communications can be made secure against direction finding; and have suggested specific techniques to achieve this goal.

Many reasons have been proposed to explain the fact that SETI researchers have detected no signals from ETs (the so-called Fermi question). For a recent review, see @universe. The model described here provides the germ of a solution to this problem. When technology has developed further than it stands today, it may be possible to design a plausible quantum search strategy for use in SETI.

Acknowledgements
================

We thank Xerxes Tata for many clarifying discussions. This work was partially supported by the U.S. Department of Energy under grant number DE-FG-03-94ER40833.

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Appendix: Information Propagation of Biphotons
==============================================

As is well known @Goodman, an electromagnetic wave cannot propagate if the transverse wave vector, q, exceeds a bound determined by the relationship, $$\hat{q}^2 + k^2_z = \left ( \frac{2 \pi}{\lambda} \right )^2$$

The pump beam has half the wavelength of the individual outgoing photons. Therefore, the maximum range of wave vector for each daughter photon is, in perturbation theory, lower than that of the pump photon. The transverse momentum in the disallowed range can, however, be carried as a net momentum of the pair of photons, depending upon the absorption of momentum in the crystal. Moreover, if the biphoton is regarded as a single quantum state, it has, in principle, the capability of carrying the same range of transverse momenta as the pump state from which it arises. This behavior has been discussed by Fonseca et al. @Fonseca, and by Jacobson, et. al. @Jacobson. This has been studied experimentally by Edamatsu, et al. @Physics. See also @Rubin.

The interesting point here is that when the photons are treated individually, they cannot carry all of the transverse momentum information of the pump photons. However, the pair can carry all of the information in the pump beam.

{'source': 'arXiv_out/quant-ph0302186'}