Abstract. We present a system to measure the distance between two parties that allows only trusted people to access the result. The security of the protocol is guaranteed by the complementarity principle in quantum mechanics. The protocol can be realized with available technology, at least as a proof of principle experiment. The following touching problem will be addressed in this paper. Alice is lost in the woods. Her friend Bob needs to find her, rescue her, and live happily ever after. On the other hand, the bad wolf Eve also wants to find her, in order to gobble her up. Suppose for simplicity an unidimensional forest, Alice and Bob need to find their relative position (ranging) without giving any hint to Eve. This intent is analogous to the one underlying cryptography, i.e. the exchange of information in a secure fashion. In this respect we will refer to it as crypto-positioning procedure. In this paper, we show how the quantum mechanical time-energy uncertainty principle can be employed to measure in a quantum cryptographically secure fashion the distance between Alice and Bob. This means that the physical limitations imposed by quantum mechanics do not allow, not even in principle in the ideal case, any kind of eavesdropping on Eve's part. The secure exchange of information, based on analogous "weird" quantum effects has been shown long ago in [1-3] and technological applications seem to be at hand [4] . Our idea to help Bob in his quest stems by joining Ekert's quantum cryptographic protocol [3] with the recently proposed quantum positioning protocol [5] , that allows one to perform ranging using frequency-entangled states. In [5, 6] we have shown how, by using N frequencyentangled photons, one can obtain an 1/ √ N accuracy enhancement in finding out the distance between Alice and Bob over the case in which the N photons are unentangled. This is a truly quantum effect that arises from the strong photon correlations between photons originating from the entanglement. This same fact, however, makes the loss of a single photon critical: as shown in [6] , when one of the entangled photons that travel from Bob to Alice is lost, the remaining photons yield no information at all on the distance between the two. Such an apparent drawback turns out to be the key feature in devising the crypto-positioning protocol. In what follows we can limit our analysis to the case N = 2. In this situation it is possible to use the state generated by cw-pumped spontaneous parametric downconversion, in which the two generated photons are anticorrelated in frequency, i.e. In Eq. (1) the notation |ν refers to a single photon state of frequency ν, the ket subscripts refer to the two distinct field modes (the signal S and the idler I) generated by the crystal, 2ω 0 is the pump frequency, and φ(ω) is the two-photon spectral function centered in ω = 0 with bandwidth ∆ω. The state |Ψ is one of the best known sources of entanglement currently available and an enormous amount of literature both theoretical and experimental is accessible (see for example [7] and citations therein). Another possibility that can be exploited is the recently proposed 'difference beam state ' [8] that displays frequency correlated photons, whereas |Ψ displays anti-correlation in frequency. The properties of the entanglement in the state |Ψ are such that, if one measures the frequency of the signal photon, he/she would obtain a random value ω 0 + ω with probability density |φ(ω)| 2 , but the subsequent measurement on the idler photon will have the predictable outcome ω 0 − ω (and vice versa if the measurements are reversed). On the other hand, it is possible to show that if one measures the time of arrival of the first photon on a detector, then he/she will be able to predict (with an accuracy of the order of ∆ω −1 ) the time of arrival of the second photon on a second detector at a distance L. In fact, the joint probability of measuring the first photon at time t 1 and the second at time t 2 is given by which exhibits a peak centered in t 2 − t 1 = L/c of width proportional to ∆ω −1 . What happens when one measures the frequency of the first photon and the time of arrival of the second? In this case it is possible to show that the outcome of the time of arrival measurement is completely unpredictable: all the timing information has been "erased" by the frequency measurement. In this respect, the measurement of the frequency on one of the photons has the same effect as the loss of such photon.
Giovannetti, Lloyd, and Maccone REF show how to measure the distance between two parties by quantum cryptographic means so that only trusted people have access to the result.
Quantum cryptographic ranging
J. Opt. B: Quantum Semiclass. Opt. 4, S413, (2002)
Physics