Quantum Communications in the Maritime Environment

Quantum Communications in the Maritime Environment Jeffrey Uhlmann Marco Lanzagorta Salvador Venegas-Andraca University of Missouri-Columbia US Naval Research Laboratory Tecnologico de Monterrey 201 EBW, Columbia, MO 65211 Washington, DC 20375 Estado de Mexico, Mexico Email: [email protected] Email: [email protected] Email: [email protected] Abstract—In this paper we describe research relating to the the underwater environment in the three major Jerlov water potential use of quantum key distribution (QKD) protocols for types. For instance, theoretical analysis has shown that, under secure underwater communications. We briefly summarize the certain conditions, a QKD protocol that guarantees the perfect BB84 QKD protocol, its implementation and use in free-space applications, and then describe recent theoretical considerations security of underwater blue-green optical communications of its use in the maritime domain [18]. We also consider appears to be feasible with a key generation rate of about 170 alternatives to QKD that offer security and bandwidth utilization kb/s over 100m in clear oceanic waters (Jerlov type I). Notice advantages when applicable. that this represents about 600 times more bandwidth than current VLF systems. Furthermore, 100m is the average depth I. INTRODUCTION of the thermocline, the required minimum depth for the stealth The deployment of efficient and secure communication navigation of an underwater vehicle. In principle, these results links with underwater vehicles is among the most significant suggest that it may be feasible to establish a quantum channel technological challenges presently confronted by the world’s between an underwater vehicle and an airborne platform. naval forces and, increasingly, by industry. To this end, recent Theoretical quantum key distribution (QKD) protocols have research efforts have explored the feasibility of free-space been developed that offer security guarantees which rest on optical communication links connecting underwater vehicles fundamental laws of physics rather than assumptions about with airborne platforms. These optical links are typically the computational limitations of potential adversaries [1], [2], implemented with blue-green lasers, which are fine-tuned to [3], [4], [5]. Physical realizations of such protocols have work at the frequency of minimal optical attenuation produced demonstrated their feasibility. Examples include QKD over op- by oceanic water. Problems related to the tracking of optical tical fiber[6], free-space communication between two ground devices and the effect of the air-water interface are currently stations [7], and free-space communications between a satellite being investigated. and a ground station [8], [9], [10]. To date, however, there At the same time, one of the major scientific thrusts in has been little if any examination of the practical feasibility recent years has been to try to harness quantum phenomena of QKD to support secure underwater communications [18]. to dramatically increase the performance of a wide variety In this paper we consider potential applications of the BB84 of classical information processing systems. These efforts QKD protocol for subsurface communications in oceanic in quantum information science have produced a variety of waters. Our objective is not to provide a complete overview promising theoretical and experimental results with a consid- of quantum information [1], [2], [3], [4], [5] and quantum erable impact on the development of perfectly secure quantum cryptography [1], [11], [12], [13]; rather, our goal is to present communications. In this regard, “perfectly secure” commu- the basic principles required to understand the operation and nication is understood to imply an encryption protocol with security of quantum cryptographic devices so that we may mathematically provable security guarantees that do not rely consider their use in underwater applications. on assumptions about the amount of computational resources A. Quantum Information available to potential adversaries. More specifically, Quantum qubit Key Distribution (QKD) protocols provide security guaranteed The fundamental unit of quantum information is the , by the laws of physics rather than assumed resource re- which has properties that generalize those of its classical quirements, e.g., to perform prime factorization, that underpin counterpart, the bit [11], [12]. A classical bit is a binary classical key distribution methods that are currently used. variable that can only assume a value of 0 or a value of 1. Its value is unique, deterministic, and unambiguous. By contrast, While the feasibility of QKD over optical fibers and the a qubit can assume a state of 0, 1, or a probabilistic mixture – atmosphere is well understood, the underwater environment or superposition – of those two states. The state of a qubit is offers a variety of new challenges. In this paper we will present represented by a pair of complex numbers, fa; bg, which are a general overview of our efforts to study the feasibility of related to classical 0-1 binary states as: an underwater quantum channel to enable QKD protocols. In particular, we will discuss how free-space QKD performs in qubit = fa; bg = a · 0bit + b · 1bit ; (1) where jaj2 is the probability that the qubit will be found qubit is therefore a complex linear combination of classical in state 0 and jbj2 is the probability that the qubit will be 0-1 states. found in state 1. By a physical process which has no classical It has been mentioned already that the state of a qubit analog, the state of a qubit only becomes equivalent to that of becomes equivalent to that of a classical bit after it has been a classical bit after it has been read/measured. measured. This is true because the quantum process of mea- Bra-ket notation is a generalization of common vector suring a superposition is unavoidably destructive and results notation in which hΨj is a row vector (read as “bra psi”) in the “collapse” of the superposition to a classical bit value. and jΨi is a complex conjugate column vector (read as “ket After this collapse of the superposition the qubit is essentially a psi”), and the inner product hΨjΨi is referred to as a “bracket” classical bit, so all subsequent read operations will produce the (which is the origin of the root terms “bra” and “ket”) [14]. same value. Prior to measurement, however, the superposition In this notation the state of a single qubit can be written as: can be non-destructively transformed in a variety of ways that permit the relative probabilities of measuring a classical 0 or 1 jΨi = aj0i + bj1i (2) to be controlled. or Another critical property of a quantum superposition is that hΨj = a∗h0j + b∗h1j ; (3) it cannot be copied/cloned. Unlike the state of a classical bit, a superposition stored in one qubit cannot be copied and stored where the condition in another qubit. The laws of quantum mechanics allow the jaj2 + jbj2 = 1 (4) state of one qubit to be moved to another qubit, but it can be shown that the process necessarily destroys the state of the is critical for the interpretation of the squared norms of a and original qubit. In other words, quantum information cannot be b as probabilities. In other words, a reading/measurement of copied – it can only be teleported from one place to another. the qubit jΨi will result in 0 with probability jaj2 and 1 with 2 Quantum cryptography exploits the destructive-measurement probability jbj . and no-cloning properties of quantum information to permit It is important to understand that j0i is not a zero vector; detection of surreptitious measurements by an eavesdropper. rather, “0” is just a label for one unit vector and “1” is a For example, assume that instead of encoding bits using the label for another unit vector, j1i. In other words, one basis computational basis, Alice and Bob communicate using an vector is arbitrarily labeled or interpreted as corresponding to encoding in which 0 and 1 values are encoded using the the classical state 0 and the other basis vector is interpreted diagonal basis: as corresponding to the classical state 1. If the context were classical True/False Boolean logic then the two basis states j0i + j1i j0iD ≡ j+i = p (9) might be labeled jTi and jFi, where the choice of which basis 2 vector is interpreted as “T”, and which is interpreted as “F”, j0i − j1i j1iD ≡ |−i = p : is entirely arbitrary. 2 Even the choice of the basis vectors is arbitrary. All that matters is that they are orthogonal and span a two-dimensional Alice and Bob could have mutually chosen any pair of space. Once they are chosen they are then referred to as the orthogonal vectors to represent 0 and 1 values, so their choice computational basis. The most common choice of vectors for of the diagonal basis instead of the computational basis (or the computational basis is: any other basis) is entirely arbitrary. Once the choice of basis has been made, Alice can send her bit-string message, say h0j = (1; 0) (5) “1001010001”, as a sequence of basis vectors: j1iD, j0iD, h1j = (0; 1) j0iD, j1iD, :::, j0iD, j1iD. In order for Bob to properly receivep her message he must know to interpret each j0i + j1i= 2 as and p a 0 and each j0i − j1i= 2 as a 1. In other words, Bob has 1 j0i = (6) to measure the qubits in the exact same basis to know what 0 logical information was sent to him by Alice. 0 Let P (xjy) denote the probability of measuring a logical j1i = : 1 bit x given receipt of a logical bit y. If Alice and Bob use the same basis, then for Bob: Because j0i and j1i are orthogonal: P (0 j0 ) = j h0j0i j2 = 1 (10) h0j0i = h1j1i = 1 (7) D D D D 2 P (0Dj1D) = jDh0j1iDj = 0 and 2 P (1Dj0D) = jDh1j0iDj = 0 h0j1i = h1j0i = 0 : (8) 2 P (1Dj1D) = jDh1j1iDj = 1 : In summary, j0i should be interpreted as being equivalent to a classical bit in the 0 state and j1i should be interpreted as On the other hand, if an eavesdropper, Eve, measures a qubit being equivalent to a classical bit in the 1 state.

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