Statistical intrusion detection and eavesdropping in quantum channels with coupling: multiple-preparation and single-preparation methods

Abstract

Classical, i.e., non-quantum, communications include configurations with multiple-input multiple-output (MIMO) channels. Some associated signal processing tasks consider these channels in a symmetric way, i.e., by assigning the same role to all channel inputs, and similarly to all channel outputs. These tasks especially include channel identification/estimation and channel equalization, tightly connected with source separation. Their most challenging version is the blind one, i.e., when the receivers have (almost) no prior knowledge about the emitted signals. Other signal processing tasks consider classical communication channels in an asymmetric way. This especially includes the situation when data are sent by Emitter 1 to Receiver 1 through a main channel, and an “intruder” (including Receiver 2) interferes with that channel so as to extract information, thus performing so-called eavesdropping, while Receiver 1 may aim at detecting that intrusion, which leads to a decision problem (existence of intrusion/no intrusion). Part of the above processing tasks have been extended to quantum channels, including those that have several quantum bits (qubits) at their input and output. For such quantum channels, beyond previously reported work for symmetric scenarios, we here address asymmetric (blind and non-blind) ones, with emphasis on intrusion detection and additional comments about eavesdropping. To develop fundamental concepts, we first consider channels with exchange coupling as a toy model. We especially use the general quantum information processing framework that we recently developed, to derive new attractive intrusion detection methods based on a single preparation of each state. Finally, we discuss how the proposed methods might be extended, beyond the specific class of channels analyzed here.

Appendices

Appendix A: Definition of a single qubit

Qubits are widely used instead of classical bits for performing computations in the field of QIP [29]. Whereas a classical bit can only take two values, usually denoted as 0 and 1, at an initial time $t_0$ a qubit with index $j$ has a quantum state expressed, for a pure state, as

$$\begin{aligned} | \psi _ j( t_0) {\rangle } = \alpha _ j| + {\rangle } + \beta _ j| - {\rangle } \end{aligned}$$

(19)

in the basis defined by the two orthonormal vectors that we hereafter³ denote $| + {\rangle }$ and $| - {\rangle }$, where $\alpha _ j$ and $\beta _ j$ are two complex-valued coefficients constrained to meet the condition

$$\begin{aligned} | \alpha _ j| ^2 + | \beta _ j| ^2 = 1 \end{aligned}$$

(20)

which expresses that the state $| \psi _ j( t_0) {\rangle }$ is normalized. In most of the literature, $\alpha _ j$ and $\beta _ j$ are deterministic, i.e., fixed, values so that $| \psi _ j( t_0) \rangle$ is a deterministic pure state. In part of our investigations dealing with BQSS and related tasks, we also considered the case when $\alpha _ j$ and $\beta _ j$ are random variables (RVs), so that $| \psi _ j( t_0) \rangle$ is a random pure state (see e.g., [10, 13, 15, 18]).

From a Quantum Physics (QP) point of view, the above abstract mathematical model especially applies to electron spins 1/2, which are quantum (i.e., non-classical) objects. The component of such a spin, with index $j$, along a given arbitrary axis Oz defines a two-dimensional linear operator $s_{ jz}.$ The two eigenvalues of this operator are equal to $+ \frac{1}{2}$ and $- \frac{1}{2}$ in normalized units, and the corresponding eigenvectors are therefore denoted as $| + {\rangle }$ and $| - {\rangle }$. The value obtained when measuring this spin component can only be $+ \frac{1}{2}$ or $- \frac{1}{2}$. Moreover, let us assume this spin is in the state $| \psi _{ j} ( t_0) {\rangle }$ defined by (19) when performing such a measurement. Then, the probability that the measured value is equal to $+ \frac{1}{2}$ (respectively, $- \frac{1}{2}$) is equal to $| \alpha _{ j} | ^2$ (respectively $| \beta _{ j} | ^2$), i.e., to the squared modulus of the coefficient in (19) of the associated eigenvector $| + {\rangle }$ (respectively $| - {\rangle }$).

The above discussion concerns the state of the considered spin at a given initial time $t_0$. This state then evolves with time. The spin is here supposed to be placed in a static magnetic field and thus coupled to it. The time interval when it is considered is assumed to be short enough for the coupling between the spin and its environment to be negligible. In these conditions, the spin has a Hamiltonian [29]. Therefore, if the spin state $| \psi _{ j} ( t_0) {\rangle }$ at time $t_0$ is defined by (19), it then evolves according to Schrödinger’s equation and its value at any subsequent time t is

$$\begin{aligned} | \psi _{ j} (t) {\rangle } = \alpha _{j } e^{- i \omega _p ( t - t_0 ) } | + {\rangle } + \beta _{j } e^{- i\omega _m ( t - t_0 ) } | - {\rangle } \end{aligned}$$

(21)

where the real (angular) frequencies $\omega _p$ and $\omega _m$ depend on the considered physical setup and i is the imaginary unit.

Appendix B: Quantum source separation problem associated with the considered quantum channels

We here detail the quantum source separation problem associated with the “mixing model” (5)–(7) . That model involves the following items. The observations are the probabilities $p_1$, ${p_2}$ and ${p_4}$ measured for each choice of the initial states (1) of the qubits. More precisely, these probabilities are not known exactly but estimated in practice. The procedure that we used to this end, e.g., in [10, 13], and that is also widely employed in the QIP literature [3, 6], operates as follows for each choice of the initial states (1) of the qubits. We repeatedly perform two operations: (i) we first initialize these qubits according to (1) and (ii) after a fixed time interval when coupling occurs, we measure the two spin components along Oz associated with the system composed of these two coupled qubits. The relative frequencies of occurrence of all four possible couples of values of spin components (i.e., $\left( +\frac{1}{2},+\frac{1}{2}\right)$ to $\left( -\frac{1}{2},-\frac{1}{2}\right)$) then yield estimates of the corresponding probabilities. This approach therefore requires a large number (typically from a few thousand up to a few hundred thousand [10, 16]) of copies of the considered two-qubit state. At this stage, we ignore the resulting estimation errors and therefore consider the exact mixing model (5)–(7). Using standard BSS notations, the observation vector is therefore $x= [ x_{1}, x_{2}, x_{3}] ^{T}$ , where $^T$ stands for transpose and⁴

$$\begin{aligned} x_{1}= {p_1},\quad x_{2}= {p_2} , \quad x_{3}= {p_4} . \end{aligned}$$

(22)

Equations (5)–(7) show that the source vector to be retrieved from these observations turns out to be $s= [ s_{1}, s_{2}, s_{3}] ^{T}$ with $s_{1}= r_1, s_{2}= {r_2}$ and $s_{3}= \varDelta _I$. The parameters ${q_j}$ are then derived from (4). The four phase parameters in (3) cannot be individually extracted from their combination $\varDelta _I$ (anyway, only the phase differences $( \phi _j - \theta _j )$ have a physical meaning [18]). The transform from the sources to the observations defined by the nonlinear mixing model (5)–(7) involves a single “mixing parameter,” namely v. As shown by (10), this parameter always meets the condition $0 \le v^2 \le 1$.

Appendix C: Communications based on photons

Communications based on photons deserve the following comments. First considering the classical framework, everyday communications use electromagnetic waves propagating either in free space or in a solid medium, e.g., an optical fiber [26]. Such media are non-magnetic, and their electric properties are classically described by the induction vector $\overrightarrow{D}$ (H. Lorentz), representing a local mean of the microscopic electric vector [24]. $\overrightarrow{E }$ being the applied field, in vacuum, $\overrightarrow{D}=\varepsilon _{0} \overrightarrow{E}$ (SI units, $\varepsilon _{0}$: vacuum permittivity), and in a dielectric medium $\overrightarrow{D}=\varepsilon _{0}\overrightarrow{E} +\overrightarrow{P}.$ The polarization $\overrightarrow{P}$ may be seen as the response to the excitation $\overrightarrow{E}$, ferroelectrics, with a spontaneous polarization, being an exception. Most dielectric media are linear, i.e., $\overrightarrow{P}$ increases linearly with the excitation (description using a scalar or more generally a tensor not depending upon the excitation). Moreover, the appearance of the laser in 1960, i.e., of intense coherent electromagnetic sources, allowed the development of nonlinear optics. Turning now to the quantum behavior associated with these phenomena, one should again make a distinction between linear and nonlinear setups. One first thinks of an electromagnetic wave propagating in vacuum space (or possibly in a linear medium): Already in 1930 Dirac [21] considered a weak electromagnetic beam and its associated photons; a device separates this beam into two partial beams, which are then made to interfere. One could then a priori think that two distinct photons possibly interfere. But, in such conditions, according to the general principles of quantum mechanics “each photon then interferes only with itself. Interference between two different photons can never occur ”. With respect to photon-based quantum communications addressed in the present paper, this entails that, if only considering communications through vacuum space or a linear medium, no entanglement is created in the transmission channel itself. This should be contrasted with the scenarios considered above, where entanglement is created by the channel itself (here with exchange coupling), whereas the original two-qubit state associated with Emitter 1 and Emitter 2 is unentangled.

The above manifestation of the superposition principle for photons, or the presence of (previously prepared) entangled states when more than one photon are implied may also be found in dielectrics, but other quantum phenomena may also be found in some optically nonlinear dielectric materials: (1) two intense laser beams at frequencies $\omega _{1}$ and $\omega _{2}$ may allow fluorescence at $\omega _{1}+\omega _{2};$ there, two photons with respective frequencies $\omega _{1}$ and $\omega _{2}$ generate a photon with frequency $\omega _{1}+\omega _{2}\$ [4]. (2) spontaneous emission may sometimes allow emission at the difference frequency: The material receives a laser beam with frequency $\omega _{p}$ (p: pump), and emits at both $\omega$ (with a material dependent value) and $\omega _{p}-\omega$ (so-called optical parametric fluorescence); here, a photon with frequency $\omega _{p}$ generates two photons, with respective frequencies $\omega$ and $\omega _{p}-\omega$ [4]. Quantum communications take profit of the superposition principle, e.g. when involving two photons in an entangled state, and of the no-cloning theorem, both specific to quantum mechanics. Future quantum communication networks should make use of quantum teleportation—which allows transport of information, presently using entangled photon qubits—and of quantum repeaters interconnecting quantum nodes [23]. An eavesdropper, trying to access the information circulating within such a network could, e.g., try and operate by interacting with a repeater. Besides, one may imagine a scenario involving transmission through quantum channels, by means of photons, be they initially entangled or not, mainly with free propagation (linear medium), but now also with a nonlinear medium inserted (e.g., by an eavesdropper) in part of the overall transmission path forming what we called the “main channel” between Emitter 1 and Receiver 1 above. One might then investigate to which extent an “intruder,” composed of Emitter 2 and Receiver 2, would thus be able to interact with the main channel so as to extract information from it (eavesdropping), and to which extent Receiver 1 would be able to detect this intrusion. The relevance and attractiveness of this scenario need to be further investigated.