Security of quantum key distribution with imperfect phase randomisation

The sequence of phases $\Phi_{1}...\Phi_{N}$ of Alice’s pulse train constitute a discrete-time stochastic process whose joint distribution can be represented by a probability density function (PDF) $f(\phi_{1}...\phi_{N})$. Our proof does not require a precise characterisation of this distribution; it requires just two pieces of knowledge, which we state as the following two assumptions:

(A1) The stochastic process $\Phi_{1}...\Phi_{N}$ has at most $l_{c}$ rounds of memory, for some finite and known $l_{c}$. That is, for all rounds $i$,

(A2) The conditional PDF of $\Phi_{i}$ given all other phases is lower bounded, i.e., for all $i$ and some known $0<q\leq 1$,

The equality in Eq. 2 follows from Eq. 1.

In addition, for simplicity, we consider that the phase randomisation process is not affected by the intensity modulation and bit-and-basis encoding processes; and for concreteness, we also consider that neither the latter processes nor Bob’s measurement setup suffer from imperfections. Precisely, we assume that (A3) Alice’s choice of intensities and the phase of her pulses are independent; (A4) Alice’s (bit-and-basis) encoding operations commute with the process that (imperfectly) randomises the phase of her pulses; (A5) Alice’s choice of bit, basis and intensity for round $i$ only affects the $i\textrm{-th}$ pulse; (A6) Alice’s encoding operations are characterised and identical for all rounds; (A7) the intensities of Alice’s pulses perfectly match her choices; and (A8) the efficiency of Bob’s measurement is independent of his basis choice. We note that previous works have investigated the security of QKD when some of these assumptions are not met [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42].

(1) For each round, Alice probabilistically selects a random intensity $\mu$ from a predetermined set and attempts to generate a PR-WCP of that intensity. Then, she selects a random bit $b$ and basis $\omega\in\{Z,X\}$, and applies an encoding operation $\hat{V}_{b_{\omega}}$ to her pulse, satisfying $\hat{V}_{b_{\omega}}^{\dagger}\hat{V}_{b_{\omega}}=\mathbb{I}$¹¹1We note that, in our description, $\hat{V}_{b_{\omega}}$ is an isometric operator, but not necessarily a unitary, which reflects the fact that the space after the encoding operation can be (and in fact typically is) larger than the original space. For example, in BB84 polarization encoding, ${\{}\hat{V}_{b_{\omega}}{\}}$ encodes a single mode of light in a prefixed input polarization into four possible outcome polarizations whose creation operators can be expressed as a linear function of those of horizontal and vertical polarizations. To compute the numerical results in Fig. 1, we have assumed that ${\{}\hat{V}_{b_{\omega}}{\}}$ are ideal BB84 $Z$- and $X$-basis encoding operators, such that, for any input coherent state $\ket{\alpha}$, $\hat{V}_{0_{Z}}\ket{\alpha}=\ket{\alpha}\ket{0}$, $\hat{V}_{1_{Z}}\ket{\alpha}=\ket{0}\ket{\alpha}$, $\hat{V}_{0_{X}}\ket{\alpha}=\ket*{\frac{\alpha}{\sqrt{2}}}\ket*{\frac{\alpha}{\sqrt{2}}}$ and $\hat{V}_{1_{X}}\ket{\alpha}=\ket*{\frac{\alpha}{\sqrt{2}}}\ket*{-\frac{\alpha}{\sqrt{2}}}$. However, we remark that our analysis is valid for any set of characterized encoding operators, regardless of the dimension of the encoding space..

In the security proof, we consider the following equivalent process for the state preparation: (1a) Alice generates $\ket{\sqrt{\nu}}^{\otimes N}$, where $\nu\geq\mu\,\,\forall\mu$; (1b) she applies an imperfect phase randomisation operation to the pulse train, obtaining

where $\hat{P}(\ket{\mkern 1.0mu\cdot\mkern 1.0mu})=\outerproduct{\mkern 1.0mu\cdot\mkern 1.0mu}{\mkern 1.0mu\cdot\mkern 1.0mu}$; (1c) she probabilistically selects all the intensities $\mu_{1},...,\mu_{N}$ and attenuates each pulse to match her selection; and (1d) she probabilistically makes all bit and basis choices $b_{\omega_{1}},...,b_{\omega_{N}}$, and applies $\hat{V}_{b_{\omega_{1}}}...\hat{V}_{b_{\omega_{N}}}$ to her pulse train. Note that, because of Assumptions (A3) and (A4), steps (1b), (1c) and (1d) commute.

(2) For each incoming signal, Bob chooses a random basis $Z$ or $X$, and measures the incoming pulse.

(3) Bob announces which rounds were detected and, for these rounds, both Alice and Bob reveal their basis choices, and Alice reveals her intensity choices. They define their sifted keys as the bit outcomes of the detected rounds in which both chose the $Z$ basis and Alice chose a certain signal intensity $\mu_{s}$. Also, they define the test rounds as the detected rounds in which Bob used the $X$ basis, and reveal their bit values for these rounds. Moreover, they assign each round $i$ to a group $w\in\{0,...,l_{c}\}$ according to the value $w=i\bmod(l_{c}+1)$. The $w\textrm{-th}$ sifted subkey is defined as the fraction of the sifted key belonging to group $w$.

(4) Alice and Bob sacrifice a small fraction of the $w\textrm{-th}$ sifted subkey to estimate its bit-error rate, and use the detection statistics of the $w\textrm{-group}$ test rounds to estimate its phase-error rate. Then, they perform error correction and privacy amplification independently for each subkey.

Let $\mathcal{G}_{w}\;(\mathcal{G}_{\overline{w}})$ be the set of rounds that belong (do not belong) to group $w$, let $\vec{\phi}_{\mathcal{G}_{w}}$ ($\vec{\phi}_{\mathcal{G}_{\overline{w}}}$) be a particular joint value for all phases in $\mathcal{G}_{w}$ ($\mathcal{G}_{\overline{w}}$), let $f(\vec{\phi}_{\mathcal{G}_{\overline{w}}})$ be the joint marginal PDF of the phases in $\mathcal{G}_{\overline{w}}$, and let $f(\vec{\phi}_{\mathcal{G}_{w}}|\vec{\phi}_{\mathcal{G}_{\overline{w}}})$ be the joint conditional PDF of the phases in $\mathcal{G}_{w}$ given $\vec{\phi}_{\mathcal{G}_{\overline{w}}}$.

After the chain of equivalences (E1)-(E4) below, the actual protocol is reduced to the $w\textrm{-th}$ alternative scenario, in which Alice’s source is characterised and uncorrelated within the rounds in $\mathcal{G}_{w}$. For the first equivalence, note that, due to Assumption (A1), the phases in $\mathcal{G}_{w}$ are conditionally independent of each other given knowledge of the phases in $\mathcal{G}_{\overline{w}}$, i.e.,

as shown in Appendix A.

(E1) Let us assume that Alice performs step (1b) in the following way. First, she chooses $\vec{\phi}_{\mathcal{G}_{\overline{w}}}$ according to the marginal PDF $f(\vec{\phi}_{\mathcal{G}_{\overline{w}}})$. Then, for each round $i$, (a) if $i\in\mathcal{G}_{\overline{w}}$, she shifts the phase of the pulse by her selected fixed value $\phi_{i}\in\vec{\phi}_{\mathcal{G}_{\overline{w}}}$; (b) if $i\in\mathcal{G}_{w}$, she shifts the phase according to the conditional PDF $f(\phi_{i}|\vec{\phi}_{\mathcal{G}_{\overline{w}}})$.

Conditioned on a specific value $\vec{\phi}_{\mathcal{G}_{\overline{w}}}$, the state generated by Alice is

and due to Eq. 4, the overall generated state is

For the next equivalence, note that Alice could attenuate her pulses before applying the phase shifts above, rather than afterwards. Also, for all $i\in\mathcal{G}_{w}$,

due to Assumptions (A1) and (A2). As a consequence, instead of shifting the $i$-th phase according to the PDF $f(\phi_{i}|\vec{\phi}_{\mathcal{G}_{\overline{w}}})$ when $i\in\mathcal{G}_{w}$, Alice could have equivalently done the following [26, 27]: to flip a biased coin $C_{i}$ such that $C_{i}=0$ with probability $q$, and (a) if $C_{i}=0$, shift the phase by a uniformly random value, (b) if $C_{i}=1$, shift it according to the PDF

The equivalence is due to

where $f(\phi_{i}|\vec{\phi}_{\mathcal{G}_{\overline{w}}},C_{i}=0)=1/2\pi$.

(E2) Instead of steps (1a) to (1c), for each round $i$, Alice probabilistically chooses an intensity $\mu$, and (a) if $i\in\mathcal{G}_{\overline{w}}$, Alice prepares $\ket*{\sqrt{\mu}}$; (b) if $i\in\mathcal{G}_{w}$, Alice prepares

where $\rho_{\rm PR}^{\mu}$ is a perfect PR-WCP of intensity $\mu$. Then, Alice chooses $\vec{\phi}_{\mathcal{G}_{\overline{w}}}$ according to the PDF $f(\vec{\phi}_{\mathcal{G}_{\overline{w}}})$ and, for each round $i$, (a) if $i\in\mathcal{G}_{\overline{w}}$, she shifts the phase by her selected fixed value $\phi_{i}\in\vec{\phi}_{\mathcal{G}_{\overline{w}}}$; (b) if $i\in\mathcal{G}_{w}$, she shifts the phase according to the PDF $f(\phi_{i}|\vec{\phi}_{\mathcal{G}_{\overline{w}}},C_{i}=1)$ in Eq. 8.

Clearly, the rounds in $\mathcal{G}_{\overline{w}}$ are identical in both (E1) and (E2). The rounds in $\mathcal{G}_{w}$ are also identical. Alice’s phase shift does not affect the $\rho_{\rm PR}^{\mu}$ term in Eq. 10, and it causes the $\outerproduct*{\sqrt{\mu}}{\sqrt{\mu}}$ term to acquire the phase distribution in Eq. 8. Thus, the overall phase distribution of the pulse after the shift is $f(\phi_{i}|\vec{\phi}_{\mathcal{G}_{\overline{w}}})$, due to Eq. 9. We can represent Alice’s probabilistic selection of $\vec{\phi}_{\mathcal{G}_{\overline{w}}}$ together with all the phase shifts described above as a single global quantum operation $\mathcal{E}_{w}$.

(E3) Same as (E2), but Alice applies her encoding operations $\hat{V}_{b_{\omega_{1}}}...\hat{V}_{b_{\omega_{N}}}$ before $\mathcal{E}_{w}$, rather than afterwards, which is possible thanks to Assumption (A4).

(E4) Since $\mathcal{E}_{w}$ is now the last operation before the quantum channel, we consider that Alice does not actually apply it. Eve may or may not apply $\mathcal{E}_{w}$ as part of her attack, putting her in a position that is never less advantageous than in the previous scenarios. Thus, if the $w\textrm{-th}$ subkey is secure in (E4), it is also secure in the actual protocol. We refer to (E4) as the $w\textrm{-th}$ alternative scenario.

As a consequence of the reduction above, when proving the security of the $w\textrm{-th}$ subkey, we can assume that, in the $w\textrm{-group}$ rounds, Alice generates the characterised and uncorrelated states $\{\rho_{\rm model}^{\mu}\}_{\mu}$. Thanks to this, it becomes straightforward to prove its security using numerical methods. In particular, flexible techniques based on semidefinite programming (SDP) have been recently proposed [44, 45, 46, 47, 48, 49, 50, 27], which can handle almost any scenario, as long as the emitted states are characterised and uncorrelated, making them well suited to our purpose. The specific approach that we have developed uses ideas from these works but is targeted to this particular scenario. Below, we provide an overview of the main ideas, and refer the reader to Appendix B for a detailed description.

Each of Alice’s generated states $\{\rho_{\rm model}^{\mu}\}_{\mu}$ can be diagonalised as

where we have omitted the dependence of the eigenvalues and eigenstates on $q$ for notational simplicity. Each set of eigenstates $\{\ket{\lambda_{n}^{\mu}}\}_{n}$ forms an orthonormal basis of the Fock space, and can be regarded as imperfect versions of the Fock states $\{\ket{n}\}_{n}$, with the two sets of states converging as $q\to 1$. Similarly, the eigenvalues $\{p_{\lambda_{n}|\mu}\}_{n}$ approach a Poisson distribution when $q\to 1$. Note that, when $q\neq 1$, the states $\{\ket{\lambda_{n}^{\mu}}\}_{n}$ depend slightly on the intensity setting $\mu$, and therefore the standard decoy-state method cannot be applied to this scenario. However, we can still assume a counterfactual scenario in which Alice holds the ancillary system that purifies $\rho_{\rm model}^{\mu}$ and measures it to learn the value of $n$ for each round. The information leakage of the $w\textrm{-th}$ sifted subkey can then be determined by estimating the fraction $q_{\lambda_{1},w}$ of its bits that originated from emissions of $\ket{\lambda_{1}^{\mu_{s}}}$, and the phase-error rate $e^{\lambda_{1},\mu_{s}}_{\textrm{ph},w}$ of these bits, as shown in Appendix C. The first can be expressed as

where $Y_{\lambda_{1},\mu_{s}}^{Z,w}$ is the yield probability of $\ket*{\lambda_{1}^{\mu_{s}}}$ when encoded in the $Z$ basis, which needs to be estimated, and $Q_{\mu_{s},w}^{Z}$ is the observed rate at which Bob obtains detections conditioned on Alice choosing the intensity $\mu_{s}$, both users choosing the $Z$ basis, and the round being in $\mathcal{G}_{w}$. On the other hand, to define the phase-error rate, we consider that, in the rounds in which both users choose the $Z$ basis and Alice prepares $\ket{\lambda_{1}^{\mu_{s}}}$, she actually generates the entangled state

and performs an $X$-basis measurement on system $A$ [51]. Equivalently, she emits

with probability $p_{\textrm{vir}\beta}=\norm*{\ket*{\tilde{\lambda}_{\textrm{vir}\beta}}}^{2}$, where $\beta\in\{0,1\}$ and $\ket{\beta_{X}}=(\ket{0_{Z}}+(-1)^{\beta}\ket{1_{Z}})/\sqrt{2}$. Also, we assume that Bob replaces his $Z$-basis measurement by an $X$-basis measurement, which is allowed due to the basis-independent detection efficiency assumption, (A8). The phase-error rate is then given by

where $Y_{\textrm{vir}\beta}^{(\beta\oplus 1)_{X}}$ is the probability that Bob obtains the measurement outcome $(\beta\oplus 1)_{X}$ conditioned on Alice emitting $\ket*{\lambda_{\textrm{vir}\beta}}$.

In Appendix B, we show how to obtain a lower bound $Y_{\lambda_{1},\mu_{s}}^{Z,w,\textrm{L}}$ and an upper bound $e^{\lambda_{1},\mu_{s},\textrm{U}}_{\textrm{ph},w}$ on $Y_{\lambda_{1},\mu_{s}}^{Z,w}$ and $e^{\lambda_{1},\mu_{s}}_{\textrm{ph},w}$, respectively, using SDP techniques. In doing so, the main hurdle to overcome is the fact that the states $\{\rho_{\rm model}^{\mu}\}_{\mu}$ are infinite-dimensional, preventing us from finding their exact eigendecompositions using numerical methods, and from constructing finite-dimensional SDPs using these states. Instead, we construct the SDPs using the finite projections of $\{\rho_{\rm model}^{\mu}\}_{\mu}$ onto the subspace with up to $M$ photons [26, 27], after numerically obtaining the eigendecompositions

where $\Pi_{M}=\sum_{n=0}^{M}\outerproduct{n}{n}$. Then, by bounding the deviation between the eigenvalues and eigenvectors of $\rho_{\rm model}^{\mu}$ and $\Pi_{M}\rho_{\rm model}^{\mu}\Pi_{M}$ using perturbation theory results, we can correct the SDP constraints and solutions, ensuring that the final bounds $Y_{\lambda_{1},\mu_{s}}^{Z,w,\textrm{L}}$ and $e^{\lambda_{1},\textrm{U}}_{\textrm{ph},w}$ apply to the original infinite-dimensional scenario. The secret-key rate obtainable per emitted $w\textrm{-group}$ pulse is then given by

where $E_{\mu_{s},w}^{Z}$ is the bit-error rate of the $w\textrm{-th}$ sifted subkey, $\epsilon_{\rm val}^{\mu_{s}}=2\sqrt{1-\textrm{Tr}[\Pi_{M}\rho_{\rm model}^{\mu_{s}}\Pi_{M}]}$ is a correction term due to the finite projection, $h(x)$ is the binary entropy function, $f$ is the error correction inefficiency, and the rest of terms have already been defined.