Finite-Key Analysis for Quantum Key Distribution with Discrete-Phase Randomization

In this section, we show how to obtain the relation between the hypothetical phase-error events and some experimental observations.

The main result obtained here is that each key bit is a successful click from a mixed state of ${\tau }_{j|2\mu },j=0,1,...,M-1$ prepared by Alice and Bob. More importantly, the number of phase-error events among these key bits corresponds to ${\tau }_{j|2\mu },j=0,2,...,M-2$. Therefore, if we denote the length of the raw key by $n_{bit}=n_{2\mu }$ and the number of successful clicks of ${\tau }_{j|2\mu }$ by $n_{j|2\mu }$, $n_{bit}=n_{2\mu }={\sum }_{j=0}^{M-1}{n}_{j|2\mu }$, the number of phase-error events $n_{ph}={\sum }_{j=0,j\in {\mathcal{N}}_0}^{M-2}{n}_{j|2\mu }$ must hold, where ${\mathcal{N}}_0$ is the set of even integers. Next, a proof is present to show how to obtain this result.

Following the symbols in [49], let us consider the evolution of the gigantic quantum state $|\mathsf{\Phi }\rangle ={|\varphi \rangle }^{\otimes N_{tot}}={(|\psi \rangle }_{A_s{A}_{c}Aa}\otimes {|\psi \rangle }_{B_s{B}_{c}Bb}{)}^{\otimes N_{tot}}$ sent to Eve. After Step 2, where Eve performs her measurement on the subsystem $ab$, the initial quantum state is transformed to ${\widehat{M}}_{eve}|\mathsf{\Phi }\rangle$, where ${\widehat{M}}_{eve}$ denotes the measurement operator of Eve. After measurement, Eve announces whether the measurement outcome is successful or not for each round. Hence, we reorder the quantum state as $|\mathsf{\Phi }\rangle ={|\varphi \rangle }^{\otimes M}{|\varphi \rangle }^{\otimes \overline{M}}$, where $M\left(\overline{M}\right)$ denotes the successful (unsuccessful) rounds. Then, in Step 3 of the virtual protocol, using measurement operators $\{{\widehat{O}}_s={(|00\rangle }_{A_s{B}_s}\langle 00|+{|11\rangle }_{A_s{B}_s}\langle 11|+{|22\rangle }_{A_s{B}_s}\langle 22|)\otimes {\sum }_{l=0}^{M-1}{(|l,l\rangle }_{A_c{B}_c}\langle l,l|+{|l,(l+M/2)modM\rangle }_{A_c{B}_c}\langle l,(l+M/2)modM|),{\widehat{O}}_d=\widehat{I}-{\widehat{O}}_s\}$, Alice and Bob measure the subsystem $A_s{A}_c$ and $B_s{B}_c$ for those rounds that are announced successful in Step 2 and retain the trials in which $A_s{A}_c$ and $B_s{B}_c$ are collapsed into ${\widehat{O}}_s$ as the final successful rounds. Hence, we reorder $|\mathsf{\Phi }\rangle ={|\varphi \rangle }^{\otimes M_s}{|\varphi \rangle }^{\otimes M_f}{|\varphi \rangle }^{\otimes \overline{M}}$ where $M_s$($M_f$) denotes the successful(unsuccessful) rounds finally. Before they measure the subsystem $AB$ to generate their sifted key in Step 3, the unnormalized quantum state is given by

Next, in Step 4, Alice and Bob measure the subsystem $A_s,B_s$$A_c,B_c$ and $A,B$ for all rounds in ${\mathcal{M}}_s$, one by one. We use $\alpha \in \{1,\cdots ,M_s\}$ to denote the different rounds in ${\mathcal{M}}_s$ and ${\xi }_{\alpha }$ to denote the measurement outcome of the $\alpha$-th subsystem. What is more, ${\widehat{M}}_{\alpha }$ is used to denote the associated operator. Hence, the unnormalized state before the measurement of the $\alpha$-th rounds in ${\mathcal{M}}_s$ is because we are only interested in the reduced state of the $\alpha$-th round in ${\mathcal{M}}_s$, we trace out the other rounds which we denote by $\overline{\alpha }$ and obtain where and the quantum states $\{|\overrightarrow{\overline{\alpha }}\rangle \}$ represent the basis for the subsystems $A_s,B_s,A_c,B_c,A,B,a,b$ of all rounds in the protocol except the $\alpha$-th round in ${\mathcal{M}}_s$.

Next, to derive the number of phase error, we expand the quantum state ${\widehat{O}}_s|\varphi \rangle$ as where and To summarize, each key bit can be viewed as an event in which Eve announces a successful click conditioned by Alice and prepares ${|\varphi \rangle }_{\mu }$ and measures $AB$ with Z-basis. Since the measurement on $AB$ made by Alice and Bob can be delayed after Eve’s announcement of a successful click, the phase error can be estimated by Alice and Bob measuring $AB$ with X-basis rather than Z-basis. To obtain the phase error of this part, we rewrite ${|\varphi \rangle }_{\mu }$ under X-bases of $AB$ as where and ${\theta }_l=\frac{l}{M}2\pi$. For the purpose of clarification, we define some quantum states below: and where ${|j\rangle }_{ab}={\sum }_{i=0}^j\sqrt{\frac{j!}{2^{j}i!(j-i)!}}{|i\rangle }_a{|j-i\rangle }_b$ and ${\mathcal{N}}_0$ is the set of even numbers. Indeed, $|j\rangle$ is a quantum state satisfying that the total photon-number of a and b is j. Moreover, another similar quantum state is defined below: and where ${|j\rangle }_{ab,even}^{\prime }={\sum }_{i=0}^j{(-1)}^i\sqrt{\frac{j!}{2^{j}i!(j-i)!}}{|i\rangle }_a{|j-i\rangle }_b$. With these definitions, we can write the quantum states ${|\psi \rangle }_{\mu ,AaBb}^l$ and $|{\psi }^{\prime }{\rangle }_{\mu ,AaBb}^l$ in a more simplified way, namely and obviously, the measurement outcome of ${|++\rangle }_{AB}$ and ${|--\rangle }_{AB}$ can be defined as phase-error event. Recall the whole density matrix ${|\varphi \rangle }_{\mu }\langle \varphi |$ given in Equation (A8); we can obtain the $ab$ part corresponding to the phase-error event, according to Equations (2.5)–(2.7) of Ref [48], ${\widehat{\rho }}_{\mu ,ph}$ can be rewritten as where ${\tilde{P}}_{j|2\mu }={\sum }_{n=0}^{\infty }{P}_{j+Mn|2\mu }$, $P_{j|2\mu }$ is the the probability of finding j photons in a Poisson source with mean photon-number $2\mu$, $|{\tilde{j}}_{2\mu }\rangle ={\sum }_{n=0}^{\infty }\frac{\sqrt{P_{j+Mn|2\mu }}}{\sqrt{{\tilde{P}}_{j|2\mu }}}{|j+Mn\rangle }_{ab}$ and $|{\tilde{j}}_{2\mu }{\rangle }^{\prime }={\sum }_{n=0}^{\infty }\frac{\sqrt{P_{j+Mn|2\mu }}}{\sqrt{{\tilde{P}}_{j|2\mu }}}{|j+Mn\rangle }_{ab}^{\prime }$, and ${\tau }_{j|2\mu }=\frac{1}{2}|{\tilde{j}}_{2\mu }{\rangle }_{ab}\langle {\tilde{j}}_{2\mu }|+\frac{1}{2}|{\tilde{j}}_{2\mu }{\rangle }_{ab}^{\prime }\langle {\tilde{j}}_{2\mu }|$.

For ease of understanding, we can easily interpret the formula of ${\widehat{\rho }}_{\mu ,ph}$. It is easy to see that ${\widehat{\rho }}_{\mu ,ph}$ is a mixture of ${\tau }_{j|2\mu }$, which consists of photon-number state ${|j+Mn\rangle }_{ab},n=0,1,2,...$, and the probability of finding $j+Mn$ photons is proportional to $P_{j+Mn|2\mu }$. Let ${\tau }_{even|2\mu }$ be the normalized ${\widehat{\rho }}_{\mu ,ph}$; then, a phase-error event for a key bit is equivalent to a successful click announced by Eve on the condition that a mixture ${\tau }_{even|2\mu }={\sum }_{j=0,j\in {\mathcal{N}}_0}^{M-2}{\tilde{P}}_{j|2\mu }{\tau }_{j|2\mu }/$$P_{even|2\mu }$ prepared by Alice and Bob, and the probability of preparing such a mixture is obviously $P_{\mu }^2{P}_{even|2\mu }2/M$.

To find a way to estimate the number of phase errors, we can give the density matrices Alice and Bob prepared in code mode and decoy mode. If we trace out $AB{A}_c{B}_c$ of the quantum state ${|\varphi \rangle }_{\mu }\langle \varphi |$, i.e., regardless of whether the measurement outcome on $AB$ is phase error or not, we have we define ${\tau }_{2\mu }={\sum }_{j=0}^{M-1}{\tilde{P}}_{j|2\mu }{\tau }_{j|2\mu }$ as the normalized ${\widehat{\rho }}_{\mu }$. The generation of a key bit is equivalent to a successful click announced by Eve conditioned on the premise that a mixture ${\tau }_{2\mu }={\sum }_{j=0}^{M-1}{\tilde{P}}_{j|2\mu }{\tau }_{j|2\mu }$ is prepared by Alice and Bob, and the probability of preparing such a mixture is obviously $P_{\mu }^{2}2/M$.

Now we are approaching a main result of above derivations. In the $N_{tot}$ rounds, Alice and Bob prepare ${\tau }_{2\mu }$ with the probability $P_{\mu }^{2}2/M$; the number of its successful rounds is denoted by $n_{2\mu }$. Of course, the key bits are generated in these rounds, thus $n_{2\mu }=n_{bit}$. Since ${\tau }_{2\mu }$ is a mixture of ${\tau }_{j|2\mu },j=0,1...,,M-1$, the $n_{2\mu }$ successful clicks are the sum of clicks by ${\tau }_{j|2\mu },j=0,1,...,M-1$. Then, $n_{2\mu }={\sum }_{j=0}^{M-1}{n}_{j|2\mu }$ holds evidently, in which $n_{j|2\mu }$ is the number of successful clicks by ${\tau }_{j|2\mu }$. Recall that the phase errors for these events are from clicks by ${\tau }_{even|2\mu }$; one can assert that the number of phase-error events $n_{ph}={\sum }_{j=0,j\in {\mathcal{N}}_0}^{M-2}{n}_{j|2\mu }$ must hold. This is a main result we have so far.