Past exam of the laser science and quantum information processing course of the University of Oxford / 2022 / 6

The two output ports are orthogonal single-photon states. Conservation of total probability at a lossless beam splitter gives $2c^2=1$. With $c$ real and positive,In the ordered input basis $(0,1^{\prime })$ and output basis $(0^{\prime },1)$, the beam splitter scattering matrix isFor the usual reciprocal lossless beam splitter, reversing propagation uses $S^{-1}=S^{†}=S$. Consequently,The relative phase in the first rule is essential to the repeated-path calculation. Overall port phases may be changed consistently; they cannot be changed on one pass alone while retaining the original scattering rules. In the following, take unit mirror reflection amplitudes by incorporating fixed reflection phases into the propagation-mode convention, and keep any unremoved propagation phase explicitly if needed.

Starting at port 1, the first beam splitter encounter sends amplitude $c$ straight out through port 0 and stores amplitude $-c$ towards M2. A return from M2 enters port $1^{\prime }$; it sends the stored amplitude times $-c$ out through port 1 and times $c$ towards M1. A return from M1 enters port $0^{\prime }$; it sends amplitude times $c$ out through port 0 and times $c$ towards M2. Both internal continuations carry $+c$, so two additional encounters multiply the surviving amplitude by $+c^2$.

With the fixed mirror and propagation phases absorbed as in a, the nonzero coefficients of the outgoing single-photon state areand $c_{0,2j}=c_{1,2j+1}=0$. The first nonzero terms are $c_{0,1}=1/\sqrt{2}$, $c_{1,2}=1/2$, $c_{0,3}=-1/(2\sqrt{2})$ and $c_{1,4}=1/4$. If fixed reflection amplitudes $r_1,r_2$ are retained instead, multiply $c_{0,2j+1}$ by $(r_1{r}_2{)}^j$ and $c_{1,2j}$ by $r_2^j{r}_1^{j-1}$, as well as the appropriate delay propagation phases. These unit-modulus factors do not alter the following probabilities.

Distinct exit events are orthogonal because they differ in output port or arrival-time bin. This requires the incident wavepacket to resolve the delay. Summing the squared moduli as geometric series givesTheir sum is one. After $n$ encounters the probability of still being inside is $2^{-n}$, which tends to zero, so no trapped component is omitted.

The source drawing places the delay line in the M2 arm. A visit to M2 crosses that line in both directions. If $\tau$ is its one-way delay, then an exit after $n$ encounters has crossed it $2\lfloor n/2\rfloor$ times:If the phrase “delay time $\tau$” instead denotes the complete M2-arm round-trip delay, replace $2\tau$ by $\tau$ in the floor formula. The wording does not explicitly distinguish these conventions. Consecutive encounters can produce the same exit time in different ports: for example $n=2,3$ both follow one M2 visit. If the incident wavepacket were much longer than the arm delay, different histories in the same port would overlap and their probability amplitudes would have to be added before squaring; the asserted time-bin expansion would then not be an orthonormal event basis.

Let $S$ denote immediate transmission through the BS1 subsystem between Bob's lower-right port and Alice's upper-left port; let $L$ denote transmission after one visit to each internal mirror. From the same beam splitter rules as in b, the corresponding amplitudes in either direction areup to a common fixed two-mirror propagation phase. The minus sign in $L$ comes from the $1^{\prime }\to 1$ or reverse $1\to 1^{\prime }$ scattering. The two paths selected by D0's time gate are $L$ on the outward journey followed by $S$ on the return, and $S$ outward followed by $L$ on the return. They have the same total delay and fixed propagation phase, so their probability amplitudes interfere.

For the first history, Alice applies $e^{i{\varphi }_A}$ to the late component, while Bob's return-time modulation is not encountered in the return delay arm. For the second, Alice's early component is unmodulated and Bob applies $e^{i{\varphi }_B}$ in the return delay arm. Bob must time his modulation to the return passage, leaving the outward reference preparation unchanged; this is implicit in the stated encoding. Each enabled modulator is traversed twice, so its $\pi /2$ single-pass phase becomes the stipulated $\pi$.

Include the amplitude $c$ to enter BS1 through BS2, and the amplitude of modulus $c$ to reach D0 through BS2 on return. Both histories therefore contain six 50:50 beam splitter factors. Ignoring one common unobservable global phase,Hence the unconditional probability per emitted photon, with no renormalization after discarding paths, isFor the allowed values $0,\pi$, this is $1/16$ when ${\varphi }_A={\varphi }_B$ and zero otherwise. The ignored trajectories exit other ports or other time bins; excluding them from D0's acceptance does not increase the success probability. Equal mirror reflection phases, channel phases and Alice's mirror phase multiply both accepted histories equally and cancel from the interference condition. A residual relative phase would need calibration.

Encode $0$ by phase zero and $1$ by phase $\pi$. Alice and Bob choose their bits independently and keep their choices private. Bob announces the trial numbers with a D0 click over an authenticated public channel. For each retained trial, c guarantees that their bits agree. In the ideal symmetric setup, the two equal-phase cases $00$ and $11$ give the same click probability, so announcing success does not itself reveal which shared bit was used.

The matching-phase cases occur with probability $1/2$, and a matching case succeeds with probability $1/16$. ThereforeFor independent ideal trials, the raw-key length has a binomial distribution with standard deviation $\sqrt{31N}/32$. These are bits before sacrificing test positions or performing privacy amplification.

A missing click does not certify unequal phases: the same phases also fail to click with probability $15/16$. With uniform choices,Thus a no-click trial supplies neither a known equal bit nor a known complemented bit. This is why it cannot be retained by the direct error-free sifting rule. The source's “cannot use” should be understood in this sense: the events are not statistically independent, so an absolute claim that they contain no information would be false.

To obtain a secret final key, reveal and discard a random subset of retained bits to test the observed error rate, then apply reconciliation and privacy amplification as required by a security analysis of the actual implementation. Resolving which time path a photon took removes the accepted histories' quantum coherence and can turn destructive interference into false D0 clicks, providing a simple disturbance test. Equal-phase click correlations alone are not a proof of security against every attack, particularly in the presence of losses and device imperfections. The supplied ideal model fixes the sifted rate $N/32$; it does not fix a positive secure final length without an adversary model and observed parameters.