Efficient Distribution of High-Dimensional through 11 km

This document provides supplementary information to "Efficient distribution of high-dimensional entanglement through 11 km fiber,"


BIPARTITE WITNESS FOR D-DIMENSIONAL ENTAN-GLEMENT
We performed quantum tomography and obtained the fidelity F after distribution. We then employed the method developed in previous work [1] to certify the 4-dimensional entanglement. In the following text, we provide a maximal overlap between the chosen high-dimensional state and states with a bounded Schmidt rank d. If the fidelity reveals a higher overlap than this bound, the justification of at least (d + 1)-dimensional entanglement is demonstrated.
The Schmidt decomposition of high dimensional state is described as |ϕ = ∑ d i=1 λ i |ii , with the coefficients in decreasing order |λ 1 | ≥ |λ 2 | ≥ · · · ≥ |λ d |. The witness for the ddimensional entanglement is constructed by comparing the two fidelities, F = Tr(ρ|ϕ ϕ|), where ρ is the density matrix after distribution and |φ d = ∑ d m,n=1 α mn |mn represents states with a bounded Schmidt rank d. A global search to maximize the F d convinced us that F > F d could not be satisfied by a d-dimensional entangled state. In other word, the generated bipartite system is entangled at least (d + 1)-dimension. We rewrite the maximal overlap as We next introduce two operators of the form where P d is a rank d-projector which always exists if B * of rank d, as |φ d is also of Schmidt rank d. Combining these equations, we have For the inner product A, B ≡ Tr(AB † ) taking advantage of Because Tr(BB † ) = ∑ D m,n=1 c mn c * nm ≤ 1 and choosing P d = ∑ d i=1 |i i|, we get the upper bound of F d for d-dimensional entangled states with a simple formula (S6) Thus we find a tight bound for witness of (d + 1)-dimensional entanglement For d = 4, this bound is found to be 3/4.

BELL AND STEERING INEQUALITIES
We used the CGLMP Bell inequality in [2] to test the non-locality of d=2, 3, 4 entangled states. Here, we show more details of the scenario, in which, we focus on a particular instance of two measurements per observer, A 1 or A 2 , and B 1 or B 2 . A state ρ is shared by Alice and Bob, who perform one of two measurements with equal probability, obtaining The correlations are joint probabilities {p(ab|xy)} a,b ≡ {p(A x = a, B y = b)}. These joint probabilities mean that Alice and Bob obtain outcomes a and b separately. They can also be expressed by using Born rule as {p(ab|xy)} a,b = Tr[ρ(M a|x ⊗ M b|y )].
The Bell inequalities we used to test nonlocality of the highdimensional entangled state are: in which The bases |a x , |b y used by Alice and Bob is where α 1 = 0, α 2 = 1/2, β 1 = 1/4, and β 2 = −1/4. To verify steering nonlocality, we introduce two sets of ddimensional mutually unbiased measurement bases for the two local subsystems: The steering parameter can be expressed as where i + j . = 0 under the second sum sign denotes equality modulo d, i.e., i + j = 0 or i + j = d, and P A i , B j , denotes the probability that Alice is holding particle in state A i , and in the meantime, Bob's state is B j , and the same to P L A i , L B j . For all unsteerable states, their results would be limited at S Therefore, if Alice could demonstrate a violation of this inequality, Bob has to admit her steerability. , we can complete the measurement of (|0 + e iϕ 1 |1 + e iϕ 3 (|2 + e iϕ 2 |3 ))/2.

CONSTRUCTION OF MEASUREMENT BASES
Here we provide the details of the measurement setup. Let us take a four-dimensional measurement as an example. The measurement bases for the four-dimensional CGLMP inequality is as follows: |2 + e −7πi/8 |3 )/2. In our experimental scheme, we use a hybrid of polarization and path coding. The angles of HWP1, HWP2, HWP3 are set at 22.5 • (Fig. S1). The function of liquid crystal (LC) is to load a phase ϕ between H-polarized and V-polarized photon. LC1, LC2 and LC3 are loaded ϕ 1 , ϕ 2 and ϕ 3 between H-polarized and V-polarized photon, respectively. The (|0 + e iϕ 1 |1 + e iϕ 3 (|2 + e iϕ 2 |3 ))/2 measurement basis can be constructed using our setup (Fig. S1). For high-dimensional QKD and CGLMP inequalities, multiple outcome projection measurements are generally required. In this section, we give the scheme of multiple outcome measurement bases for high-dimensional QKD and CGLMP inequalities.

MULTIPLE OUTCOME MEASUREMENT BASES FOR
Computational (|0 , |1 , ..., |d − 1 ) and Fourier bases (∑ d−1 k=0 e i 2π d |i / √ d) are needed for high-dimensional QKD. We need to set LCs at some fixed angles as shown in Fig. S2. When LC is loaded with zero voltage, LC does not affect the photon's polarization. At this time, the measurement of the computational basis is completed. When LC is loaded with half-wave voltage, the LC's effect is equivalent to an HWP, and the measurement of the Fourier basis is completed. By controlling the voltage of LCs, we can realize the fast switch of these two bases.
For CGLMP inequality, its measurement basis is a set of highdimensional MUB (Eq. S9). As shown in Fig. S3, we first construct a set of fixed Fourier bases using HWPs, QWPs, BDs, and PBSs. After that, by adjusting the phase of each dimension through LCs, the switch of other MUBs can be completed.

MULTIDIMENSIONAL QUANTUM KEY DISTRIBUTION
In our experiment we use a generalized entanglement-based version of the BB84 protocol for high dimensional systems, initially proposed and analyzed in Ref. [3], and test it for different dimensions (d = 2, 3, 4). In Table S1, we report experimental correlations data for high-dimensional mutually unbiased bases required for QKD (computational and Fourier bases). The final secret key rate (per coincidence) can be obtained as [4]: where H d (e) represents the d-dimensional Shannon entropy, given as a function of the fildelity F QKD and the dimension d by In Table S1, we report the maximal values of the QBER, given by the infidelity Q BER = 1 − F QKD , required to achieve a positive key rate in the cases where Eve is allowed coherent attacks QBER T Co h h . It can be observed that in general higher dimensionality corresponds to higher tolerance to noise and higher photon information efficiency.

PHASE STABILITY AND FIBER LOCKING SYSTEM
In our experiment, we use 11 km multicore fiber ( MCF) to distribute path-polarization entanglement. There were two challenges, one is the polarization preservation through longdistance fibers, the other is the phase stability between different cores. Firstly, we lay all the fibers including fan-in and fan-out on the optical table. We place an HWP in each export of the fanin/fan-out to maintain H-and V-polarization, and a tilt quarterwave plate to compensate for the differential birefringence of Table S1. Security analysis for a device-dependent QKD (BB84-type) in different dimensions. Reported are the experimentally measured fidelity F QKD , the associated experimental QBER Exp , the theoretical maximal QBER bounds QBER Coh Th for coherent attacks, experimental secure key rate R Exp and theoretical secure key rate bound R Bound from dimension 2 to 4.  These results are worse than those using single-mode fibers in a length of a few meters due to the polarization mode dispersion and group velocity dispersion. For an outdoor environment, an active feedback system is feasible to compensate for the polarization mode dispersion [5]. Then, we use a fiber locking system (FLS) to lock the relative phase between core1 and core2 used in our experiment. Fig. S5 shows the phase drift between the two cores without active feedback. Different from Ref [6], where the phase drift is very slow in 300 m MCFs (the time scale of 2π phase change is in an order of hours without active feedback), the phase drift in our experiment is terrible without feedback. The difference is caused by the coupling methods. In our experiment, we use packaged fan-in and fan-out where the spatially separated fiber pigtails are approximate 10 m in length, and the phase drift between different paths is increased dramatically. Combined with 11 km MCF, the time scale of 2π phase change between different paths is in an order of minutes without active feedback as shown in Fig. S5. As shown in Fig. S6, the system is stable after the active feedback system (FLS given in Fig. 1, the frequency of the feedback system is 5 Hz) is turned on. To test the feedback system, we measured the visibility between different cores (core1-core2 and core1-core3), and the average visibility of the two cases in one hour is V = 0.915 ± 0.001, and 0.907 ± 0.001. These results prove that it is feasible to use more cores to distribute high-dimensional path entanglement. The brightness of reference light used in the locking system is 10 6 photons/s. . Phase drift between core1 and core2 without active feedback. Due to the influence of fan-in and fan-out pigtails, the phase drift is faster than that of the multicore fiber itself [6]. Here, P = N Detection /N Total refers to the proportion of photons detected in one arm of the interferometer. To reduce the fluctuation, we use a total photon rate of 10 6 photons/s.