N-dimensional measurement-device-independent quantum key distribution with N + 1 un-characterized sources: zero quantum-bit-error-rate case

We study N-dimensional measurement-device-independent quantum-key-distribution protocol where one checking state is used. Only assuming that the checking state is a superposition of other N sources, we show that the protocol is secure in zero quantum-bit-error-rate case, suggesting possibility of the protocol. The method may be applied in other quantum information processing.


Results
For the protocol, each user prepares N encoding states. Let the states prepared by Alice and Bob denoted by |ϕ m 〉 and ϕ ′ m , respectively, where m = 0, 1, 2, …, N − 1. Here nothing is assumed for the encoding states so they are completely un-characterized. Each user also prepares a checking state which is assumed to be a superposition of the encoding states. Alice's and Bob's checking states are, respectively, , respectively, and θ m and θ ′ m are real. The protocol is as follows.
(1) Alice generates a random number i where i = 0, 1, 2, …, N. She sends a state |ϕ i 〉 to Charlie. Here Charlie can be anyone. So Charlie can be either Eve or users themselves. (2) Bob independently generates a random number j where j = 0, 1, 2, …, N. He sends a state ϕ | ′〉 j to Charlie. (3) Charlie performs a measurement on set of the states |ϕ i 〉 and ϕ | ′〉 j . The measurement can be any one which finally gives two outcomes 0 and 1. Charlie announces the outcome. (4) When the outcome is 0, users discard the data. Otherwise, they keep the data. By sacrificing some of the data for public discussion, users estimate, p(1|ij) ≡ p ij , conditional probability to get outcome 1 for each i, j. (5) For each measurement, if both i and j are less than N, the i and j become raw key. Otherwise, the data are used only for checking purposes. Then users do post-processing to get final key. Now let us consider Eve's (Charlie's) measurement on the states |ϕ i 〉 and ϕ | ′〉 j . In the most general collective attack, Eve attaches an ancilla |e〉 to the states and then applies a unitary operation to them 17 Eve gets the outcome by measuring the quantum state indexed by M in basis of |0〉 and |1〉. Now let us consider the attack from Eve's viewpoint. Clearly she can get no information about key from the data with outcome 0 which are not used by the users. Thus she analyze the states for outcome 1, |Γ ij1 〉's. For convenience, let us omit 1, |Γ ij1 〉 ≡ |Γ ij 〉. We can see that Eqs (1) and (2) give constraints

n g t h i s w i t h t h e t h i r d o f E q . ( 3 ) , w e o b t a i n
Eq. (6) means that all |Γ mm 〉's are essentially identical and that righthandside terms in Eqs (4) and (5) are zero, implying the security. Moreover, combining = ′ = c c N 1/ m n , Eq. (1), and normalization condition, we obtain that all |ϕ m 〉's are orthogonal with one another and the same property holds for ϕ ′ m 's. We can also see that the states |ϕ N 〉 and ϕ ′ N are expected ones. Let us consider another set of conditional probabilites p mn = δ mn and p iN = p Nj = p NN = 1/N. This corresponds to a case when Eve performs measurement in the encoding bases on each quantum states received, and announces 1 (0) when the same (different) outcomes are obtained. Analogously we get = ′ = . Combined by normalization condition, we get that all |Γ mm 〉's are orthogonal each other. The righthandside terms in Eqs (4) and (5) are N and there is no security clearly.

Discussion and Conclusion
In principle, the method is applicable to other set of conditional probabilities physically realizable. Within Eq. (3) with given conditional probabilities, optimize the bounds in Eqs (4) and (5). However, it does not seem to be feasible because of its complexity. Robustness of the method can be shown by the fact that functions involved here are all continuous. If the set of conditional probabilities are arbitrarily close to the ones discussed above, the bounds are also arbitrarily close to the given ones. Only with the assumption about dimensionality, security was obtained. It seems to be worthwhile to search for application of the method in other tasks in quantum information processing.
Good candidates for real implementation of the N-dimensional states seems to be time-bins of single photons which are adapted in the phase-reference-free MDI QKD 15,19 and round-robin-differential-phase-shift QKD [20][21][22] .
Here the checking state can be made by opening optical switches such that all time-bins have non-zero possibility to contain a photon. Also spatial-bins may be a good candidate 23 .
In conclusion, we studied N-dimensional MDI QKD where one checking state is used. With the assumption about dimensionality, Eq. (1), we showed that the protocol is secure in zero QBER case. In the case when Eve's does full measurement attack, the method also works. This suggests possibility of the protocol.