Research ArticleQuantum Mechanics

High-dimensional quantum cloning and applications to quantum hacking

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Science Advances  03 Feb 2017:
Vol. 3, no. 2, e1601915
DOI: 10.1126/sciadv.1601915
  • Fig. 1 Simplified sketch of the experimental design.

    The input quantum state |ψ〉 is imprinted on a single photon using an SLM-A. The single photon is subsequently sent to the cloning machine for optimal cloning. The cloning machine consists of a delay line (DL), to adjust the arrival time of the input photon, a second photon that is in a completely mixed state when exiting SLM-B, and a first beam splitter (BS1). The two photons are made to arrive at the beam splitter simultaneously using the DL. The two photons exiting one of the output ports of the first beam splitter together are separated at a second beam splitter (BS2) and are sent out of the cloning machine. The cloned photons are then detected and characterized using detectors (D1 and D2) and SLMs (SLM-C and SLM-D), respectively. (A to C) Examples of Hong-Ou-Mandel (HOM) coalescence curves for input photons of Embedded Image respectively (top to bottom). The curve is obtained by recording the coincidences between the output ports of BS2 for various delays of one of the input photons. Examples of enhancement peaks of Embedded Image, Embedded Image, and Embedded Image are obtained experimentally, and agree with the theoretical value of Rth = 2, corresponding to a visibility of Embedded Image.

  • Fig. 2 Optimal cloning fidelity for various dimensions.

    (A) Experimental values of the cloning fidelities are shown for each d number of elements of the logical basis, along with theoretical values, for various dimensions d. (B) The average cloning fidelities (blue dots) are plotted for various dimensions, along with probability matrices Embedded Image of detecting a cloned photon in any output state |i〉 of the OAM logical basis, given an input state |ψ〉 of the same basis. The diagonal elements of the probability matrices correspond to the cloning fidelity of each element of the basis. The light and dark gray shaded areas correspond to fidelities not accessible by state estimation and 1 → 2 optimal symmetric UQCM, respectively. In quantum cryptography, a more effective class of quantum hacking, namely, coherent attacks (1), yields larger fidelities illustrated by the dim gray shaded area.

  • Fig. 3 Cloning fidelities for various MUBs and cloning of Gaussian states in dimension d = 7.

    (A) Probability Embedded Image of detection of an output cloned state |i〉 given an input state |ψ〉, where |i〉 and |ψ〉 belong to a specific MUB. This set of measurement is repeated for all d + 1 MUBs (I) to (VIII), in dimension 7. The on-diagonal elements represent the cloning fidelities for each element of a given basis. (B) Theoretical and experimental high-dimensional cloning of a Gaussian state. The cloned fidelity is obtained by calculating the overlap of the reduced density matrix of the cloned state with the input state. The experimental reduced density matrix of the cloned state is obtained by full quantum state tomography. The experimentally reconstructed density matrices of the Gaussian state before and after cloning are shown along with their theoretical counterparts.

  • Fig. 4 High-dimensional QKD without and with quantum hacking.

    (A) Experimental probability matrices obtained from projective measurements are shown on the left side. The bases selected by Alice and Bob are indicated on the vertical and horizontal axes, respectively. On the right, we show Alice’s initial message and Bob’s decrypted message. (B) Experimental probability matrices with the presence of an eavesdropper having access to a symmetric optimal UQCM. Similarly, Alice’s initial message is shown along with the decrypted message obtained by both Bob and Eve. One may note that for the BB84 protocol, the symmetric UQCM does not lead to the optimal individual attack. Rather, our UQCM results in the optimal individual attack for the QKD protocol exploiting all d + 1 available MUBs. In the simpler case of the BB84 protocol, the optimal attack consists of the asymmetric Fourier-covariant cloner (1, 26), which cannot be straightforwardly implemented in our experimental setup.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/2/e1601915/DC1

    Supplementary Text

    fig. S1. Detailed experimental setup.

    fig. S2. Experimental cloning fidelities for every element of each MUB in dimension 7.

    fig. S3. Projective measurements of the input and cloned Gaussian state.

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Text
    • fig. S1. Detailed experimental setup.
    • fig. S2. Experimental cloning fidelities for every element of each MUB in dimension 7.
    • fig. S3. Projective measurements of the input and cloned Gaussian state.

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