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High-dimensional quantum cloning and applications to quantum hacking
Frédéric Bouchard, Robert Fickler, Robert W Boyd, Ebrahim Karimi
TL;DR
Optimal high-dimensional photonic cloning had not yet been experimentally demonstrated, despite its relevance to quantum information. This work implements a universal cloner by symmetrization, tests its fidelity and universality, and applies it to a high-dimensional BB84 attack.
Problem
Optimal quantum cloning had been experimentally realized for low-dimensional photonic states, but not for high-dimensional systems, which can encode more information on a carrier.
Source: p. 1Method
The symmetrization method sends an arbitrary input state and a dimension-matched completely mixed state into a beam splitter, yielding two symmetric cloned photons.
Source: p. 1Results
Fidelities agree near-perfectly with the theoretical 1→2 symmetric optimal UQCM across d=2–7 and all d+1 OAM MUBs; dimension-7 Gaussian tomography estimates 0.40 ± 0.01, and a d=7 BB84 cloning attack lowers fidelity.
Source: pp. 2–4Takeaways & Limitations
The observed decrease in cloning fidelity with increasing dimension offers an intuitive explanation of the superiority of high-dimensional quantum cryptography.
Source: p. 2
Abstract
from arXiv · showhide
Attempts at cloning a quantum system result in the introduction of imperfections in the state of the copies. This is a consequence of the no-cloning theorem, which is a fundamental law of quantum physics and the backbone of security for quantum communications. Although such perfect copies are prohibited, a quantum state may be copied with maximal accuracy via various optimal cloning schemes. Optimal quantum cloning, which lies at the border of the physical limit imposed by the no-signalling theorem and the Heisenberg uncertainty principle, has been experimentally realized for low dimensional photonic states. However, an increase in the dimensionality of quantum systems is greatly beneficial to quantum computation and communication protocols. Nonetheless, no experimental demonstration of optimal cloning machines has hitherto been shown for high-dimensional quantum systems. Here, we perform optimal cloning of high-dimensional photonic states by means of the symmetrization method. We show the universality of our technique by conducting cloning of numerous arbitrary input states, and fully characterize our cloning machine by performing quantum state tomography on \emph{cloned} photons. In addition, a cloning attack on a Bennett and Brassard (BB84) quantum key distribution protocol is experimentally demonstrated in order to reveal the robustness of high-dimensional states in quantum cryptography.
Supplementary Materials Materials and Methods, Supplementary Text, and Figs. S1 and S2.
The supplementary materials acknowledge support from Canadian research and innovation programs and fellowships.
Supplementary Materials Materials and Methods, Supplementary Text, and Figs. S1 and S2.
- The work acknowledges support from the Canada Research Chairs Program, the Canada Foundation for Innovation, and several graduate and postdoctoral fellowships.F.B. and R.F. are individually associated with the Vanier and Banting fellowships, respectively.Source: Supplementary Materials Materials and… · p. 5
Part 1 METHOD:
The experiment uses two-photon interference at a beam splitter to implement symmetrization, then separates and characterizes the cloned photons. Hong-Ou-Mandel measurements show interference and coalescence close to their theoretical values.
Part 1 METHOD:
- Indistinguishable photons entering opposite ports of a 50:50 beam splitter ideally bunch into one output port, providing the two-photon interference underlying symmetrization.The paths must be equidistant, adjusted with a translational stage to tens-of-microns precision.Source: Part 1 METHOD · p. 6
- The measured Hong-Ou-Mandel visibility is 0.89 ± 0.01, and coalescence peaks are 1.97 ± 0.08, 2.02 ± 0.08, and 1.99 ± 0.09 for ℓ=−1, 0, and 1, respectively.The visibility is compared with Vth = 1 and the peaks with Rth = 2.Source: Figure S1 · p. 6
- After the first beam splitter, a second beam splitter separates the bunched photons for coincidence detection and characterization using spatial light modulators and single-mode fibres.Photons are made indistinguishable in arrival time, polarization, and frequency while retaining spatial modes as the quantum-state degree of freedom.Source: Part 1 METHOD · pp. 6–7
Part 2 CLONING FIDELITIES FOR VARIOUS MUTUALLY UNBIASED BASES:
The section defines mutually unbiased bases and describes their construction, then tests cloning fidelity for their elements; Figure S2 displays the dimension-7 measurements.
Part 2 CLONING FIDELITIES FOR VARIOUS MUTUALLY UNBIASED BASES:
- Different mutually unbiased bases have squared inner products of 1/d between their elements; dimensions that are powers of primes have d+1 such bases.The section gives explicit expressions for the basis elements.Source: Part 2 CLONING FIDELITIES FOR VARIOUS… · p. 7
- The first mutually unbiased basis is the logical basis, while the remaining bases with α≥2 are specified by explicit expressions.The supplied passage introduces these expressions but does not provide their full readable form.Source: Part 2 CLONING FIDELITIES FOR VARIOUS… · p. 7
- Cloning fidelity was measured for every element of every mutually unbiased basis; Figure S2 shows dimension-7 results across all d+1 bases and their transverse profiles.The profiles combine transverse phase and intensity to visualize both patterns.Source: Part 2 CLONING FIDELITIES FOR VARIOUS… · p. 7, Figure S2 · p. 8