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# Unscrambling entanglement through a complex medium

## Abstract

The transfer of quantum information through a noisy environment is a central challenge in the fields of quantum communication, imaging and nanophotonics. In particular, high-dimensional quantum states of light enable quantum networks with significantly higher information capacities and noise robustness as compared with qubits. However, although qubit entanglement has been distributed over large distances through free space and fibre, the transport of high-dimensional entanglement is hindered by the complexity of the channel, which encompasses effects such as free-space turbulence or mode mixing in multimode waveguides. Here, we demonstrate the transport of six-dimensional spatial-mode entanglement through a 2-m-long, commercial multimode fibre with 84.4% fidelity. We show how the entanglement can itself be used to measure the transmission matrix of the complex medium, allowing the recovery of quantum correlations that were initially lost. Using a unique property of entangled states, the medium is rendered transparent to entanglement by carefully ‘scrambling’ the photon that did not enter it, rather than unscrambling the photon that did. Our work overcomes a primary challenge in the fields of quantum communication and imaging, and opens a new pathway towards the control of complex scattering processes in the quantum regime.

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## Data availability

Data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

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## Acknowledgements

We thank M. Huber, N. Friis, D. Phillips, S. Leedumrongwatthanakun and A. Fedrizzi for helpful discussions. This work was made possible by financial support from the QuantERA ERA-NET Co-fund (FWF Project I3773-N36) and the UK Engineering and Physical Sciences Research Council (EP/P024114/1). H.D. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 840958.

## Author information

Authors

### Contributions

M.M. conceived the research and supervised the project. M.M. and H.D. designed the experiment. N.H.V. and S.G. performed the experiment. All authors developed theoretical methods, analysed the data and contributed to writing the manuscript.

### Corresponding authors

Correspondence to Natalia Herrera Valencia or Mehul Malik.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 ‘Scrambled’ correlations of the output state $$\left|{\psi }_{\text{MMF}}\right\rangle$$.

Two-photon coincidence counts in the 7-dimensional Pixel basis $${\{\left|m\right\rangle ,\left|n\right\rangle \}}_{m,n}$$, and its 7 mutually unbiased bases $${\{\left|{f}_{m}\right\rangle ,\left|{f}_{n}\right\rangle \}}_{m,n}$$ at the output of the multimode fibre. For this set of measurements we obtained a fidelity to the maximally entangled state of $$\tilde{F}(\rho_T ,{\psi }^{+})=5.4\pm 1.0 \%$$, that is, no entanglement can be certified.

### Extended Data Fig. 2 S matrix measurement.

Examples of computer-generated holograms displayed on the SLMs corresponding to (a) Alice and (b) Bob, when measuring a given matrix element $${R}_{mn}^{\theta }$$. Changing the relative phase θ, we obtain matrices Rθ that allow us to calculate the elements Smn through Eq. (17). The absolute value of the matrix S is shown in (c).

### Extended Data Fig. 3 E matrix measurement.

Examples of computer-generated holograms displayed on the SLMs corresponding to (a) Alice and (b) Bob, when measuring matrix elements $${R}_{m}^{\theta }$$. We change the relative phase θ via a phase-stepping process to determine the diagonal elements (m = n) of the matrix E. The absolute value of this diagonal matrix is shown in (c).

## Supplementary information

### Supplementary Information

Supplementary discussion and methods, Figs. 1–6, equations (1)–(23) and Tables 1 and 2.

## Source data

### Source Data Fig. 2

Transmission matrix.

### Source Data Fig. 3

Correlations measured in two bases in the absence and presence of the MMF.

### Source Data Fig. 4

Recovered correlations in the Pixel and seven ‘tilted’ Pixel bases.

### Source Data Extended Data Fig. 1

Scrambled correlations in eight mutually unbiased bases in the presence of the MMF.

### Source Data Extended Data Fig. 2

Absolute value of S matrix.

### Source Data Extended Data Fig. 3

Absolute value of E matrix.

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Valencia, N.H., Goel, S., McCutcheon, W. et al. Unscrambling entanglement through a complex medium. Nat. Phys. 16, 1112–1116 (2020). https://doi.org/10.1038/s41567-020-0970-1

• Accepted:

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• DOI: https://doi.org/10.1038/s41567-020-0970-1

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