# An Improved Sample Complexity Lower Bound for (Fidelity) Quantum State Tomography

Henry Yuen

Columbia University

### Abstract

We show that $\Omega(rd/\epsilon)$ copies of an unknown rank-$r$, dimension-$d$ quantum mixed state are necessary in order to learn a classical description with $1 – \epsilon$ fidelity. This improves upon the tomography lower bounds obtained by Haah, et al. and Wright (when closeness is measured with respect to the fidelity function).

This paper presents a sharper lower bound on the number of copies of a quantum state needed to learn a classical description of it.

### ► References

[1] Dagmar Bruß and Chiara Macchiavello. Optimal state estimation for $d$-dimensional quantum systems. Physics Letters A, 253 (5-6): 249–251, 1999. https:/​/​doi.org/​10.1016/​S0375-9601(99)00099-7.
https:/​/​doi.org/​10.1016/​S0375-9601(99)00099-7

[2] Jeongwan Haah, Aram W Harrow, Zhengfeng Ji, Xiaodi Wu, and Nengkun Yu. Sample-optimal tomography of quantum states. IEEE Transactions on Information Theory, 63 (9): 5628–5641, 2017. https:/​/​doi.org/​10.1145/​2897518.2897585.
https:/​/​doi.org/​10.1145/​2897518.2897585

[3] Michael Keyl and Reinhard F Werner. Optimal cloning of pure states, testing single clones. Journal of Mathematical Physics, 40 (7): 3283–3299, 1999. https:/​/​doi.org/​10.1063/​1.532887.
https:/​/​doi.org/​10.1063/​1.532887

[4] Ryan O'Donnell and John Wright. Efficient quantum tomography. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 899–912, 2016. https:/​/​doi.org/​10.1145/​2897518.2897544.
https:/​/​doi.org/​10.1145/​2897518.2897544

[5] Reinhard F Werner. Optimal cloning of pure states. Physical Review A, 58 (3): 1827, 1998. https:/​/​doi.org/​10.1103/​PhysRevA.58.1827.
https:/​/​doi.org/​10.1103/​PhysRevA.58.1827

[6] Andreas Winter. Coding theorem and strong converse for quantum channels. IEEE Transactions on Information Theory, 45 (7): 2481–2485, 1999. https:/​/​doi.org/​10.1109/​18.796385.
https:/​/​doi.org/​10.1109/​18.796385

[7] John Wright. How to learn a quantum state. PhD thesis, Carnegie Mellon University, 2016.

### Cited by

[1] Nic Ezzell, Elliott M. Ball, Aliza U. Siddiqui, Mark M. Wilde, Andrew T. Sornborger, Patrick J. Coles, and Zoë Holmes, "Quantum Mixed State Compiling", arXiv:2209.00528, (2022).

[2] Joran van Apeldoorn, Arjan Cornelissen, András Gilyén, and Giacomo Nannicini, "Quantum tomography using state-preparation unitaries", arXiv:2207.08800, (2022).

[3] Ming-Chien Hsu, En-Jui Kuo, Wei-Hsuan Yu, Jian-Feng Cai, and Min-Hsiu Hsieh, "Quantum state tomography via non-convex Riemannian gradient descent", arXiv:2210.04717, (2022).

[4] Srinivasan Arunachalam, Sergey Bravyi, Arkopal Dutt, and Theodore J. Yoder, "Optimal algorithms for learning quantum phase states", arXiv:2208.07851, (2022).

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