Quantum algorithms for Uhlmann transformation

TL;DR

This work provides quantum algorithms to realize the Uhlmann transformation across three common access models (purified query, purified sample, and mixed sample), achieving polynomial-time performance for low-rank states and offering exponential improvements over tomography-based methods. The authors combine block-encoding, quantum singular value transformation, and density-matrix exponentiation to implement the optimal local B-operations implied by Uhlmann's theorem, while establishing upper and lower bounds on query and sample complexities. They demonstrate practical value by applying the Uhlmann transformation to square-root fidelity estimation and to information-theoretic tasks via decoupling, including entanglement transmission, quantum state merging, and Petz recovery maps. The results significantly advance the practical deployment of these foundational quantum-information protocols, clarifying algorithmic costs and illuminating avenues for further optimization and complexity-theoretic connections.

Abstract

Uhlmann's theorem is a central result in quantum information theory, which associates the closeness of two quantum states with that of their purifications. The theorem also well characterizes a fundamental task: how close a pure quantum state can be transformed into another state via local operations acting only on its subsystem. The optimal transformation for this task is called the Uhlmann transformation, which has broad applications in various information-processing tasks. However, its quantum circuit implementation and computational cost have remained unclear, limiting the utility of the transformation. In this work, we fill this gap by proposing quantum algorithms that realize the Uhlmann transformation in query and sample access models. Notably, our Uhlmann transformation algorithms can be polynomial-time for low-rank states, exhibiting an exponential improvement over the previous approach and other naive approaches based on quantum state tomography. In addition, we derive a lower bound on the query and sample complexities of the Uhlmann transformation for a deeper understanding of its algorithmic features. We apply our Uhlmann transformation algorithms to fidelity estimation between two states, and substantially improve the previous best query and sample complexities. We further discuss other applications to several information-theoretic tasks, including entanglement transmission, quantum state merging, and the algorithmic implementation of the Petz recovery map, providing a comprehensive evaluation of their computational costs. These results, hence, contribute to the practical realization of such widely recognized and useful protocols.

Figures (4)

• Figure 1: A quantum circuit preparing the state $\Upsilon$ in Eq. \ref{['inteq:64']}. The systems ${\mathsf{C}}_1$, ${\mathsf{C}}_2$, and ${\mathsf{C}}_3$ are each one-qubit systems. Open circles indicate that the gate is applied when the control qubit is in the state $|0\rangle$, while closed circles indicate control for a qubit in the state $|1\rangle$. The gate $H$ is the one-qubit Hadamard gate. The double vertical lines represent that the qubits of that system are traced out. By convention, the objects with a cross mark and a circle with a cross represent the swap operator and the NOT operator, respectively.
• Figure 2: A diagram of entanglement transmission. Alice applies a Haar random unitary $U^{{\mathsf{A}}{\mathsf{G}}}$ to encode ${\mathsf{A}}$ with ${\mathsf{G}}$, which is part of a possibly pre-shared entangled state $|\Phi\rangle^{{\mathsf{G}}\hat{{\mathsf{G}}}}$. Alice then transmits ${\mathsf{A}}{\mathsf{G}}$ to Bob through a noisy quantum channel ${\mathcal{N}}^{{\mathsf{A}}{\mathsf{G}}\rightarrow{\mathsf{B}}}$. After receiving ${\mathsf{B}}$, Bob applies a decoding map ${\mathcal{D}}^{{\mathsf{B}}\hat{{\mathsf{G}}}\rightarrow\hat{{\mathsf{A}}}}$ on ${\mathsf{B}}\hat{{\mathsf{G}}}$ to obtain a final state that approximates the maximally entangled state $|\Phi\rangle^{{\mathsf{R}}\hat{{\mathsf{A}}}}$.
• Figure 3: A diagram of the Stinespring dilation of ${\mathcal{N}}^{{\mathsf{A}}{\mathsf{G}}\rightarrow{\mathsf{B}}}$ induced by the canonical purification. We denote by $V_{{\mathcal{N}}, \tau_{\rm c}}^{{\mathsf{A}}{\mathsf{G}}\rightarrow{\mathsf{B}}{\mathsf{R}}"{\mathsf{B}}"\hat{{\mathsf{G}}}"}$ the Stinespring isometry associated with $|\tau_{\rm c}\rangle^{{\mathsf{R}}{\mathsf{B}}\hat{{\mathsf{G}}}{\mathsf{R}}"{\mathsf{B}}"\hat{{\mathsf{G}}}"}$. Then, the Stinespring isometry corresponding to $|\Psi_{\rm c}\rangle^{{\mathsf{R}}{\mathsf{B}}\hat{{\mathsf{G}}}{\mathsf{R}}"{\mathsf{B}}"\hat{{\mathsf{G}}}"}$ is given by $(U^{{\mathsf{R}}"\hat{{\mathsf{G}}}"})^\dag V_{{\mathcal{N}}, \tau_{\rm c}}^{{\mathsf{A}}{\mathsf{G}}\rightarrow{\mathsf{B}}{\mathsf{R}}"{\mathsf{B}}"\hat{{\mathsf{G}}}"}$, which is denoted as $V_{{\mathcal{N}}, \Psi_{\rm c}}^{{\mathsf{A}}{\mathsf{G}}\rightarrow{\mathsf{B}}{\mathsf{R}}"{\mathsf{B}}"\hat{{\mathsf{G}}}"}$ and shown in a red dotted box. The complementary channels determined from these isometries are generally not identical, so Eq. \ref{['inteq:136']} cannot be applied directly. To resolve this issue, it suffices for Bob to apply $U^{{\mathsf{R}}"\hat{{\mathsf{G}}}"}$ to copies of $|\Psi_{\rm c}\rangle^{{\mathsf{R}}{\mathsf{B}}\hat{{\mathsf{G}}}{\mathsf{R}}"{\mathsf{B}}"\hat{{\mathsf{G}}}"}$ at hand in the algorithm, thereby canceling the extra $(U^{{\mathsf{R}}"\hat{{\mathsf{G}}}"})^\dag$.
• Figure 4: A diagram of quantum state merging. The goal is to transfer Alice's subsystem, which is part of a joint quantum state $|\omega\rangle^{{\mathsf{R}}{\mathsf{A}}{\mathsf{B}}}$, to Bob, while simultaneously distilling as much entanglement as possible between ${\mathsf{S}}$ and $\hat{{\mathsf{S}}}$, using only local operations and classical communication. The double line connecting Alice and Bob represents a noiseless classical communication channel.

Theorems & Definitions (62)

• Theorem 1: Uhlmann transformation algorithm in the purified query access models
• Theorem 2: Uhlmann transformation algorithm in the purified sample access models
• Theorem 3: Uhlmann transformation algorithm in the mixed sample access models
• Theorem 4: Algorithm for a variant of the Uhlmann transformation in the mixed sample access models
• Corollary 4: Lower bound on the query complexity for the Uhlamnn transformation
• Theorem 5: Square root fidelity estimation using the Uhlmann transformation algorithms
• Theorem 6: Uhlmann's theorem uhlmann1976transition
• Proposition 7: Explicit form of the Uhlmann partial isometry Jozsa1994FidelityforMixedQuantumStates
• Theorem 8: Quantum singular value transformation with a real polynomial gilyen2019qsvt; see also Ref. Gilyn2019thesis
• Proposition 9: QSVT with the sign function gilyen2019qsvt; see also Ref. Gilyn2019thesis