POVM-induced Coherence Measure in Terms of Fidelity

C. L. Liu, Da-Jian Zhang, Xiao-Dong Yu +2 more

A. Hayashi, T. Hashimoto, M. Horibe

A direct derivation is given for the optimal mean fidelity of quantum state estimation of a $d$-dimensional unknown pure state with its $N$ copies given as input, which was first obtained by Hayashi in terms of an infinite set of covariant positive operator valued measures (POVM's) and by Bru\ss{} and Macchiavello establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we construct optimal estimators with finite POVMs using exact quadratures on a hypersphere. These finite optimal estimators are not generally universal, where universality means the fidelity is independent of input states. However, any optimal estimator with finite POVM for $M(>N)$ copies is universal if it is used for $N$ copies as input.

Meng‐Li Guo, Jin‐Min Liang, Bo Li +2 more

Quantum coherence is a fundamental feature of quantum physics and plays a significant role in quantum information processing. By generalizing the resource theory of coherence from von Neumann measurements to positive operator-valued measures (POVMs), POVM-based coherence measures have been proposed with respect to the relative entropy of coherence, the l 1 norm of coherence, the robustness of coherence and the Tsallis relative entropy of coherence. We derive analytically the lower and upper bounds on these POVM-based coherence of an arbitrary given superposed pure state in terms of the POVM-based coherence of the states in superposition. Our results can be used to estimate range of quantum coherence of superposed states. Detailed examples are presented to verify our analytical bounds.

Ties-A. Ohst, Sebastian Schlösser, Roope Uola

Quantifying the disturbance caused by a quantum measurement typically requires detailed knowledge of the underlying measurement channel. In this work, we introduce a statistical disturbance bound, which connects the statistical properties of a quantum measurement to the state disturbance induced by any compatible measurement channel. Specifically, we show that the average fidelity between input and output with respect to an arbitrary ensemble of pure input states is fundamentally bounded in terms of the measurement, described as a positive operator-valued measure (POVM). We further develop the weighted state exclusion technique, which enables an experimental determination of the statistical disturbance bound without requiring explicit knowledge of the measurement effects. To see the advantages of our approach over existing information-disturbance relations, we show that our bound distinguishes between measurements with equivalent informativeness. Furthermore, we demonstrate that the weighted state exclusion technique can detect and quantify measurement-induced disturbance using state preparations that are insufficient for tomographic reconstruction of the measurement operators. Finally, we illustrate how disturbance bounds defined with respect to specific input ensembles can be used to bound an eavesdropper's guessing probability in a simple protocol for quantum randomness generation.

Anna Vershynina

Dan-Yang Chen, Ming Yang, Yuanqin Yu

High-dimensional hyperentangled states, serving as crucial physical resources in quantum communication, have attracted significant attention due to their exceptional properties (e.g., higher information density) compared to conventional two-dimensional single-degree-of-freedom entanglement. However, practical quantum entanglement systems inevitably suffer from environmental interactions, resulting in their degradation into partially entangled pure states or even mixed states, thereby significantly compromising the efficiency and security of quantum communication. Consequently, improving entanglement preservation in noisy systems emerges as a pivotal challenge in realizing fault-tolerant quantum networks. We propose two universal distillation schemes based on local positive-operator-valued measures (POVMs) for single-copy $(2\ifmmode\times\else\texttimes\fi{}4\ifmmode\times\else\texttimes\fi{}4)$-dimensional hyperentangled photons. These schemes effectively suppress photon vacuum errors induced by transmission losses and are constructed using spatial-mode and polarization-based POVM measurements, respectively. Our results demonstrate that the spatial-mode-based scheme outperforms its polarization-based counterpart in terms of fidelity and success probability. Remarkably, we discover that with appropriate POVM parameters, the fidelity range of amplitude-damped noisy states can be expanded to cover almost all parameter spaces. Due to the universal applicability of POVM measurements, which require neither auxiliary entanglement nor complex detectors and operate on single copies of imperfect states in each purification round, our scheme provides an experimentally feasible solution for distilling high-dimensional hyperentangled photonic states.