Meaning updating of density matrices

脉冲量子光的密度矩阵

Density Matrices of Pulsed Quantum Light

本文通过密度矩阵统一描述脉冲量子光，证明有限探测时间下测量等效性，揭示腔长物理意义及光谱起源，并推广至高斯态以关联计数率与光谱。

The Scalar Curvature of the Bures Metric on the Space of Density Matrices

The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we determine its scalar curvature and Ricci tensor and prove a lower bound for the curvature on the submanifold of trace one matrices. This bound is achieved for the maximally mixed state, a further hint for the quantum statistical meaning of the scalar curvature.

Random matrices: Application to quantum paradoxes

Recently, a geometric embedding of the classical space and classical phase space of an n-particle system into the space of states of the system was constructed and shown to be physically meaningful. Namely, the Newtonian dynamics of the particles was recovered from the Schroedinger dynamics by constraining the state of the system to the classical phase space submanifold of the space of states. A series of theorems related to the embedding and the Schroedinger evolution with a random Hamiltonian was proven and shown to be applicable to the process of measurement in classical and quantum mechani…

Giving Operational Meaning to the Superposition of Causal Orders

In this work, we give rigorous operational meaning to superposition of causal orders. This fits within a recent effort to understand how the standard operational perspective on quantum theory could be extended to include indefinite causality. The mainstream view, that of "process matrices", takes a top-down approach to the problem, considering all causal correlations that are compatible with local quantum experiments. Conversely, we pursue a bottom-up approach, investigating how the concept of indefiniteness emerges from specific characteristics of generic operational theories. Specifically, w…

The Inferential Design of Entropy and its Application to Quantum Measurements

This thesis synthesizes probability and entropic inference with Quantum Mechanics (QM) and quantum measurement [1-6]. It is shown that the standard and quantum relative entropies are tools designed for the purpose of updating probability distributions and density matrices, respectively [1]. The derivation of the standard and quantum relative entropies are completed in tandem and follow from the same inferential principle - the principle of minimal updating [21,66]. As the quantum maximum entropy method is derived using the standard quantum mechanical formalism, the quantum maximum entropy meth…

Peculiarities of Quantum Mechanics: Origins and Meaning

The most peculiar, specifically quantum, features of quantum mechanics --- quantum nonlocality, indeterminism, interference of probabilities, quantization, wave function collapse during measurement --- are explained on a logical-geometrical basis. It is shown that truths of logical statements about numerical values of quantum observables are quantum observables themselves and are represented in quantum mechanics by density matrices of pure states. Structurally, quantum mechanics is a result of applying non-Abelian symmetries to truth operators and their eigenvectors --- wave functions. Wave fu…

Dynamics of quantum Fisher information in the two-qubit systems constructed from the Yang-Baxter matrices

By using the quantum Yang-Baxterization approach to the three different Hamiltonians, we investigate the behavior of the quantum Fisher information (QFI) under the actions of these Hamiltonians on the different two-qubit input states and by estimating the meaningful parameter $φ$. We address the overall estimation properties by evaluating the QFI for the whole system undergone different unitary evolution. The results show that the behavior of the QFI depends on the choice of the initial states. Choosing the optimal input states can improve the precision of quantum parameter estimation. On the …

The ZX-calculus is complete for stabilizer quantum mechanics

The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics, meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound, i.e. any equality that can be derived graphically can also be derived using matrix mechanics. Here, we show that the ZX-calculus is complete for pure qubit stabilizer quantum mechanics, meaning any equality that can be derived using matrices can also be derived pictorially. The proof rel…

The quantum Chernoff bound as a measure of distinguishability between density matrices: application to qubit and Gaussian states

Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability when discriminating two hypotheses given a large number of observations. Recently the combined work of Audenaert et al. [Phys. Rev. Lett. 98, 160501] and Nussbaum and Szkola [quant-ph/0607216] has proved the quantum analog of this bound, which applies when the hypotheses correspond to two quantum states. Based on the quantum Chernoff bound, we define a physically meaningful distinguishabilit…

A recipe for topological observables of density matrices

Meaningful topological invariants for mixed quantum states are challenging to identify as there is no unique way to define them, and most choices do not directly relate to physical observables. Here, we propose a simple pragmatic approach to construct topological invariants of mixed states while preserving a connection to physical observables, by continuously deforming known topological invariants for pure (ground) states. Our approach relies on expectation values of many-body operators, with no reference to single-particle (e.g., Bloch) wavefunctions. To illustrate it, we examine extensions t…

从部分信息细化集体$N$-可表性问题

Refining ensemble $N$-representability of one-body density matrices from partial information

本文提出一种层次化的集体一阶$N$-可表性问题，利用部分已知的一阶集体缩减密度矩阵信息，通过自然职业数向量简化问题，并结合Horn问题的扩展获得显式解。