Teiko Heinosaari, Juha-Pekka Pellonpää
We show that the canonical phase measurement is pure in the sense that the corresponding positive operator valued measure (POVM) is extremal in the convex set of all POVMs. This means that the canonical phase measurement cannot be interpreted as a noisy measurement, even if it is not a projection valued measure.
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Three-qubit pure-state canonical forms
在本文中，我们分析了任何纯三量子比特态可以铸造成的规范形式。最小的形式，即具有最小数量的产品状态建立从本地基地，也提出了，并导致一个完整的分类纯三量子比特状态。这种分类与局部酉变换下多项式不变量的值一一对应。
Three-qubit pure-state canonical forms for Perfect Teleportation
最近，Agrawal和Pati [Phys.Rev.A74，062320（2006）]给出了一类可用于完美隐形传态的W态。利用变换算符给出了理想量子信道的两种典型形式，GHZ态和W态是这两种典型形式的特例。进一步给出了正交完备测度基。
We construct a canonical form for pure states in $\bwe^3(\bC^6)$, the three-fermion system with six single particle states, under local unitary (LU) transformations, i.e., the unitary group $\Un(6)$. We also construct a minimal set of generators of the algebra of polynomial $\Un(6)$-invariants on $\bwe^3(\bC^6)$. It turns out that this algebra is isomorphic to the algebra of polynomial LU-invariants of three-qubits which are additionally invariant under qubit permutations. As a consequence of this surprising fact, we deduce that there is a one-to-one correspondence between the $\Un(6)$-orbits …
Using the example of the harmonic oscillator, we illustrate the use of hybrid dynamical brackets in analyzing quantum-classical interaction. We only assume that a hybrid dynamical bracket exists, is bilinear, and reduces to the pure quantum/classical bracket when acting on pure quantum/classical variables. Any hybrid bracket obeying these natural requirements will produce the same dynamics for pure classical or quantum variables, given a hybrid Hamiltonian. Backreaction is manifested in the evolution of a nonvanishing commutator between classical variables. The more massive the classical syste…
Canonical steering ellipsoids of pure symmetric multiqubit states with two distinct spinors and volume monogamy of steering
本文研究含两个不同旋量的纯对称多量子比特态，通过规范导引椭球几何刻画其 SLOCC 不等价族，并分析 W 类及相关态的体积单配性。
We consider a system weakly interacting with a bath as a thermodynamic setting to establish a quantum foundation of statistical physics. It is shown that even if the composite system is initially in an arbitrary nonequilibrium pure quantum state, the unitary dynamics of a generic weak interaction almost always drives the subsystem into the canonical ensemble, in the usual sense of typicality. A crucial step is taken by assuming that the matrix elements of the interaction Hamiltonian have random phases, while their amplitudes are left unrestricted.
Canonical and micro-canonical typical entanglement of continuous variable systems
我们提出了一个框架，符合统计力学的一般规范原则，定义连续变量系统的纯高斯态集上的措施。在这样一个框架内，我们定义了两个具体的措施，被称为“微正则”和“正则”，并应用它们来系统地研究的统计特性的二分纠缠的n模纯高斯态（量化的熵的子系统）。我们严格证明了“浓度的措施”周围的一个有限的平均值，发生在规范和微规范的方法在量子极限的纠缠。对于有限的n，我们确定分析的平均值和标准偏差的纠缠（量化的减少纯度）之间的一个模式和所有其他模式。此外，我们数值研究更一般的情况下，清楚地表明，开始的浓度的措施已经发生在相对较小的n。
A thermal equilibrium state of a quantum many-body system can be represented by a typical pure state, which we call a thermal pure quantum (TPQ) state. We construct the canonical TPQ state, which corresponds to the canonical ensemble of the conventional statistical mechanics. Only a single realization of the TPQ state gives all variables of statistical-mechanical interest. We show that TPQ states corresponding to different ensembles give identical thermodynamic results, and they are related by simple analytic transformations. We have thus completed the TPQ formulation of statistical mechanics.…
Statistical mechanics from relational complex time with a pure state
本研究提出无需熵最大化或状态计数，仅通过系统与环境最大纠缠纯态的关系复时间演化，即可导出系统的正则密度矩阵，为统计力学提供新的纯态基础。
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy an…
The eigenvalues of a pure quartic oscillator are computed, applying a canonical operator formulation, generalized from the harmonic oscillator. Solving a 10x10 secular equation produces eigenvalues in agreement, to at least 4 significant figures, with accurate computations given in the literature.
We investigate the time evolution of a generic and finite isolated quantum many-body system starting from a pure quantum state. We find the kinematical general canonical principle proposed by Popescu-Short-Winter for statistical mechanics can be built in a more solid ground by studying the thermalization, i.e. comparing the density matrices themselves rather than the measures of distances. In particular, this allows us to explicitly identify that, from any instantaneous pure state after thermalization, the state of subsystem is like from a microcanonical ensemble or a generalized Gibbs ensembl…
Measurements of single-mode phase observables are studied in the spirit of the quantum theory of measurement. We determine the minimal measurement models of phase observables and consider methods of measuring such observables by using a double homodyne detector. We show that, in principle, the canonical phase distribution of the signal state can be measured via double homodyne detection by first processing the state using a two-mode unitary channel.
We present a joint-measurement uncertainty relation for a pair of mean square deviations of canonical variables averaged over Gaussian distributed quantum optical states. Our Bayesian formulation is free from the unbiasedness assumption, and enables us to quantify experimentally implemented joint-measurement devices by feeding a moderate set of coherent states. Our result also reproduces the most informative bound for quantum estimation of phase-space displacement in the case of pure Gaussian states.
Exponential Moments of Canonical Phase: Homodyne Measurements
本文提出了一种从平衡零拍探测数据中直接采样正则相位指数矩的方法。采样函数的解析表达式示出了这是有效的任意状态。数值模拟表明了该方法的适用性，并与双零差法直接测量相位进行了比较。
Continuous measurement of canonical observables and limit Stochastic Schrodinger equations
本文导出了以正则位置和动量为观测量的连续弱测量极限下的随机薛定谔方程。为此，我们扩展了一个参数，由于Smolianov和杜鲁门从冯诺依曼模型的间接测量的位置，阿瑟和凯利模型的位置和动量的同时测量。我们只需要探测器状态的无偏性和足以确保中心极限效应的可积性条件。尽管采取了弱相互作用，而不是弱测量极限，所得到的随机波动方程是相同的形式，在最近的论文中推导出的联合高斯状态的特定情况下，由斯科特和米尔本。
Canonical Phase Measurements in the Presence of Photon Loss
我们分析了Berry和Wiseman [Phys. Rev. Lett { \bf 85}，5098，（2000）]给出的最佳状态，在光子损失的情况下，在正则相位测量下。光子损失模型是一个通用的虚拟分束器，我们提出了完整的密度矩阵计算，这是更直接的，不涉及任何近似。我们发现，对于一个给定的损失量的上限输入光子数，产生一个子散粒噪声估计。
Physical origins and limitations of canonical quantum measurement behavior
本文回顾并扩展了作者关于正则量子测量行为物理起源与局限性的工作，通过分析真实测量装置的微观结构，特别是莫特问题，得出一个由实验支持的涌现的、近似的玻恩规则，并指出该规则在极小概率密度下可能失效，且该图景适用于多种测量场景。
The quantum weak value draws many attentions recently from theoretical curiosity to experimental applications. Now we design an unusual weak measuring procedure as the pre-selection, mid-selection and post-selection to study the correlation function of two weak values, which we called the weak correlation function. In this paper, we proposed an weak measurement experiment to measure the canonical commutator $[\hat{x},\hat{p}]=i\hbar$ in quantum mechanics. Furthurmore, we found the intriguing equivalence between the canonical commutation relation and Riemann hypothesis, and then obtained the we…
We propose a mechanism for system-induced spontaneous time reparametrization symmetry breaking in canonical gravity. We consider a model of extended canonical gravity, based on Sen's spinorial expression of the spatial-diffeomorphism-gauge independent part of the shift vector, with one additional long-range self-interacting massive spin-$1/2$ particle whose spin is coupled to Sen's spinor in the Hamiltonian. This additional particle plays the role of creating the system-induced time reparametrization invariant potential. We use this potential to break time reparametrization symmetry spontaneou…
We address the implementation of the positive operator-valued measure (POVM) describing the optimal M-outcomes discrimination of the polarization state of a single photon. Initially, the POVM elements are extended to projective operators by Naimark theorem, then the resulting projective measure is implemented by a Knill-Laflamme-Milburn scheme involving an optical network and photon counters. We find the analytical expression of the Naimark extension and the detection scheme that realise it for an arbitrary number of outcomes M = 2^N.
An entanglement bound based on local measurements is introduced for multipartite pure states. It is the upper bound of the geometric measure and the relative entropy of entanglement. It is the lower bound of minimal measurement entropy. For pure bipartite states, the bound is equal to the entanglement entropy. The bound is applied to pure tripartite qubit states and the exact tripartite relative entropy of entanglement is obtained for a wide class of states.
Implementation of a canonical phase measurement with quantum feedback
本文通过对 Josephson 参量放大器施加量子反馈，在单光子波包上实现自适应单次正则相位测量，性能超越外差检测并达到单次相位估计最优。
We examine the problem of finding the minimum number of Pauli measurements needed to uniquely determine an arbitrary $n$-qubit pure state among all quantum states. We show that only $11$ Pauli measurements are needed to determine an arbitrary two-qubit pure state compared to the full quantum state tomography with $16$ measurements, and only $31$ Pauli measurements are needed to determine an arbitrary three-qubit pure state compared to the full quantum state tomography with $64$ measurements. We demonstrate that our protocol is robust under depolarizing error with simulated random pure states. …
In this paper, a characterization of maps between quantum states that preserve pure states and strict convex combinations is obtained. Based on this characterization, a structural theorem for maps between multipartite quantum states that preserve separable pure states and strict convex combinations is established. Then these results are applied to characterize injective (local) quantum measurements and answer some conjectures proposed in [J.Phys.A:Math.Theor. 45 (2012) 205305].
Features of biphoton polarization-frequency ququarts are considered. Their wave functions are defined as functions of both polarization and frequency variables of photons with the symmetry obligatory for two-boson states taken into account. In experiments, biphoton ququarts can display different features in dependence on whether experiments involve purely polarization or (alternatively) polarization-frequency measurements. If in experiments one uses only polarization measurements, the originally pure states of ququarts can be seen as mixed biphoton polarization states. Features of such states …
In a recent work [Phys. Rev. Lett. 120, 170502 (2018)], Pallister et al. proposed an optimal strategy to verify non-maximally entangled two-qubit pure states under the constraint that the accessible measurements being locally projective and non-adaptive. Their nice result leads naturally to the question: What is the optimal strategy among general LOCC measurements? In this Letter, we answer this problem completely for two-qubit pure states. To be specific, we give the optimal strategy for each of the following available classes of measurements: (i) local operations and one-way classical commun…
For any finite dimensional Hilbert space, we construct explicitly five orthonormal bases such that the corresponding measurements allow for efficient tomography of an arbitrary pure quantum state. This means that such measurements can be used to distinguish an arbitrary pure state from any other state, pure or mixed, and the pure state can be reconstructed from the outcome distribution in a feasible way. The set of measurements we construct is independent of the unknown state, and therefore our results provide a fixed scheme for pure state tomography, as opposed to the adaptive (state dependen…
Optimal scheme for estimating a pure qubit state via local measurements
我们提出的最佳方案估计一个纯量子比特状态的本地测量N个相同的副本。我们给出了低N的显式例子。对于大的N，我们表明，保真度饱和的集体测量的顺序1/N。当信号状态位于布洛赫球的子午线上时，我们证明了这可以在没有经典通信的情况下实现。
Quantum state tomography (QST) aims at estimating a quantum state from averaged quantum measurements made on copies of the state. Most quantum algorithms rely on QST at some point and it is a well explored topic in the literature, mostly for mixed states. In this paper we focus on the QST of a pure quantum state using parallel unentangled measurements. Pure states are a small but useful subset of all quantum states, their tomography requires fewer measurements and is essentially a phase recovery problem. Parallel unentangled measurements are easy to implement in practice because they allow the…
We generalize the thermal pure quantum (TPQ) formulation of statistical mechanics, in such a way that it is applicable to systems whose Hilbert space is infinite dimensional. Assuming particle systems, we construct the grand-canonical TPQ (gTPQ) state, which is the counterpart of the grand-canonical Gibbs state of the ensemble formulation. A single realization of the gTPQ state gives all quantities of statistical-mechanical interest, with exponentially small probability of error. This formulation not only sheds new light on quantum statistical mechanics but also is useful for practical computa…
Paper withdrawn. Results are similar to V. P. Belavkin, "Optimum distinction of non-orthogonal quantum signals," Radio Engineering and Electronic Physics, vol. 20, June 1975, p. 39-47.
Generalized uncertainty relations and efficient measurements in quantum systems
我们考虑两个变种的量子统计推广的Cramer-Rao不等式，建立了一个不变的下限上的均方误差的广义量子测量。这个不等式的复杂变形导致了任意状态的广义不确定原理的精确表述，与Helstrom的对称变形相反，在Helstrom的对称变形中，这些关系只针对纯态获得。引入了正则态的概念，并给出了正则态参数，特别是李群正则参数估计的均方误差下界。结果表明，这些边界是全球可达到的，只有对规范的状态，存在有效的测量或quasimeasurements。
Optimal reconstruction of a pure qubit state with local measurements
我们分析了一个未知的纯量子比特态的重建。我们得出的最佳猜测，可以推断出从任何一组测量的N个相同的副本系统的保真度作为一个数字的优点。我们详细研究了估计过程中与个人冯诺依曼测量，并证明他们是非常有竞争力的（复杂的）集体测量相比。我们计算的大$N$的保真度的表达式，并表明，个人的测量方案可以在渐近制度的最佳执行。
Proposal for Direct, Local Measurement of Entanglement for Pure Bipartite Systems of Arbitrary Dimension
基于复合系统的纠缠度与子系统纯度之间的互补关系，提出了一种简单的测量纠缠度的方法。该方法可应用于任意维数的纯态二分系统。它只需要单量子点旋转和直接的概率测量的子系统之一，因此可以很容易地实现实验使用线性光学设备。
One main goal of quantum control is to steer a quantum system toward an expected state or dynamics. For measurement-induced quantum control, measurements serve as the only control, which is like the cases in quantum Zeno and anti-zeno effects. In this paper, this scenario will be investigated in a general $N$-level quantum system. It is proved that, when the intial and expected states are both pure, the control space could be reduced to an effective subspace spanned by these two states only. This result will greatly simplify the measurement-induced control strategy of an $N$-level quantum syst…
The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continu…
Nearly Time-Optimal Pure State Tomography with Pauli Measurements
本文提出一种近最优的纯态量子态重构算法，利用单量子比特测量，在时间复杂度和复制复杂度上接近最优，输出高保真度的量子态估计。
Asymptotic estimation theory for a finite dimensional pure state model
研究了n个纯样品的最优化测量。将n个样本的最优测量的误差渐近地与由一个样本的最优测量给出的n个数据的极大似然估计的误差进行比较。
Minimal Informationally Complete Measurements for Pure States
我们考虑测量，描述了一个积极的运营商价值的措施（POVM），其结果概率确定一个任意的纯态的D维量子系统。我们称这样的测量为纯态信息完备（PSI-complete）POVM。我们表明，2D-1结果的测量不能是PSI完全的，然后我们构建了一个POVM的2D结果，这就足够了，从而表明，一个最小的PSI完全POVM有2D的结果。我们还考虑了仅具有秩一POVM元素的PSI完全POVM，并构造了具有3D-2结果的示例，这是量子比特的四面体测量的推广。在秩1的PSI-完全POVM中的最小数量的元素的问题是开放的。
Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. In the fixed measurement scheme four orthonormal bases are required even though there are far less parameters in a pure state. Peres conjectured that two unbiased bases suffice to determine a pure state up to some finite ambiguities. Here we shall at first prove Peres conjecture in the case of $d=3,4$, namely, two unbiased measurements determine a pure state up to to 6 and 16 candidates for a qutrit and ququad, respectively. And then, taking Peres' conjecture for established, w…
Preparation of pure and mixed polarization qubits and the direct measurement of figures of merit
非经典联合测量可以极大地提高量子系统的某些品质因数的测量效率。我们使用这样的测量来确定偏振量子比特的特定品质因数，即纯度。在这个过程中，我们强调了一些微妙的共同方法，在量子光学产生退相干。
Certifying unknown-basis pure-state entanglement dimensionality with few measurement settings
通过随机Pauli采样方法，提出了一种认证多量子比特纯态Schmidt数的高效算法，实现了样本复杂度的显著改进。
Sharing Bell nonlocality of bipartite high-dimensional pure states using only projective measurements
本文证明仅使用投影测量即可共享任意高维纯两体态的Bell非定域性，推广了已有两量子比特纯态的结果，深化了对高维量子态非定域性共享的理解。
We address measurement schemes where certain observables are chosen at random within a set of non-degenerate isospectral observables and then measured on repeated preparations of a physical system. Each observable has a given probability to be measured, and the statistics of this generalized measurement is described by a positive operator-valued measure (POVM). This kind of schemes are referred to as quantum roulettes since each observable is chosen at random, e.g. according to the fluctuating value of an external parameter. Here we focus on quantum roulettes for qubits involving the measureme…
Energy Measurements and Preparation of Canonical Phase States of a Nano-Mechanical Resonator
我们表明，一个连续的量子非爆破测量的能量的纳米机械谐振器可以实现通过监测谐振器的量子点接触通过库珀对盒。这种技术可以进一步用于制备两个谐振器的高度非经典态，例如正则相位参考态和所谓的“正午”态。
We propose a practical method for finding the canonical forms of arbitrary dimensional multipartite entangled states, either pure or mixed. By extending the technique developed in one of our recent works, the canonical forms for the mixed $N$-partite entangled states are constructed where they have inherited local unitary symmetries from their corresponding $N+1$ pure state counterparts. A systematic scheme to express the local symmetries of the canonical form is also presented, which provides a feasible way of verifying the local unitary equivalence for two multipartite entangled states.
If the system is known to be in one of two non-orthogonal quantum states, $|ψ_1\rangle$ or $|ψ_2\rangle$, $\mathcal{PT}$-symmetric quantum mechanics can discriminate them, \textit{in principle}, by a single measurement. We extend this approach by combining $\mathcal{PT}$-symmetric and Hermitian measurements and show that it's possible to distinguish an arbitrary number of pure quantum states by an appropriate choice of the parameters of $\mathcal{PT}$-symmetric Hamiltonian.
As an application of the simultaneous and continuous measurement of noncommutative observables formulated in our previous paper [C. Jiang and G. Watanabe, Phys. Rev. A 102, 062216 (2020)], we propose a scheme to generate the pure ideal quadrature-squeezed state in a one-dimensional harmonic oscillator system by the feedback control based on such type of measurement of noncommutative quadrature observables. We find that, by appropriately setting the strengths of the measurement and the feedback control, the pure ideal quadrature-squeezed state with arbitrary squeezedness can be produced. This i…
Continuous quantum measurement with observer: pure wavefunction evolution instead of decoherence
本文讨论了用弱耦合探测器（隧道点接触）对二能级系统（双点）的连续测量。虽然通常的治疗会导致逐渐的系统退相干由于测量，我们表明，测量结果的知识可以恢复纯波函数在任何时候（这可以通过实验验证）。该形式主义允许写一个简单的朗之万方程的随机演化的系统密度矩阵，这是反映和引起的随机检测器输出。该方程自然地描述了渐变波函数“突变”和量子芝诺效应。
Continuous quantum measurement with particular output: pure wavefunction evolution instead of decoherence
本文讨论了用弱耦合探测器（隧道点接触）对二能级系统（双点）的连续测量。虽然通常的治疗会导致逐渐的系统退相干由于测量，我们表明，测量结果的知识可以恢复纯波函数在任何时候（这可以通过实验验证）。该形式主义允许写一个简单的朗之万方程的随机演化的系统密度矩阵，这是反映和引起的随机检测器输出。该方程自然地描述了渐变波函数“突变”和量子芝诺效应。
We give a complete characterization for pure quantum measurements, i.e., for POVMs which are extremals in the convex set of all POVMs. Such measurements are free from classical noise. The characterization is valid both in discrete and continuous cases, and also in the case of an infinite Hilbert space. We show that sharp measurements are clean, i.e. they cannot be irreversibly connected to another POVMs via quantum channels and thus they are free from any additional quantum noise. We exhibit an example which demonstrates that this result could also be approximately true for pure measurements.
Canonical Typicality For Other Ensembles Than Micro-Canonical
本文将 Lévy 引理推广至 GAP 测度，证明了当密度矩阵最大特征值较小时测度集中成立，从而将量子统计力学中的正则典型性与动力学典型性推广至微正则系综以外的更一般系综。
Reconstructing a pure state of a spin s through three Stern-Gerlach measurements
考虑在纯状态下准备的旋转。结果表明，一般情况下，（2s+1）自旋分量沿空间三个方向的模量确定明确的状态。这些概率可以通过标准的Stern-Gerlach装置在实验上获得。因此，基于3（2s+1）个测量的强度来重建纯自旋态是可能的。
Hawking-Page phase transition for pure Lovelock black holes
研究纯Lovelock引力中静态球对称AdS黑洞的热力学性质，发现最小温度与HP相变温度的关系受维度和阶数修正，且归一化Ruppeiner标量曲率在HP相变处为仅依赖时空维度的普适常数。
The Gibbs canonical state, as a maximum entropy density matrix, represents a quantum system in equilibrium with a thermostat. This state plays an essential role in thermodynamics and serves as the initial condition for nonequilibrium dynamical simulations. We solve a long standing problem for computing the Gibbs state Wigner function with nearly machine accuracy by solving the Bloch equation directly in the phase space. Furthermore, the algorithms are provided yielding high quality Wigner distributions for pure stationary states as well as for Thomas-Fermi and Bose-Einstein distributions. The …
Two measurement bases are asymptotically informationally complete for any pure state tomography
本文证明对于有限深度Clifford+T电路生成的态及代数纯态，仅需两个测量基即可实现纯态层析的渐近信息完备性，为容错量子计算提供了理论保证。
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