C. A. M. de Melo, B. M. Pimentel, J. A. Ramirez
Here we deal in a pedagogical way with an approach to construct an algebraic structure for the Quantum Mechanical measurement processes from the concept of \emph{Measurement Symbol}. Such concept was conceived by Julian S. Schwinger and constitutes a fundamental piece in his variational formalism and its several applications.
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The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.
Graph states are the main computational building blocks of measurement-based computation and a useful tool for error correction in the gate model architecture. The graph states form a class of quantum states which are eigenvectors for the abelian group of stabilizer operators. They own topological properties, arising from their graph structure, including the presence of highly connected nodes, called hubs. Starting from hub nodes, we demonstrate how to efficiently express a graph state through the generators of the stabilizer group. We provide examples by expressing the ring and the star topol…
In this paper we attempt to explicate Salmon's idea of a causal process, as defined in terms of the mark method, in the context of C*-dynamical systems. We prove two propositions, one establishing mark manifestation infinitely many times along a given interval of the process, and, a second one, which establishes continuous manifestation of mark with the exception of a countable number of isolated points. Furthermore, we discuss how these results can be implemented in the context of Haag-Araki theories of relativistic quantum fields in Minkowski spacetime.
It's argued that Information-Theoretical restrictions for the systems selfdescription are important for Quantum Measurement Problem. They are described by O information system restricted states R formalism by Breuer and can be obtained also in Algebraic QM considering Segal Algebra of O observables. From Segal theorem it's shown that R describes the random measurement outcomes in the individual events.
We analyze quantum state tomography in scenarios where measurements and states are both constrained. States are assumed to live in a semi-algebraic subset of state space and measurements are supposed to be rank-one POVMs, possibly with additional constraints. Specifically, we consider sets of von Neumann measurements and sets of local observables. We provide upper bounds on the minimal number of measurement settings or outcomes that are required for discriminating all states within the given set. The bounds exploit tools from real algebraic geometry and lead to generic results that do not only…
We define a class of stochastic processes based on evolutions and measurements of quantum systems, and consider the complexity of predicting their long-term behavior. It is shown that a very general class of decision problems regarding these stochastic processes can be efficiently solved classically in the space-bounded case. The following corollaries are implied by our main result: (1) Any space O(s) uniform family of quantum circuits acting on s qubits and consisting of unitary gates and measurement gates defined in a typical way by matrices of algebraic numbers can be simulated by an unboun…
We study the mathematical structure of the notion of measurement space, which extends aspects of noncommutative topology that are based on quantale theory. This yields a geometric model of physical measurements that provides a realist picture, yet also operational, such that measurements and classical information arise interdependently as primitive concepts. A derived notion of classical observer caters for a mathematical formulation of Bohr's classical/quantum divide. Two important classes of measurement spaces are obtained, respectively from C*-algebras and from second-countable locally comp…
Okay, Roberts, Bartlett and Raussendorf recently introduced a new cohomological approach to contextuality in measurement based quantum computation. We give an abstract description of their obstruction and the algebraic structure it exploits, using the sheaf theoretic framework of Abramsky and Brandenburger. At this level of generality we contrast their approach to the Cech cohomology obstruction of Abramsky, Mansfield and Barbosa and give a direct proof that Cech cohomology is at least as powerful.
Markov processes play an important role in physics and the theory of open systems in particular. In this paper we study the asymptotic evolution of trace-nonincreasing homogenous quantum Markov processes (both types, discrete quantum Markov chains and continues quantum dynamical semigroups) equipped with a so-called strictly positive T -state in the Schrodinger and the Heisenberg picture. We derive a fundamental theorem specifying the structure of the asymptotic and uncover a rich set of transformations between attractors of quantum Markov processes in both pictures. Moreover, we generalize th…
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。
Von Neumann projections are the main operations by which information can be extracted from the quantum to the classical realm. They are however static processes that do not adapt to the states they measure. Advances in the field of adaptive measurement have shown that this limitation can be overcome by "wrapping" the von Neumann projectors in a higher-dimensional circuit which exploits the interplay between measurement outcomes and measurement settings. Unfortunately, the design of adaptive measurement has often been ad hoc and setup-specific. We shall here develop a unified framework for desi…
On the operational and algebraic quantum correlations
本文探讨了量子测量侵入性对相关函数定义的影响，提出新的不确定性关系，并揭示了操作性和代数相关性一致的等价条件。
Generalized symmetric informationally complete (SIC) measurements are SIC measurements that are not necessarily rank one. They are interesting originally because of their connection with rank-one SICs. Here we reveal several merits of generalized SICs in connection with quantum state tomography and Lie algebra that are interesting in their own right. These properties uniquely characterize generalized SICs among minimal IC measurements although, on the face of it, they bear little resemblance to the original definition. In particular, we show that in quantum state tomography generalized SICs ar…
Quantum Measurement and Continuous Markov Processes
本讲义介绍扩散型量子测量仪器，涵盖量子测量理论、连续马尔可夫过程及其在量子系统中的应用。
In this work, we consider the task of faithfully simulating a quantum measurement, acting on a joint bipartite quantum state, in a distributed manner. In the distributed setup, the constituent sub-systems of the joint quantum state are measured by two agents, Alice and Bob. A third agent, Charlie receives the measurement outcomes sent by Alice and Bob. Charlie uses local and pairwise shared randomness to compute a bivariate function of the measurement outcomes. The objective of three agents is to faithfully simulate the given distributed quantum measurement acting on the given quantum state wh…
Examples of symmetric informationally complete positive operator valued measures (SIC-POVMs) have been constructed in every dimension less than or equal to 67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl(d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characte…
Algebraic paradoxes in adaptive quantum computation
该研究解决了自适应测量量子计算中代数语境性的开放问题，证明自适应$\boldsymbol{\boldsymbol{Z}_2}$线性MBQC协议计算非仿射布尔函数时会出现代数悖论，可通过上同调检测，为量子优势的代数分析提供了关键方法。
Measurement-induced phase transition in the presence of competition between projective measurement and random unitary evolution has attracted increasing attention due to the rich phenomenology of entanglement structures. However, in open quantum systems with free fermions, a generalized measurement with conditional feedback can induce skin effect and render the system short-range entangled without any entanglement transition, meaning the system always remains in the ``area law'' entanglement phase. In this work, we demonstrate that the power-law long-range hopping does not alter the absence of…
Lattice systems with certain Lie algebraic or quantum Lie algebraic symmetries are constructed. These symmetric models give rise to series of integrable systems. As examples the $A_n$-symmetric chain models and the SU(2)-invariant ladder models are investigated. It is shown that corresponding to these $A_n$-symmetric chain models and SU(2)-invariant ladder models there are exactly solvable stationary discrete-time (resp. continuous-time) Markov chains with transition matrices (resp. intensity matrices) having spectra which coincide with the ones of the corresponding integrable models.
Algebraic Kolmogorov--Arnold representation theorem for quantum measurement
提出代数有界多项式版本的Kolmogorov-Arnold表示定理，证明无纠缠多量子比特乘积态的物理性质可精确分解为局部内观测量和浅层单变量多项式架构，并证明该表示对对抗性扰动具有稳定性。
We identify an unconventional algebraic scaling phase in the quantum dynamics of free fermions with long range hopping, which are exposed to continuous local density measurements. The unconventional phase is characterized by an algebraic entanglement entropy growth, and by a slow algebraic decay of the density-density correlation function, both with a fractional exponent. It occurs for hopping decay exponents $1< p \lesssim 3/2$ independently of the measurement rate. The algebraic phase gives rise to two critical lines, separating it from a critical phase with logarithmic entanglement growth a…
It is proposed the scheme of quantum mechanics, in which a Hilbert space and the linear operators are not primary elements of the theory. Instead of it certain variant of the algebraic approach is considered. The elements of noncommutative algebra (observables) and the nonlinear functionals on this algebra (physical states) are used as the primary constituents. The functionals associate with results of a particular measurement. It is suggested to consider certain ensembles of the physical states as quantum states of the standart quantum mechanics. It is shown that in such scheme the mathematic…
Parallel algebraic surgery for qLDPC codes with constant shuttling depth
本文提出提升积码的代数手术方法，协同设计逻辑测量、辅助码和原子穿梭，在中性原子处理器上实现常数穿梭深度的并行逻辑测量，显著加速典型逻辑操作并保持竞争性量子比特开销。
In this paper, we attempt to establish quantum measurement theory in the Heisenberg picture. First, we review foundations of quantum measurement theory, that is usually based on the Schrödinger picture. The concept of instrument is introduced there. Next, we define the concept of system of measurement correlations and that of measuring process. The former is the exact counterpart of instrument in the (generalized) Heisenberg picture. In quantum mechanical systems, we then show a one-to-one correspondence between systems of measurement correlations and measuring processes up to complete equival…
When a measurement is compatible with each of two other measurements that are incompatible with one another, these define distinct contexts for the given measurement. The Kochen-Specker theorem rules out models of quantum theory that satisfy a particular assumption of context-independence: that sharp measurements are assigned outcomes both deterministically and independently of their context. This notion of noncontextuality is not suited to a direct experimental test because realistic measurements always have some degree of unsharpness due to noise. However, a generalized notion of noncontextu…
Learning Hidden Structures in Open Quantum Dynamics
本文提出一种机器学习方法，用于在实验限制下识别开放量子动力学的隐藏结构，通过*代数描述可测量算子，从多时间测量序列中估计最大似然，识别非平凡的中间代数结构。
Generalized Catability of Relativistic Quantum States Measurement in a Unified Lie-Algebraic Foldy-Wouthuysen (FW) Framework
本文在统一李代数折叠-沃图伊斯恩框架中构建了广义可消性理论，研究了任意自旋相对量子系统的相干结构与量子干涉特性，并通过广义折叠-沃图伊斯恩变换分析了相对论费米子的可消性。
Operator quantum error-correction is a technique for robustly storing quantum information in the presence of noise. It generalizes the standard theory of quantum error-correction, and provides a unified framework for topics such as quantum error-correction, decoherence-free subspaces, and noiseless subsystems. This paper develops (a) easily applied algebraic and information-theoretic conditions which characterize when operator quantum error-correction is feasible; (b) a representation theorem for a class of noise processes which can be corrected using operator quantum error-correction; and (c)…
Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator algebra is introduced, which allows the construction of a Hilbert space as a representation space of a Heisenberg algebra, giving a noncommutative operator algebraic variant of the Koopman-von Neumann approach. In this form, the measurement theory for unary classical mechanics can be the same as and inform that for quantum mechanics, expanding classical mechan…
In this paper, effects of the regular measurements on a noisy channel has been investigated. The strategy introduced by A. Bayat, and Y. Omar [New J. Phys. 17, 103041 (2015)] is followed to suppress dephasing and dissipation effects in a noisy spin channel and generate long distance entanglement by global measurement on the channel. A regular global measurements performed on spin channel weakly coupled to the sender and receiver qubits via $XX$ interaction. This scheme is applied for the dephasing and dissipation in non-zero temperature processes separately and the results show that amounts of…
Lie algebraic quantum phase reduction based on heterodyne detection
本研究将量子相位约化方法扩展至外差检测，通过对所有正交可观测量进行均匀连续测量平均化测量反作用，实现了量子振荡器间无偏的同步评估，并揭示相位簇数量受玻色子能级限制。
In 1985, Leggett and Garg formulated a class of inequalities for testing the compatibility between macrorealism and quantum mechanics. In this paper, we point out that based on the same assumptions of macrorealism that are used in the derivation of Leggett-Garg inequalities (LGIs) , there is a scope of formulating another class of inequalities different from standard LGIs. By considering the three-time measurement scenario in a dichotomic system, we first propose an interesting variant of standard LGIs and show that its quantum violation is larger than the standard LGI. By extending this formu…
In this mainly pedagogical article, we discuss under what circumstances measurements play a special role in quantum processes. In particular we discuss the following facts which appear to be a common area of confusion: i) from a fundamental point of view measurements play no special role whatsoever: all dynamics that can be generated by measurements can be generated by unitary processes (for which post-selection is no exception), ii) from a purely physical point of view, measurements are not "outside" of quantum mechanics, iii) the only difference between the abilities of measurement-based pro…
In this paper we introduce a novel notion of probabilistic bisimulation for quantum processes and prove that it is congruent with respect to various process algebra combinators including parallel composition even when both classical and quantum communications are present. We also establish some basic algebraic laws for this bisimulation. In particular, we prove uniqueness of the solutions to recursive equations of quantum processes, which provides a powerful proof technique for verifying complex quantum protocols.
In this contribution, we show that the use of conditional measurements in the resonant interaction of two quantized electromagnetic fields gives rise to nonclassical multiphoton processes. Furthermore, we demonstrate that this phenomenon may enable a robust integrated-optics protocol to engineer quantum states containing a high number of photons, thus making it a potentially appealing platform for exploring mesoscopic quantum phenomena.
Using a very simple Gedankenexperiment, I remind the reader that (contrary to what happens in classical mechanics) the energy of a quantum system is inevitably increased just by performing (some) textbook measurements on it. As a direct conclusion, this means that some measurements require the expenditure of a finite amount of energy to be carried out. I also argue that this makes it very difficult to regard measurements as disembodied, immaterial, informational operations, and it forces us to look at them as physical processes just like any other one.
In this paper, we investigate how to reduce the number of measurement configurations needed for sufficiently precise entanglement quantification. Instead of analytical formulae, we employ artificial neural networks to predict the amount of entanglement in a quantum state based on results of collective measurements (simultaneous measurements on multiple instances of the investigated state). This approach allows us to explore the precision of entanglement quantification as a function of measurement configurations. For the purpose of our research, we consider general two-qubit states and their ne…
Quantum measurement not only can destroy coherence but also can create it. Here, we estimate the maximum amount of coherence, one can create under a complete non-selective measurement process. For our analysis, we consider projective as well as POVM measurements. Based on our observations, we characterize the measurement processes into two categories, namely, the measurements with the ability to induce coherence and the ones without this ability. Our findings also suggest that the more POVM elements present in a measurement that acts on the quantum system, the less will be its coherence creati…
One of the broadest concepts of measurement in quantum theory is the generalized measurement. Another paradigm of measurement--arising naturally in quantum optics, among other fields--is that of continuous-time measurements, which can be seen as the limit of a consecutive sequence of weak measurements. They are naturally described in terms of stochastic processes, or time-dependent random variables. We show that any generalized measurement can be decomposed as a sequence of weak measurements with a mathematical limit as a continuous stochastic process. We give an explicit construction for any …
Information-Theoretical restrictions on the information transfer are applied to Quantum Measurements. For the measurement of quantum object S by information system O this restrictions are calculated in Algebraic QM formalism as the inference map to the (sub)algebra of O observables.
Exclusive-or encoded algebraic structure for efficient quantum dynamics
本研究提出基于异或编码的量子代数形式体系ORQA，利用李代数元素索引的二进制表示简化运算，结合动态截断实现稀疏密度算符，可近线性高效计算多种量子动力学，并通过22比特最大独立集量子退火模拟验证了概念可行性。
The sensitivity of a repeated measurement-based purification scheme to additional undesired couplings is analyzed, focusing on the very simple and archetypical system consisting of two two-level systems interacting with a repeatedly measured one. Several regimes are considered and in the strong coupling (i.e., when the coupling constant of the undesired interaction is very large) the occurrence of a quantum Zeno effect is proven to dramatically jeopardize the efficiency of the purification process.
We propose a new structure of ensembles in quantum theory, based on the recently introduced intrinsic properties of electrons and photons. On this statistical basis the spreading of a wave-packet, collapse of the wave function, the quantum eraser, and interaction-free measurements are re-analyzed and the usual conceptual problems removed.
In this paper we will give a short presentation of the quantum Levy-Khinchin formula and of the formulation of quantum continual measurements based on stochastic differential equations, matters which we had the pleasure to work on in collaboration with Prof. Holevo. Then we will begin the study of various entropies and relative entropies, which seem to be promising quantities for measuring the information content of the continual measurement under consideration and for analysing its asymptotic behaviour.
Quantum Algebraic Diversity: Single-Copy Density Matrix Estimation via Group-Structured Measurements
将代数多样性框架扩展到量子测量，提出量子代数多样性定理，通过群结构POVM从单副本量子态估计密度矩阵，实现高保真度，并建立经典-量子对偶映射和最优性继承定理。
Lyapunov Densities For Markov Processes: An Application To Quantum Systems With Non-Demolition Measurements
基于对偶 Lyapunov 方法分析离散时间马尔可夫过程的随机收敛性，证明正常次不变函数可作为依概率收敛的证书，并将其应用于非破坏性测量量子系统以证明扫出性。
We outline a framework for describing photo-activated biological reactions as generalized quantum measurements of external fields, for which the biological system takes on the role of a quantum meter. By using general arguments regarding the Hamiltonian that describes the measurement interaction, we identify the cases where it is essential for a complex chemical or biological system to exhibit non-equilibrium quantum coherent dynamics in order to achieve the requisite functionality. We illustrate the analysis by considering measurement of the solar radiation field in photosynthesis and measure…
Statistical Interpretation of the Procedures Measurement of Physical Quantities
本文通过重新组织和综合现有模型，提出量子物理基础的框架，强调实验可行程序和实验室实践的统计模型，为量子力学的公理化重述提供基础。
When an experimentalist measures a time series of qubits, the outcomes generate a classical stochastic process. We show that measurement induces high complexity in these processes in two specific senses: they are inherently unpredictable (positive Shannon entropy rate) and they require an infinite number of features for optimal prediction (divergent statistical complexity). We identify nonunifilarity as the mechanism underlying the resulting complexities and examine the influence that measurement choice has on the randomness and structure of measured qubit processes. We introduce new quantitat…
This brief article gives an overview of quantum mechanics as a {\em quantum probability theory}. It begins with a review of the basic operator-algebraic elements that connect probability theory with quantum probability theory. Then quantum stochastic processes is formulated as a generalization of stochastic processes within the framework of quantum probability theory. Quantum Markov models from quantum optics are used to explicitly illustrate the underlying abstract concepts and their connections to the quantum regression theorem from quantum optics.
Efficient Gradient Estimation for Parameterized Quantum Systems with Lie Algebraic Symmetries
提出一种利用李代数结构和Hadamard测试的梯度估计框架，通过阴影层析技术实现测量次数随参数对数增长，指数级降低测量成本并多项式加速时间。
Tomography by Design: An Algebraic Approach to Low-Rank Quantum States
提出一种代数算法，通过测量特定可观测量估计密度矩阵的结构化条目，并利用低秩假设通过数值线性代数完成矩阵，实现高效且确定性的量子态层析。
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