Quantum Logic

Turing machines based on unsharp quantum logic

In this paper, we consider Turing machines based on unsharp quantum logic. For a lattice-ordered quantum multiple-valued (MV) algebra E, we introduce E-valued non-deterministic Turing machines (ENTMs) and E-valued deterministic Turing machines (EDTMs). We discuss different E-valued recursively enumerable languages from width-first and depth-first recognition. We find that width-first recognition is equal to or less than depth-first recognition in general. The equivalence requires an underlying E value lattice to degenerate into an MV algebra. We also study variants of ENTMs. ENTMs with a class…

Experimental characterisation of unsharp qubit observables and sequential measurement incompatibility via quantum random access codes

Unsharp measurements are increasingly important for foundational insights in quantum theory and quantum information applications. Here, we report an experimental implementation of unsharp qubit measurements in a sequential communication protocol, based on a quantum random access code. The protocol involves three parties; the first party prepares a qubit system, the second party performs operations which return both a classical and quantum outcome, and the latter is measured by the third party. We demonstrate a nearly-optimal sequential quantum random access code that outperforms both the best …

On the central limit theorem for unsharp quantum random variables

In this letter we study the weak-convergence properties of random variables generated by unsharp quantum measurements. More precisely, for a sequence of random variables generated by repeated unsharp quantum measurements, we study the limit distribution of relative frequency. We provide the de Finetti-type representation theorem for all separable states, showing that the distribution can be well approximated by mixture of normal distributions. No symmetry restrictions, such as the permutational invariance were needed. Furthermore, we investigate the convergence rates and show that the relative…

Unsharp Quantum Reality

The positive operator (valued) measures (POMs) allow one to generalize the notion of observable beyond the traditional one based on projection valued measures (PVMs). Here, we argue that this generalized conception of observable enables a consistent notion of unsharp reality and with it an adequate concept of joint properties. A sharp or unsharp property manifests itself as an element of sharp or unsharp reality by its tendency to become actual or to actualize a specific measurement outcome. This actualization tendency - or potentiality - of a property is quantified by the associated quantum p…

Precise quantum control via unsharp measurements and feedback operations

In this paper, we propose a scheme to eliminate the influence of noises on system dynamics, by means of a sequential unsharp measurements and unitary feedback operations. The unsharp measurements are carried out periodically during system evolution, while the feedback operations are well designed based on the eigenstates of the density matrices of the exact (noiseless) dynamical states and its corresponding post-measurement states. For illustrative examples, we show that the dynamical trajectory errors caused by both static and non-static noises are successfully eliminated in typical two-level…

利用连续量子非锐测量攻击窃听量子密钥分发网络

Eavesdropping a Quantum Key Distribution network using sequential quantum unsharp measurement attacks

该研究分析多个连续窃听者对单边设备无关QKD网络实施量子非锐测量攻击，表明他们可在保留Bob正密钥率的同时获取秘密信息，特定情形下窃听者数量甚至可为无限。

Semi-device-independent self-testing of unsharp measurements

Unsharp quantum measurements provide a resource in scenarios where one faces the trade-off between information gain and disturbance. In this work we introduce a prepare-transform-measure scenario in which two-outcome unsharp measurements outperform their sharp counterparts, as well as any stochastic strategy involving dichotomic projective measurements. Based on that, we propose a scheme for semi-device-independent self-testing of unsharp measurements and show that all two-outcome qubit measurements can be characterized in a robust way. Along with the main result, in this work we introduce a m…

Robust certification of unsharp instruments through sequential quantum advantages in a prepare-measure communication game

Communication games are one of the widely used tools that are designed to demonstrate quantum supremacy over classical resources. In that, two or more parties collaborate to perform an information processing task to achieve the highest success probability of winning the game. We propose a specific two-party communication game in the prepare-measure scenario that relies on an encoding-decoding task of specific information. We first demonstrate that quantum theory outperforms the classical preparation non-contextual theory, and the optimal quantum success probability of such a communication game…

Certification of three black boxes with unsharp measurements using $3 \rightarrow 1 $ sequential quantum random access codes

Unsharp measurements play an increasingly important role in quantum information theory. In this paper, we study a three-party prepare-transform-measure experiment with unsharp measurements based on $ 3 \rightarrow 1 $ sequential random access codes (RACs). We derive optimal trade-off between the two correlation witnesses in $ 3 \rightarrow 1 $ sequential quantum random access codes (QRACs), and use the result to complete the self-testing of quantum preparations, instruments and measurements for three sequential parties. We also give the upper and lower bounds of the sharpness parameter to comp…

Noise-robust preparation contextuality shared between any number of observers via unsharp measurements

Multiple observers who independently harvest nonclassical correlations from a single physical system share the system's ability to enable quantum correlations. We show that any number of independent observers can share the preparation contextual outcome statistics enabled by state ensembles in quantum theory. Furthermore, we show that even in the presence of any amount of white noise, there exists quantum ensembles that enable such shared preparation contextuality. The findings are experimentally realised by applying sequential unsharp measurements to an optical qubit ensemble which reveals th…

利用不锐非局域乘积测量激活量子导引共享

Activation of quantum steering sharing with unsharp nonlocal product measurements

针对局域不锐测量难以支持非局域量子导引共享的问题，本文提出基于量子椭球构造最优不锐非局域乘积测量的方法，并证明其可在 GHZ 态中激活导引共享且无需复制态或损耗其他关联。

A sequence of unsharp measurements enabling a real time visualization of a quantum oscillation

The normalized state $\ket{ψ(t)}=c_1(t)\ket{1}+c_2(t)\ket{2}$ of a single two-level system performs oscillations under the influence of a resonant driving field. It is assumed that only one realization of this process is available. We show that it is possible to approximately visualize in real time the evolution of the system as far as it is given by $|c_2(t)|^2$. For this purpose we use a sequence of particular unsharp measurements separated in time. They are specified within the theory of generalized measurements in which observables are represented by positive operator valued measures (POVM…

Unsharp continuous measurement of a Bose-Einstein condensate: full quantum state estimation and the transition to classicality

We study a Bose-Einstein condensate (BEC) in a double-well potential subject to an unsharp continuous measurement of the atom number in one of the two wells. We investigate the back action of the measurement on the quantum dynamics and the viability to monitor the ensuing time evolution. For vanishing inter-atomic interactions, mainly the expectation values of the measured local observable can be inferred from the measurement record. Conversely, in the presence of moderate inter-atomic interactions, the entire many-body state --modified by the measurement-- is monitored with unit fidelity and,…

Probing various formulations of macrorealism for unsharp quantum measurements

Standard Leggett and Garg inequalities (SLGIs) were formulated for testing the incompatibility between the classical worldview of macrorealism and quantum mechanics. In recent times, various other formulations, such as Wigner form of LGIs (WLGIs), entropic LGIs (ELGIs) and the no-signaling in time (NSIT) condition have also been proposed. It is also recently argued that no set of SLGIs can provide the necessary and sufficient conditions for macrorealism but a suitable conjunction of NSIT conditions provides the same. In this paper, we first provide a comparative study of the various formulatio…

非锐测量的设备无关自检测

Device-independent self-testing of unsharp measurements

本文利用序贯 Bell 测试和平方和方法，实现非锐仪器及非锐参数的设备无关自检测，并证明其对噪声鲁棒。

Insolubility of the Quantum Measurement Problem for Unsharp Observables

The quantum mechanical measurement problem is the difficulty of dealing with the indefiniteness of the pointer observable at the conclusion of a measurement process governed by unitary quantum dynamics. There has been hope to solve this problem by eliminating idealizations from the characterization of measurement. We state and prove two `insolubility theorems' that disappoint this hope. In both the initial state of the apparatus is taken to be mixed rather than pure, and the correlation of the object observable and the pointer observable is allowed to be imperfect. In the {\it insolubility the…

EPR-Bell Tests with Unsharp Observables and Relativistic Quantum Measurement

In this contribution I will review the analysis of the Einstein-Podolsky-Rosen argument, Bell's inequalities and of associated experiments for spins in terms of positive operator valued measures. Specifically, I will explore the relation between the Clauser-Horne-Shimony-Holt inequality and a fundamental \emph{classicality} property of observables -- their \emph{coexistence}. A derivation of Bell's inequalities for unsharp spins will be given which follows a reconstruction by Mittelstaedt and Stachow of the original EPR argument. In this context we discuss the need for a consistent relativisti…

Unsharp eigenvalues and quantum contextuality

The Kochen-Specker theorem, Bell inequalities, and several other tests that were designed to rule out hidden-variable theories, assume the existence of observables having infinitely sharp eigenvalues. A paradigmatic example is spin-1/2. It is measured with a Stern-Gerlach array whose outputs are divided into two classes, spin-up and spin-down, in correspondence to the two spots observed on a detection screen. The spot's finite size is attributed to imperfections of the measuring device. This assumption turns the experimental output into a dichotomic, discrete one, thereby allowing the assignme…

利用非精确测量实现量子图像阈值化的新方法

A Novel Approach to Threshold Quantum Images by using Unsharp Measurements

提出一种基于直方图峰谷和UM参数的混合量子阈值化方法，通过迭代多阈值和高效QC实现灰度图像二值化，降低算法复杂度。

Unsharp pointer observables and the structure of decoherence

The theory of decoherence attempts to explain the emergent classical behaviour of a quantum system interacting with its quantum environment. In order to formalize this mechanism we introduce the idea that the information preserved in an open quantum evolution (or channel) can be characterized in terms of observables of the initial system. We use this approach to show that information which is broadcast into many parts of the environment can be encoded in a single observable. This supports a model of decoherence where the pointer observable can be an arbitrary positive operator-valued measure (…

Wigner's form of the Leggett-Garg inequality, No-Signalling in Time, and Unsharp Measurements

Wigner's form of the local realist inequality is used to derive its temporal version for an oscillating two-level system involving two-time joint probabilities. Such an inequality may be regarded as a novel form of the Leggett-Garg inequality (LGI) constituting a necessary condition for macrorealism. The robustness of its quantum mechanical (QM) violation against unsharpness of measurement is investigated by using a suitable model of unsharp measurements. It is found that there exists a range of values of the sharpness parameter (characterizing precision of the relevant measurements) for which…

Unsharp Localization and Causality in Relativistic Quantum Theory

The conflict between relativistic causality and localizability is analyzed in the light of the existence of unsharp localization observables. A theorem due to S. Schlieder is generalized, showing that the assumption of local commutativity implies the localization observable in question to be unsharp in a strong sense. Furthermore, a recent generalization of a theorem of Lüders is applied to demonstrate that local commutativity is a necessary consequence of Einstein causality even in the case of unsharp observables if they admit local measurements. These findings raise the question whether loca…

Quantum Computational Logics. A Survey

Quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, a system of qubits, representing a possible pure state of a compound quantum system. The generalization to mixed states, which might be useful to analyse entanglement-phenomena, is due to Gudder. Quantum computational logics represent non standard examples of unsharp quantum logic, where the non-contradiction principle is violated, while conjunctions and disjunctions are strongly non-idempotent. I…

Unsharp measurements, joint measurability and classical distributions for some qudits

Classicality associated with joint measurability of operators manifests through a valid classical joint probability distribution on measurement outcomes. For qudits in dimension $n$, where $n$ is prime or power of prime, we present a method to construct unsharp versions of projective measurement operators which results in a geometric description of the set of quantum states for which the operators engender a classical joint probability distribution, and are jointly measurable. Specifically, within the setting of a generalised Bloch sphere in $n^2-1$ dimensions, we establish that the constructe…

No-Cloning Theorem on Quantum Logics

This paper discusses the no-cloning theorem in a logico-algebraic approach. In this approach, an orthoalgebra is considered as a general structure for propositions in a physical theory. We proved that an orthoalgebra admits cloning operation if and only if it is a Boolean algebra. That is, only classical theory admits the cloning of states. If unsharp propositions are to be included in the theory, then a notion of effect algebra is considered. We proved that an atomic Archimedean effect algebra admitting cloning operation is a Boolean algebra. This paper also presents a partial result indicati…

通过序贯噪声测量恢复资源

Recovery of resources through sequential noisy measurements

本文提出在量子网络中利用辅助量子比特序贯执行最优非锐利噪声测量，可像投影测量一样实现节点间纠缠局域化并缓解噪声影响，为制备量子资源态及提升安全通信控制权提供了新方法。

Joint measurability in nonequilibrium quantum thermodynamics

In this Letter we investigate the concept of quantum work and its measurability from the viewpoint of quantum measurement theory. Very often, quantum work and fluctuation theorems are discussed in the framework of projective two-point measurement (TPM) schemes. According to a well known no-go theorem, there is no work observable which satisfies both (i) an average work condition and (ii) the TPM statistics for diagonal input states. Such projective measurements represent a restrictive class among all possible measurements. It is desirable, both from a theoretical and experimental point of view…

Quantum Physics and Classical Physics - in the Light of Quantum Logic

In contrast to the Copenhagen interpretation we consider quantum mechanics as universally valid and query whether classical physics is really intuitive and plausible. - We discuss these problems within the quantum logic approach to quantum mechanics where the classical ontology is relaxed by reducing metaphysical hypotheses. On the basis of this weak ontology a formal logic of quantum physics can be established which is given by an orthomodular lattice. By means of the Soler condition and Piron's result one obtains the classical Hilbert spaces. - However, this approach is not fully convincing.…

Qubit semantics and quantum trees

In the qubit semantics the \emph{meaning} of any sentence $α$ is represented by a \emph{quregister}: a unit vector of the $n$--fold tensor product $\otimes^n \C^2$, where $n$ depends on the number of occurrences of atomic sentences in $α$. The logic characterized by this semantics, called {\it quantum computational logic} (QCL), is {\it unsharp}, because the non-contradiction principle is violated. We show that QCL does not admit any logical truth. In this framework, any sentence $α$ gives rise to a \emph{quantum tree}, consisting of a sequence of unitary operators. The quantum tree of $α$ can…

热时间作为一种非锐观测量

Thermal time as an unsharp observable

本文证明量子谐振子的Connes-Rovelli热时间可视为非锐观测量（正算子值测度），并推广至一维自由无质量相对论粒子及非交换积分平衡态系统。

The complementarity of quantum observables: theory and experiment

Three notions of complementarity - operational, probabilistic, and value complementarity - are reanalysed with respect to the question of joint measurements and compared with reference to some examples of canonically conjugate observables. It is shown that the joint measurability of noncommuting observables is a consequence of the quantum formalism if unsharp observables are taken into account; a fact not in conflict with the idea of complementarity, which, in its strongest version, was originally formulated only for sharp observables. As an illustration of the general theory, the wave-particl…

Unsharp Values, Domains and Topoi

The so-called topos approach provides a radical reformulation of quantum theory. Structurally, quantum theory in the topos formulation is very similar to classical physics. There is a state object, analogous to the state space of a classical system, and a quantity-value object, generalising the real numbers. Physical quantities are maps from the state object to the quantity-value object -- hence the `values' of physical quantities are not just real numbers in this formalism. Rather, they are families of real intervals, interpreted as `unsharp values'. We will motivate and explain these aspects…

Maintaining Quantum Coherence in the Presence of Noise through State Monitoring

Unsharp POVM measurements allow the estimation and tracking of quantum wavefunctions in real-time with minimal disruption of the dynamics. Here we demonstrate that high fidelity state monitoring, and hence quantum control, is possible even in the presence of classical dephasing and amplitude noise, by simulating such measurements on a two-level system undergoing Rabi oscillations. Finite estimation fidelity is found to persist indefinitely long after the decoherence times set by the noise fields in the absence of measurement.

The status of determinism in proofs of the impossibility of a noncontextual model of quantum theory

In order to claim that one has experimentally tested whether a noncontextual ontological model could underlie certain measurement statistics in quantum theory, it is necessary to have a notion of noncontextuality that applies to unsharp measurements, i.e., those that can only be represented by positive operator-valued measures rather than projection-valued measures. This is because any realistic measurement necessarily has some nonvanishing amount of noise and therefore never achieves the ideal of sharpness. Assuming a generalized notion of noncontextuality that applies to arbitrary experiment…

The Nondemolition Measurement of Quantum Time

The problem of time operator in quantum mechanics is revisited. The unsharp measurement model for quantum time based on the dynamical system-clock interaction, is studied. Our analysis shows that the problem of the quantum time operator with continuous spectrum cannot be separated from the measurement problem for quantum time.

模糊测量与自适应高斯POVM用于量子启发式图像处理

Unsharp Measurement with Adaptive Gaussian POVMs for Quantum-Inspired Image Processing

本文提出了一种数据自适应的概率强度重映射框架，用于保持结构的灰度图像变换。通过将强度变换建模为连续的数据驱动重映射过程，该方法在传统直方图方法基础上实现了更平滑的过渡。通过高斯加权方法概率性地分配每个像素到多个组件，从而在保持结构特征的同时实现平滑过渡。

Saturation of repeated quantum measurements

We study sequential measurement scenarios where the system is repeatedly subjected to the same measurement process. We first provide examples of such repeated measurements where further repetitions of the measurement do not increase our knowledge on the system after some finite number of measurement steps. We also prove, however, that repeating the Lüders measurement of an unsharp two-outcome observable never saturates in this sense, and we characterize the observable measured in the limit of infinitely many repetitions. Our result implies that a repeated measurement can be used to correct the…

量化测量客观性：一种回溯预测方法

Quantifying Measurement Objectivity: A Retrodictive Approach

该研究针对量子测量结果何时可解读为揭示预先存在的客观属性的问题，提出基于量子测量回溯预测的定量方法，构造量化非客观性的双线性型并证明其分解性质，得出总非客观性在特定映射下不降低的结论。

What is orthodox quantum mechanics?

What is called "orthodox" quantum mechanics, as presented in standard foundational discussions, relies on two substantive assumptions --- the projection postulate and the eigenvalue-eigenvector link --- that do not in fact play any part in practical applications of quantum mechanics. I argue for this conclusion on a number of grounds, but primarily on the grounds that the projection postulate fails correctly to account for repeated, continuous and unsharp measurements (all of which are standard in contemporary physics) and that the eigenvalue-eigenvector link implies that virtually all interes…

Quantum Logic and Quantum Reconstruction

Quantum logic understood as a reconstruction program had real successes and genuine limitations. This paper offers a synopsis of both and suggests a way of seeing quantum logic in a larger, still thriving context.

Quantum logic as a sum over classical logic gates

It is shown that certain natural quantum logic gates, {\it i.e.} unitary time evolution matrices for spin-\frac{1}{2} quantum spins, can be represented as sums, with appropriate phases, over classical logic gates, in a direct analogy with the Feynman path integral representation of quantum mechanics. On the other hand, it is shown that a natural quantum gate obtained by analytically continuing the transfer matrix of the anisotropic nearest-neighbour Ising model to imaginary time, does not admit such a representation.

Quantum logic network for probabilistic cloning the quantum states

We construct efficient quantum logic network for probabilistic cloning the quantum states used in implemented tasks for which cloning provides some enhancement in performance.

Hoare Logic for Quantum Programs

Hoare logic is a foundation of axiomatic semantics of classical programs and it provides effective proof techniques for reasoning about correctness of classical programs. To offer similar techniques for quantum program verification and to build a logical foundation of programming methodology for quantum computers, we develop a full-fledged Hoare logic for both partial and total correctness of quantum programs. It is proved that this logic is (relatively) complete by exploiting the power of weakest preconditions and weakest liberal preconditions for quantum programs.

Dynamic Quantum Logic for Quantum Programs

We present a way to apply quantum logic to the study of quantum programs. This is made possible by using an extension of the usual propositional language in order to make transformations performed on the system appear explicitly. This way, the evolution of the system becomes part of the logical study. We show how both unitary operations and two-valued measurements can be included in this formalism and can thus be handled logically.

Logical structures underlying quantum computing

In this work we advance a generalization of quantum computational logics capable of dealing with some important examples of quantum algorithms. We outline an algebraic axiomatization of these structures.

Model-Theoretic Investigations into Consequence Operation (Cn) in Quantum Logics. An Algebraic Approach

In this paper we present the fundamentals of the so-called algebraic approach to propositional quantum logics. We define the set of formulas describing quantum reality as a free algebra freely generated by the set of quantum propositional variables. We define the general notion of logic as a structural consequence operation. Next we introduce the concept of logical matrices understood as a models of quantum logics. We give the definitions of two quantum consequence operations defined in these models.

Simulation of Markovian quantum dynamics on quantum logic networks

We study how generators of Markovian dynamics of a qubit can be simulated using a programmable quantum processor.

Superlogic Manifolds and Geometric approach to Quantum Logic

The main purpose of this paper is to present a new approach to logic or what we will call superlogic. This approach constitutes a new way of looking at the connection between quantum mechanics and logic. It is a {\it geometrisation} of the quantum logic. Note that this superlogic is not distributive reflecting a good propriety to describe quantum mechanics, non commutative spaces and contains a nilpotent element.

Relation of the Bell inequalities with quantum logic, hidden variables and information theory

I review the relation of the Bell inequalities - characteristic of (classical) probabilities defined on Boolean logics - with noncontextual and local hidden variables theories of quantum mechanics and with quantum information.

量子结构作为生成分数：分区逻辑、生成逻辑与审美形式

Quantum Structures as Generative Scores: Partition Logic, Generative Logic, and Aesthetic Form

通过将有限分区逻辑转化为基于Prolog的简单生成逻辑语法，连接分区逻辑与生成逻辑。以五原子V-逻辑L_{12}生成模块化视觉艺术品量子正方形，实现逻辑结构与视觉、文本或声音表现的分离。

Fuzzy Propositional Logic Associated with Quantum Computational Gates

We apply residuated structures associated with fuzzy logic to develop certain aspects of information processing in quantum computing from a logical perspective. For this purpose, we introduce an axiomatic system whose natural interpretation is the irreversible quantum Poincare structure.

A Practical Method of Constructing Quantum Combinational Logic Circuits

We describe a practical method of constructing quantum combinational logic circuits with basic quantum logic gates such as NOT and general $n$-bit Toffoli gates. This method is useful to find the quantum circuits for evaluating logic functions in the form most appropriate for implementation on a given quantum computer. The rules to get the most efficient circuit are utilized best with the aid of a Karnaugh map. It is explained which rules of using a Karnaugh map are changed due to the difference between the quantum and classical logic circuits.

A Lecture on Quantum Logic Gates

In this note we make a short review of constructions of n-repeated controlled unitary gates in quantum logic gates.

A Weakly Intuitionistic Quantum Logic

In this paper we motivate and study the possibility of an intuitionistic quantum logic. An explicit investigation of the application of the theory of Bruns and Lakser on distributive hulls on traditional quantum logic (as suggested in [Coe]) leads us to a small modification of this scheme. In this way we obtain a weak Heyting algebra (cf. [CJ]) for describing the language of quantum mechanics.