This article considers quantum systems described by a finite-dimensional complex Hilbert space $H$. We first define the concept of a finite observable on $H$. We then discuss ways of combining observables in terms of convex combinations, post-processing and sequential products. We also define complementary and coexistent observables. We then introduce finite instruments and their related compatible observables. The previous combinations and relations for observables are extended to instruments and their properties are compared. We present four types of instruments; namely, identity, trivial, Lüders and Kraus instruments. These types are used to illustrate different ways that instruments can act. We next consider joint probabilities for observables and instruments. The article concludes with a discussion of measurement models and the instruments they measure.
Incompatibility of quantum devices is one of the cornerstones of quantum theory, and the incompatibility of quantum measurements and channels has been linked to quantum advantage in certain information theoretic tasks. In this work, we focus on the less well-explored question of the incompatibility of quantum instruments, that is, devices that describe the measurement process in its entirety, accounting for both the classical measurement outcome and the quantum post-measurement state. In particular, we focus on the recently introduced notion of parallel compatibility of instruments, which has …
In quantum information processing quantum operations are often processed alongside measurements which result in classical data. Due to the information gain of classical measurement outputs non-unitary dynamical processes can take place on the system, for which common quantum channel descriptions fail to describe the time evolution. Quantum measurements are correctly treated by means of so-called quantum instruments capturing both classical outputs and post-measurement quantum states. Here we present a general recipe to characterize quantum instruments alongside its experimental implementation …
Studying sequential measurements is of the utmost importance to both the foundational aspects of quantum theory and the practical implementations of quantum technologies, with both of these applications being abstractly described by the concatenation of quantum instruments into a sequence of certain length. In general, the choice of instrument at any given step in the sequence can be conditionally chosen based on the classical results of all preceding instruments. For two instruments in a sequence we consider the conditional second instrument as an effective way of post-processing the first in…
We introduce the concepts of dual instruments and sub-observables. We show that although a dual instruments measures a unique observable, it determines many sub-observables. We define a unique minimal extension of a sub-observable to an observable and consider sequential products and conditioning of sub-observables. Sub-observable effect algebras are characterized and studied. Moreover, the convexity of these effect algebras is considered. The sequential product of instruments is discussed. These concepts are illustrated with many examples of instruments. In particular, we discuss Lüders, Hole…
This article points out that observables and instruments can be combined in many ways that have natural and physical interpretations. We shall mainly concentrate on the mathematical properties of these combinations. Section~1 reviews the basic definitions and observables are considered in Section~2. We study parts of observables, post-processing, generalized convex combinations, sequential products and tensor products. These combinations are extended to instruments in Section~3. We consider properties of observables measured by combinations of instruments. We introduce four special types of in…
Quantum Random Access Codes (QRACs) are key tools for a variety of protocols in quantum information theory. These are commonly studied in prepare-and-measure scenarios in which a sender prepares states and a receiver measures them. Here, we consider a three-party prepare-transform-measure scenario in which the simplest QRAC is implemented twice in sequence based on the same physical system. We derive optimal trade-off relations between the two QRACs. We apply our results to construct semi-device independent self-tests of quantum instruments, i.e. measurement channels with both a classical and …
We formally extend the notion of Markov order to open quantum processes by accounting for the instruments used to probe the system of interest at different times. Our description recovers the classical Markov order property in the appropriate limit: when the stochastic process is classical and the instruments are non-invasive, \emph{i.e.}, restricted to orthogonal, projective measurements. We then prove that there do not exist non-Markovian quantum processes that have finite Markov order with respect to all possible instruments; the same process exhibits distinct memory effects with respect to…
Resource theory of interactive quantum instruments
H. H. Wills Physics Laboratory, University of Bristol（布里斯托大学）；Physics Department and NanoLund, Lund University（隆德大学）；Department of Physics, National Taiwan Normal University（台湾师范大学）
In Ref. [Phys. Rev. A 100, 062317 (2019)], the authors reported an algorithm to implement, in a circuit-based quantum computer, a general quantum measurement (GQM) of a two-level quantum system, a qubit. Even though their algorithm seems right, its application involves the solution of an intricate non-linear system of equations in order to obtain the angles determining the quantum circuit to be implemented for the simulation. In this article, we identify and discuss a simple way to circumvent this issue and implement GQMs on any $d$-level quantum system through quantum state preparation algori…
Observables and instruments have played significant roles in recent studies on the foundations of quantum mechanics. Sequential products of effects and conditioned observables have also been introduced. After an introduction in Section~1, we review these concepts in Section~2. Moreover, it is shown how these ideas can be unified within the framework of measurement models. In Section~3, we illustrate these concepts and their relationships for the simple example of a qubit Hilbert space. Conditioned observables and their distributions are studied in Section~4. Section~5 considers joint probabili…
Instrument-based quantum resources: quantification, hierarchies and constructing resource theories
The Institute of Mathematical Sciences（数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Indian Institute of Technology Bhubaneswar（布巴内斯瓦尔印度理工学院）；Korea Research Institute of Standards and Science（韩国标准科学研究院）
Among certification techniques, those based on the violation of Bell inequalities are appealing because they do not require assumptions on the underlying Hilbert space dimension and on the accuracy of calibration methods. Such device-independent techniques have been proposed to certify the quality of entangled states, unitary operations, projective measurements following von Neumann's model and rank-one positive-operator-valued measures (POVM). Here, we show that they can be extended to the characterization of quantum instruments with post-measurement states that are not fully determined by th…
The duration, strength and structure of memory effects are crucial properties of physical evolution. Due to the invasive nature of quantum measurement, such properties must be defined with respect to the probing instruments employed. Here, using a photonic platform, we experimentally demonstrate this necessity via two paradigmatic processes: future-history correlations in the first process can be erased by an intermediate quantum measurement; for the second process, a noisy classical measurement blocks the effect of history. We then apply memory truncation techniques to recover an efficient de…
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…
According to the subjective Bayesian interpretation of quantum mechanics (QBism), the instruments used to measure quantum systems are to be regarded as an extension of the senses of the agent who is using them, and quantum states describe the agent's expectations for what they will experience through these extended senses. How can QBism then account for the fact that (i) instruments must be calibrated before they can be used to `sense' anything; (ii) some instruments are more precise than others; (iii) more precise instruments can lead to discovery of new systems? Furthermore, is the agent `in…
Robust certification of quantum instruments through a sequential communication game
S. N. Bose National Centre for Basic Sciences（S. N. Bose 基础科学国家中心）；Centre for Quantum Science and Technology, Chennai Institute of Technology（金奈理工学院量子科学与技术中心）
Our basic structure is a finite-dimensional complex Hilbert space $H$. We point out that the set of effects on $H$ form a convex effect algebra. Although the set of operators on $H$ also form a convex effect algebra, they have a more detailed structure. W introduce sequential products of effect and operations. Although these have already been studied, we introduce the new concept of sequential products of effects with operations and operations with effects. We then consider various special types of operations. After developing properties of these concepts, the results are generalized to includ…
We first show that every operation possesses an unique dual operation and measures an unique effect. If $a$ and $b$ are effects and $J$ is an operation that measures $a$, we define the sequential product of $a$ then $b$ relative to $J$. Properties of the sequential product are derived and are illustrated in terms of Lüders and Holevo operations. We next extend this work to the theory of instruments and observables. We also define the concept of an instrument (observable) conditioned by another instrument (observable). Identity, state-constant and repeatable instruments are considered. Sequenti…
Non-Markovian quantum processes exhibit different memory effects when measured in different ways; an unambiguous characterization of memory length requires accounting for the sequence of instruments applied to probe the system dynamics. This instrument-specific notion of quantum Markov order displays stark differences to its classical counterpart. Here, we explore the structure of quantum stochastic processes with finite length memory in detail. We begin by examining a generalized collision model with memory, before framing this instance within the general theory. We detail the constraints tha…
Incompatibility of quantum devices is a useful resource in various quantum information theoretical tasks, and it is at the heart of some fundamental features of quantum theory. While the incompatibility of measurements and quantum channels is well-studied, the incompatibility of quantum instruments has not been explored in much detail. In this work, we revise a notion of instrument compatibility introduced in the literature that we call traditional compatibility. Then, we introduce the new notion of parallel compatibility, and show that these two notions are inequivalent. Then, we argue that t…
We present the novel approach to mathematical modeling of information processes in biosystems. It explores the mathematical formalism and methodology of quantum theory, especially quantum measurement theory. This approach is known as {\it quantum-like} and it should be distinguished from study of genuine quantum physical processes in biosystems (quantum biophysics, quantum cognition). It is based on quantum information representation of biosystem's state and modeling its dynamics in the framework of theory of open quantum systems. This paper starts with the non-physicist friendly presentation …
Thermal channels -- the free processes allowed in the resource theory of quantum thermodynamics -- are generalised to thermal instruments, which we interpret as implementing thermodynamically free quantum measurements; a Maxwellian demon using such measurements never violates the second law of thermodynamics. Further properties of thermal instruments are investigated and, in particular, it is shown that they only measure observables commuting with the Hamiltonian, and they thermalise the measured system when performing a complete measurement, the latter of which indicates a thermodynamically i…
In an analogue quantum simulation, an experimentally accessible quantum system is controlled and measured precisely in order to learn about the properties of another quantum system. As such, analogue quantum simulation is a novel tool of scientific inference standing between computation simulation and conventional experiment. In this book we undertake a comprehensive appraisal of the epistemology of analogue quantum simulation. In particular, we consider the types of understanding that analogue quantum simulation can yield. We draw a distinction between analogue quantum computations and analog…
