Pekka Lahti
This talk is a survey of the question of joint measurability of coexistent observables and its is based on the monograph Operational Quantum Physics [1] and on the papers [2,3,4].
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One of the hallmarks of quantum theory is the realization that distinct measurements cannot in general be performed simultaneously, in stark contrast to classical physics. In this context the notions of coexistence and joint measurability are employed to analyze the possibility of measuring together two general quantum observables, characterizing different degrees of compatibility between measurements. It is known that two jointly measurable observables are always coexistent, and that the converse holds for various classes of observables, including the case of observables with two outcomes. He…
We show that a rank-1 quantum observable (POVM) M is jointly measurable with a quantum observable M' exactly when M' is a smearing of M. If M is extreme, rank-1 and discrete then M and M' are coexistent if and only if they are jointly measurable.
Recently a problem concerning the equivalence of joint measurability and coexistence of quantum observables was solved [15]. In this paper we generalize two known joint measurability results from sharp observables to the class of extreme observables and study relationships between coexistence, joint measurability, and post-processing of quantum observables when an extreme observable is involved. We also discuss another notion of compatibility and provide a counterexample separating this from the former notions.
Quantum operations are used to describe the observed probability distributions and conditional states of the measured system. In this paper, we address the problem of their joint measurability (coexistence). We derive two equivalent coexistence criteria. The two most common classes of operations - Luders operations and conditional state preparators - are analyzed. It is shown that Luders operations are coexistent only under very restrictive conditions, when the associated effects are either proportional to each other, or disjoint.
The conceptual relation between the measurability of quantum mechanical observables and the computability of numerical functions is re-examined. A new formulation is given for the notion of measurability with finite precision in order to reconcile the conflict alleged by M. A. Nielsen [Phys. Rev. Lett. 79, 2915 (1997)] that the measurability of a certain observable contradicts the Church-Turing thesis. It is argued that any function computable by a quantum algorithm is a recursive function obeying the Church-Turing thesis, whereas any observable can be measured in principle.
The notion coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measurable (in one sense of the phrase). We demonstrate the possibility of measuring together (in another sense of the phrase) noncoexistent observables. This leads us to a reconsideration of the connection between joint measurability and noncommutativity of…
The measurability by means of continuous measurements, of an observable $\A(t_0)$, at an instant, and of a time averaged observable, $\bar \A=1/T\int \A(t')dt'$, is examined for linear and in particular for non-linear quantum mechanical systems. We argue that only when the exact (non-perturbative) solution is known, an exact measurement may be possible. A perturbative approach is shown to fail in the non-linear case for measurements with accuracy $Δ\bar \A < Δ\bar \A_{min}(T)$, giving rise to a restriction on the accuracy. Thus, in order to prepare an initial pure state of a non-linear system,…
The fact that not all measurements can be carried out simultaneously is a peculiar feature of quantum mechanics and responsible for many key phenomena in the theory, such as complementarity or uncertainty relations. For the special case of projective measurements quantum behavior can be characterized by the commutator but for generalized measurements it is not easy to decide whether two measurements can still be understood in classical terms or whether they show already quantum features. We prove that generalized measurements which do not fulfill the notion of joint measurability are nonclassi…
Quantum mechanics predicts the joint probability distributions of the outcomes of simultaneous measurements of commuting observables, but the current formulation lacks the operational definition of simultaneous measurements. In order to provide foundations of joint statistics of local general measurements on entangled systems in a general theoretical framework, the question is answered as to under what condition the outputs of two measuring apparatuses satisfy the joint probability formula for simultaneous measurements of their observables. For this purpose, all the possible state changes caus…
The localization problem: an antinomy between measurability and causal dynamics
针对相对论量子理论中存在七十余年、易被忽视且困扰学界的局域化问题，本文通过历史回顾厘清文献误解，揭示可局域性与相对论因果性的根本冲突，凸显因果动力学与局域可观测量的二律背反。
The question of quantifying the sharpness (or unsharpness) of a quantum mechanical effect is investigated. Apart from sharpness, another property, bias, is found to be relevant for the joint measurability or coexistence of two effects. Measures of bias will be defined and examples given.
We establish connections between the requirement of measurability of a probability space and the principle of complimentarity in quantum mechanics. It is shown that measurability of a probability space implies the dependence of results of quantum measurement not only on the properties of a quantum object under consideration, but also on the classical characteristics of the measuring device which is used. We show that if one takes into account the requirement of measurability in a quantum case, the Bell inequality does not follow from the hypothesis about the existence of an objective reality.
In this paper we identify a hidden premise in Bell's theorem: measurability of the underlying space. But our system (the space of all paths, SP) is not measurable, although it replicates the predictions of standard quantum mechanics. Using it we present three counterexamples to Bell's theorem and also show why Bell-like arguments for more than two particles cannot be carried out in this model. Moreover, we show that the result places severe constraints on possible viable interpretations of quantum mechanics: Either an interpretation must in some form represent a quantum system in terms of all …
Some measurements in quantum mechanics disturb each other. This has puzzled physicists since the formulation of the theory, but only in recent decades has the incompatibility of measurements been analyzed in depth and detail, using the notion of joint measurability of generalized measurements. In this Colloquium joint measurability and incompatibility are reviewed from the perspective of quantum information science. The Colloquium starts by discussing the basic definitions and concepts. An overview on applications of incompatibility, such as in measurement uncertainty relations, the characteri…
Loophole-free test of macroscopic realism via high-order correlations of measurement
本文提出基于测量高阶关联的普适不等式，在弱信号极限下实现无解释性漏洞和非侵入可测量性漏洞的宏观实在性检验，并用量子自旋模型证明其可被违背。
The fact that there are quantum observables without a simultaneous measurement is one of the fundamental characteristics of quantum mechanics. In this work we expand the concept of joint measurability to all kinds of possible measurement devices, and we call this relation compatibility. Two devices are incompatible if they cannot be implemented as parts of a single measurement setup. We introduce also a more stringent notion of incompatibility, strong incompatibility. Both incompatibility and strong incompatibility are rigorously characterized and their difference is demonstrated by examples.
During the recent developments of quantum theory it has been clarified that the observable quantities (like energy or position) may be represented by operators (with real spectra) which are manifestly non-Hermitian. The mathematical consistency of the resulting models of stable quantum systems requires a reconstruction of an alternative, amended, physical inner product of states. We point out the less known fact that for more than one observable the task is not always feasible. The difficulty is re-analyzed and its elementary linear-algebraic interpretation and treatment are outlined.
The coexistence between Quantum Mechanics and Special Relativity is usually formulated in terms of the no-signaling condition. Several authors have even suggested that this condition should be included between the basic postulates of Quantum Theory. However, there are several scenarios where signaling is, in principle, possible: based on previous results and the analysis of the relation between unitarity and signaling we present an example of a two-particle interferometric arrangement for which the dynamics is, in principle, compatible with superluminal transmission of information. This type o…
It is shown that in the Rovelli relational interpretation of quantum mechanics, in which the notion of absolute or observer independent state is rejected, the conclusion of the ordinary EPR argument turns out to be frame-dependent, provided the conditions of the original argument are suitably adapted to the new interpretation. The consequences of this result for the `peaceful coexistence' of quantum mechanics and special relativity are briefly discussed.
According to Relational Quantum Mechanics (RQM) the wave function $ψ$ is considered neither a concrete physical item evolving in spacetime, nor an object representing the absolute state of a certain quantum system. In this interpretative framework, $ψ$ is defined as a computational device encoding observers' information; hence, RQM offers a somewhat epistemic view of the wave function. This perspective seems to be at odds with the PBR theorem, a formal result excluding that wave functions represent knowledge of an underlying reality described by some ontic state. In this paper we argue that RQ…
We examine the measurability of the temporal ordering of two events, as well as event coincidences. In classical mechanics, a measurement of the order-of-arrival of two particles is shown to be equivalent to a measurement involving only one particle (in higher dimensions). In quantum mechanics, we find that diffraction effects introduce a minimum inaccuracy to which the temporal order-of-arrival can be determined unambiguously. The minimum inaccuracy of the measurement is given by dt=1/E where E is the total kinetic energy of the two particles. Similar restrictions apply to the case of coincid…
Coexistence of CHSH Nonlocality and KCBS Contextuality in a Single Quantum State
研究在单个量子态中纠缠与非定域性共存的机制，通过混合CHSH-KCBS场景展示上下文性与非定域性在参数空间中的分离，揭示优化KCBS违反与抑制CHSH违反的区域。
Despite the great success of quantum mechanics, questions regarding its application still exist and the boundary between quantum and classical mechanics remains unclear. Based on the philosophical assumptions of macrorealism and noninvasive measurability, Leggett and Garg devised a series of inequalities (LG inequalities) involving a single system with a set of measurements at different times. Introduced as the Bell inequalities in time, the violation of LG inequalities excludes the hidden-variable description based on the above two assumptions. We experimentally investigated the single photon…
Quantum key distribution (QKD) uniquely allows distribution of cryptographic keys with security verified by quantum mechanical limits. Both protocol execution and subsequent applications require the assistance of classical data communication channels. While using separate fibers is one option, it is economically more viable if data and quantum signals are simultaneously transmitted through a single fiber. However, noise-photon contamination arising from the intense data signal has severely restricted both the QKD distances and secure key rates. Here, we exploit a novel temporal-filtering effec…
Quantum uncertainty is described here in two guises: indeterminacy with its concomitant indeterminism of measurement outcomes, and fuzziness, or unsharpness. Both features were long seen as obstructions of experimental possibilities that were available in the realm of classical physics. The birth of quantum information science was due to the realization that such obstructions can be turned into powerful resources. Here we review how the utilization of quantum fuzziness makes room for a notion of approximate joint measurement of noncommuting observables. We also show how from a classical perspe…
Optimal universal entanglement processes are discussed which entangle two quantum systems in an optimal way for all possible initial states. It is demonstrated that the linear character of quantum theory which enforces the peaceful coexistence of quantum mechanics and relativity imposes severe restrictions on the structure of the resulting optimally entangled states. Depending on the dimension of the one-particle Hilbert space such a universal process generates either a pure Bell state or mixed entangled states. In the limit of very large dimensions of the one-particle Hilbert space the von-Ne…
As is well known, the macroscopic realism and the noninvasive measurability together lead to Leggett-Garg inequalities violated by quantum mechanics. We consider tests of the Leggett-Garg type with use of the $q$-entropies. For all $q\geq1$, quantum mechanics predicts violations of an entire family of $q$-entropic inequalities of the Leggett-Garg type. Violations are exemplified with a quantum spin-$s$ system. In general, entropic Leggett-Garg inequalities give only necessary conditions that some probabilistic model is compatible with the macrorealism in the broader sense. The presented $q$-en…
The inequalities that separate contextual from noncontextual correlations for the n-cycle scenario (consisting of n dichotomic observables Oj, with j = 0, ..., n-1 and such that Oj and Oj+1 are jointly measurable) have been recently identified [arXiv:1206.3212 (2012)]. Here we report the results of an experiment designed to reach the maximum quantum violation of these inequalities for any even number of observables ranging from 4 to 14. The four dimensional Hilbert space required for the test was spanned by two photonic qubits encoded in the transversal path of photon pairs, and the joint meas…
I start from the fundamental principles of non-relativistic quantum mechanics, without probability, and interpret them using the notion of coexistence: a quantum state can be read, not uniquely, as a coexistence of other quantum states, which are pairwise orthogonal. In this formalism, I prove that a conscious observer is necessarily a physical object that can memorize local events by setting one of its parts in an exactly specified constant quantum state (hypotheses H1, H2 and H3). Then I define the probability of a future event as the proportion of initial observers, all identical, who will …
From Classical Stochastic to Monitored Quantum Dynamics: Dynamical Phase Coexistence in East Circuit Models
研究监控量子电路模型在经典随机与单位元量子动力学之间的动力学相共存现象，通过测量记录探索量子 regime 中的动力学相共存持续性。
Quantum key distribution (QKD) provides information-theoretic security based on the laws of quantum mechanics. The desire to reduce costs and increase robustness in real-world applications has motivated the study of coexistence between QKD and intense classical data traffic in a single fiber. Previous works on coexistence in metropolitan areas have used wavelength-division multiplexing, however, coexistence in backbone fiber networks remains a great experimental challenge, as Tbps data of up to 20 dBm optical power is transferred, and much more noise is generated for QKD. Here we present for t…
A new quantum ontology of quantum mechanics has been proposed recently. This ontology is based on impossible to realize measurements which need to be performed repeatedly on the same single physical system or on the same pair of physical systems. We agree that quantum mechanics is a contextual theory and that the experimental contexts have to be a part of any description of quantum phenomena but in our opinion this new ontology is neither convincing nor useful. In particular the authors claim that their ontology explains the peaceful coexistence between quantum mechanics and relativity in spin…
This paper posits that the cosmic digital code as the law of all physical laws contains two mutually exclusive values: attachment space attaching to object and detachment space detaching from object. However, the cosmic physical system could not start with mutually exclusive attachment space and detachment space at the same time in the beginning. The way out of this impasse is that the complete cosmic system consists of both the cosmic physical system and the unphysical cosmic digital code. The unphysical cosmic digital code allows the coexistence of attachment space and detachment space at th…
Classical chimera states are paradigmatic examples of partial synchronization patterns emerging in nonlinear dynamics. These states are characterized by the spatial coexistence of two dramatically different dynamical behaviors, i.e., synchronized and desynchronized dynamics. Our aim in this contribution is to discuss signatures of chimera states in quantum mechanics. We study a network with a ring topology consisting of N coupled quantum Van der Pol oscillators. We describe the emergence of chimera-like quantum correlations in the covariance matrix. Further, we establish the connection of chim…
Quantum mechanics imposes 'monogamy' constraints on the sharing of entanglement. We show that, despite these limitations, entanglement can be fully 'promiscuous', i.e. simultaneously present in unlimited two-body and many-body forms in states living in an infinite-dimensional Hilbert space. Monogamy just bounds the divergence rate of the various entanglement contributions. This is demonstrated in simple families of N-mode (N >= 4) Gaussian states of light fields or atomic ensembles, which therefore enable infinitely more freedom in the distribution of information, as opposed to systems of indi…
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