Willem M. de Muynck
A theory of joint nonideal measurement of incompatible observables is used in order to assess the relative merits of quantum tomography and certain measurements of generalized observables, with respect to completeness of the obtained information. A method is studied for calculating a Wigner distribution from the joint probability distribution obtained in a joint measurement.
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We study the sum uncertainty relations based on variance and skew information for arbitrary finite N quantum mechanical observables. We derive new uncertainty inequalities which improve the exiting results about the related uncertainty relations. Detailed examples are provided to illustrate the advantages of our uncertainty inequalities.
Allowing Wigner's friend to sequentially measure incompatible observables
本文提出一种新的维格纳的朋友场景，其中超级观察者可重复与朋友交互，并证明在此类场景下局部友好性不等式与贝尔不等式等价。
Bell conjectured that a positive Wigner function does not allow violation of the inequalities imposed by local hidden variable theories. A requirement for this conjecture is "when phase space measurements are performed". We introduce the theory-independent concept of "operationally local transformations" which refers to the change of the switch on a local measurement apparatus. We show that two separated parties, performing only phase space measurements on a composite quantum system with a positive Wigner function and performing only operationally local transformations that preserve this posit…
We analyze the validity of Bell and Kochen-Specker theorems under local (or noncontextual) realism but avoiding an assumption of the existence of a joint probability distribution for incompatible observables. We formulate a realist model which complies with this requirement. This is obtained by employing divergent sequences that nevertheless have marginals which are convergent. We find that under standard reasonable assumptions this possibility does not lead to a loophole of those theorems, by deriving a short of CHSH inequality valid for any finite size ensemble. Moreover, we analyze a Hardy'…
It is well known that jointly measurable observables cannot lead to a violation of any Bell inequality - independent of the state and the measurements chosen at the other site. In this letter we prove the converse: every pair of incompatible quantum observables enables the violation of a Bell inequality and therefore must remain incompatible within any other no-signaling theory. While in the case of von Neumann measurements it is sufficient to use the same pair of observables at both sites, general measurements can require different choices. The main result is obtained by showing that for arbi…
We study the problem of performing orthogonal qubit measurements simultaneously. Since these measurements are incompatible, one has to accept additional imprecision. An optimal joint measurement is the one with the least possible imprecision. All earlier considerations of this problem have concerned only joint measurability of observables, while in this work we also take into account conditional state transformations (i.e., instruments). We characterize the optimal joint instrument for two orthogonal von Neumann instruments as being the Luders instrument of the optimal joint observable.
Exploiting the tension between the two dynamics of quantum theory (QT) in the Wigner's Friend thought experiment, we point out that the standard QT leads to inconsistency in observed probabilities of measurement outcomes between two super-observers - Wigner and his Student. To avoid such inconsistent predictions of QT, we hypothesize two distinct perspectives. The first one is "Absoluteness of measurement (AoM)," that is, any measurement process is an absolute event irrespective of other observers and yields a single outcome. The other is "Non-absoluteness of measurement (NoM)" as the negation…
A joint measurement of two observables is a {\it simultaneous} measurement of both quantities upon the {\it same} quantum system. When two quantum-mechanical observables do not commute, then a joint measurement of these observables cannot be accomplished by projective measurements alone. In this paper we shall discuss the use of quantum cloning to perform a joint measurement of two components of spin associated with a qubit system. We introduce a cloning scheme which is optimal with respect to this task. This cloning scheme may be thought to work by cloning two components of spin onto its outp…
Being one of the centroidal concepts in quantum theory, the fundamental constraint imposed by Heisenberg uncertainty relations has always been a subject of immense attention and challenging in the context of joint measurements of general quantum mechanical observables. In particular, the recent extension of the original uncertainty relations has grabbed a distinct research focus and set a new ascendent target in quantum mechanics and quantum information processing. In the present work we explore the joint measurements of three incompatible observables, following the basic idea of a newly propo…
We investigate the optimal approximation to triple incompatible quantum measurements within the framework of statistical distance and joint measurability. According to the lower bound of the uncertainty inequality presented in [Physical Review A 99, 312107 (2019)], we give the analytical expressions of the optimal jointly measurable approximation to two kinds of triple incompatible unbiased qubit measurements. We also obtain the corresponding states which give the minimal approximation errors in measuring process. The results give rise to plausible experimental verifications of such statistica…
The quantification of the "measurement uncertainty" aspect of Heisenberg's Uncertainty Principle---that is, the study of trade-offs between accuracy and disturbance, or between accuracies in an approximate joint measurement on two incompatible observables---has regained a lot of interest recently. Several approaches have been proposed and debated. In this paper we consider Ozawa's definitions for inaccuracies (as root-mean-square errors) in approximate joint measurements, and study how these are constrained in different cases, whether one specifies certain properties of the approximations---na…
We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e. g. estimate the ensemble average of an observable by measuring other observable. A main ingredient in this approach is that of informationally complete (infocomplete in short) measurements, which allow to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of AB-informationally complete measureme…
Uncertainty principle is the basis of quantum mechanics. It reflects the basic law of the movement of microscopic particles. Wigner-Yanase skew information, as a measure of quantum uncertainties, is used to characterize the intrinsic features of the state and the observable. In this paper, we mainly investigate the sum uncertainty relations for both quantum mechanical observables and quantum channels based on skew information. We establish a new uncertainty relation in terms of Wigner-Yanase skew information for $n$ observables, which is saturated (thus it holds as equality) for two incompatib…
Heisenberg's uncertainty principle is one of the main tenets of quantum theory. Nevertheless, and despite its fundamental importance for our understanding of quantum foundations, there has been some confusion in its interpretation: although Heisenberg's first argument was that the measurement of one observable on a quantum state necessarily disturbs another incompatible observable, standard uncertainty relations typically bound the indeterminacy of the outcomes when either one or the other observable is measured. In this paper, we quantify precisely Heisenberg's intuition. Even if two incompat…
Robustness-based bounds for approximate joint realizations of incompatible quantum channels
本文研究了量子通道不相容性对近似联合实现精度的影响，提出通用稳健性作为衡量通道不相容性的资源量化器，并展示了其在测量通道中的统一框架。
The existence of maximally incompatible quantum observables in the sense of a minimal joint measurability region is investigated. Employing the universal quantum cloning device it is argued that only infinite dimensional quantum systems can accommodate maximal incompatibility. It is then shown that two of the most common pairs of complementary observables (position and momentum; number and phase) are maximally incompatible.
Incompatible observables underlie pillars of quantum physics such as contextuality and entanglement. The Heisenberg uncertainty principle is a fundamental limitation on the measurement of the product of incompatible observables, a `joint' measurement. However, recently a method using weak measurement has experimentally demonstrated joint measurement. This method [Lundeen, J. S., and Bamber, C. Phys. Rev. Lett. 108, 070402, 2012] delivers the standard expectation value of the product of observables, even if they are incompatible. A drawback of this method is that it requires coupling each obser…
The notion of measurements is central for many debates in quantum mechanics. One critical point is whether a measurement can be regarded as an absolute event, giving the same result for any observer in an irreversible manner. Using ideas from the gedankenexperiment of Wigner's friend it has been argued that, when combined with the assumptions of locality and no-superdeterminism, regarding a measurement as an absolute event is incompatible with the universal validity of quantum mechanics. We consider a weaker assumption: is the measurement event realised relatively to the observer when he only …
Relational Quantum Mechanics is Still Incompatible with Quantum Mechanics
本文回应了CDR对作者先前研究的批评，通过在GHZ态上的Wigner-Friend型顺序测量场景，重申了关系量子力学的相对事实概念与量子力学不相容，并指出该批评无效且引发原则性问题。
Joint or simultaneous measurements of non-commuting quantum observables are possible at the cost of increased unsharpness or measurement uncertainty. Many different criteria exist for defining what an "optimal" joint measurement is, with corresponding different tradeoff relations for the measurements. Understanding the limitations of such measurements is of fundamental interest and relevant for quantum technology. Here, we experimentally test a tradeoff relation for the sharpness of qubit measurements, a relation which refers directly to the form of the measurement operators, rather than to er…
Whereas complementarity manifests itself via two incompatible observables, quantum contextuality can only be revealed via the joint measurements among at least three observables. By incorporating unsharp measurements and joint measurements into a realistic model, we reestablish an inequality due to Liang, Spekkens, and Wiseman rigorously based on the assumption of noncontextuality alone. Its violation therefore unambiguously pinpoints the quantum contextuality of a two-level system. The maximal violation is attained by three triplewise jointly measurable observables that are pairwise jointly m…
Joint Measurability, Einstein-Podolsky-Rosen Steering, and Bell Nonlocality
本文研究了量子测量不相容性与非定域性的关系，证明了任何不相容测量集合均可演示EPR导向，表明两者等价；同时通过展示不违反Bell不等式的不相容测量集合，证明了Bell非定域性与联合可测性不等价。
We investigate the measurement uncertainties of a triple of positive operator-valued measures (POVMs) based on statistical distance, and formulate state-independent tight uncertainty inequalities satisfied by the three measurements in terms of triple-wise joint measurability. Particularly, uncertainty inequalities for three unbiased qubit measurements are presented with analytical lower bounds which relates to the necessary and sufficient condition of the triple-wise joint measurability of the given triple. We show that the measurement uncertainties for a triple measurement are essentially dif…
We formulate the necessary and sufficient conditions for the existence of a pair of maximally incompatible two-outcome measurements in a finite dimensional General Probabilistic Theory. The conditions are on the geometry of the state space, they require existence of two pairs of parallel exposed faces with additional condition on their intersections. We introduce the notion of discrimination measurement and show that the conditions for a pair of two-outcome measurements to be maximally incompatible are equivalent to requiring that a (potential, yet non-existing) joint measurement of the maxima…
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.
Stronger Reverse Uncertainty Relation for Multiple Incompatible Observables
针对现有反向不确定关系上界易趋于无穷的问题，构建新的更强反向不确定关系，证明其与常规不确定关系统一于同一框架，可实现任意紧度的多可观测量反向不确定关系，并探讨了其在纯度检测中的应用。
Quintino et. al. (Phys. Rev. Lett. 113, 160402 (2014)) and Uola et. al. (Phys. Rev. Lett. 113, 160403 (2014)) have recently established an intrinsic relation between non-joint measurability and Einstein-Podolsky- Rosen steering. They showed that a set of measurements is incompatible (i.e., not jointly measurable) if and only if it can be used for the demonstration of steering. In this paper, we prove the temporal analog of this result viz., a set of measurements are incompatible if and only if it exhibits temporal steering.
The Heisenberg-Robertson uncertainty relation quantitatively expresses the impossibility of jointly sharp preparation of incompatible observables. However it does not capture the concept of incompatible observables because it can be trivial even for two incompatible observables. We experimentally demonstrate the new stronger uncertainty relations proposed by Maccone and Pati [Phys. Rev. Lett. 113, 260401 (2014)] relating on that sum of variances are valid in a state-dependent manner and the lower bound is guaranteed to be nontrivial for two observables being incompatible on the state of the sy…
One of the most intriguing aspects of Quantum Mechanics is the impossibility of measuring at the same time observables corresponding to non-commuting operators. This impossibility can be partially relaxed when considering joint or sequential weak values evaluation. Indeed, weak measurements have been a real breakthrough in the quantum measurement framework that is of the utmost interest from both a fundamental and an applicative point of view. Here we show how we realized, for the first time, a sequential weak value evaluation of two incompatible observables on a single photon.
Complementarity restricts the accuracy with which incompatible quantum observables can be jointly measured. Despite popular conception, the Heisenberg uncertainty relation does not quantify this principle. We report the experimental verification of universally valid complementarity relations, including an improved relation derived here. We exploit Einstein-Poldolsky-Rosen correlations between two photonic qubits, to jointly measure incompatible observables of one. The product of our measurement inaccuracies is low enough to violate the widely used, but not universally valid, Arthurs-Kelly rela…
The notorious quantum measurement problem brings out the difficulty to reconcile two quantum postulates: the unitary evolution of closed quantum systems and the wave-function collapse after a measurement. This problematics is particularly highlighted in the Wigner's friend thought experiment, where the mismatch between unitary evolution and measurement collapse leads to conflicting quantum descriptions for different observers. A recent no-go theorem has established that the (quantum) statistics arising from an extended Wigner's friend scenario is incompatible when one try to hold together thre…
We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice's pair of non-commuting observables in the presence of Bob's entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice-Bob state, implementing incompatible POVMs at Bob's end is …
I describe a method of inferring the past of quantum observables given the initial state and the subsequent measurement results using Wigner quasi-probability representations. The method is proved to be compatible with logic for large subclasses of quantum systems, including those that involve incompatible observables and can still exhibit some quantum features, such as the uncertainty relation, measurement backaction, and entanglement.
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators, which is a distribution uniquely characterized by the property that the marginals for all linear combinations of the given operators agree with the quantum mechanical distributions. Its role as a joint quasi-probability distribution is underlined by the property that its support always lies in the set of expectation value tuples of the operators. We characterize the set of singularities and positivity, and provide some basic examples.
We study a class of phase-space distribution functions that is generated from a Gaussian convolution of the Wigner distribution function. This class of functions represents the joint count probability in simultaneous measurements of position and momentum. We show that, using these functions, one can determine the expectation value of a certain class of operators accurately, even if measurement data performed only with imperfect detectors are available. As an illustration, we consider the eight-port homodyne detection experiment that performs simultaneous measurements of two quadrature amplitud…
This article was written for the Logic in Computer Science column in the February 2015 issue of the Bulletin of the European Association for Theoretical Computer Science. The intended audience is general computer science audience. The uncertainty principle asserts a limit to the precision with which position x and momentum p of a particle can be known simultaneously. You may know the probability distributions of x and p individually but the joint distribution makes no physical sense. Yet Wigner exhibited such a joint distribution f(x,p). There was, however, a little trouble with it: some of it…
Wigner function is a quasi-distribution that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between Wigner function and appropriate measurements involving the system position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system position and momentum. We show that one can relate Wigner function to Kirkwood joint quasi-distribution of position and momentum, the latter, in turn, being a par…
Most incompatible measurements and sum-of-squares optimisation
通过平方和优化框架构造通用父测量，证明反交换可观测量集产生最不相容的二值测量，并改进高维量子导引的认证。
Quantum physics constrains the accuracy of joint measurements of incompatible observables. Here we test tight measurement-uncertainty relations using single photons. We implement two independent, idealized uncertainty-estimation methods, the 3-state method and the weak-measurement method, and adapt them to realistic experimental conditions. Exceptional quantum state fidelities of up to 0.99998(6) allow us to verge upon the fundamental limits of measurement uncertainty.
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators. We show that for any collection of hermitian operators A1...An , and any quantum state there is a unique joint distribution on R^n, with the property that the marginals of all linear combinations of the operators coincide with their quantum counterpart. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution, because for position and momentum this property defines the standard Wigner function. …
Derivation and experimental violation of Bell-like inequalities involve the measurement of in-compatible observables. Simple complementarity forbids the existence of such joint probabilitydistribution. Moreover, the measurement of incompatible observables requires different experimen-tal procedures, which no necessarily must share a common joint statistics. In this work, we avoidthese difficulties by proposing a joint simultaneous measurement experiment. We can obtain theexact individual statistics of all the observables involved in the Bell inequalities after a suitable datainversion. A lack …
Qualitative equivalence between incompatibility and Bell nonlocality
本文证明任何有限顶点的非平凡联合可测性结构都可通过量子实现支持贝尔不等式违反，从而表明测量不相容性不仅是贝尔非定域性的必要条件，也是充分条件。
Wigner's friend's memory and the no-signaling principle
本文研究了维格纳的朋友实验中记忆变化与无通信原则的冲突，指出朋友对记忆变化的意识可通过额外寄存器建模，但该模型违反了无通信条件。
In a joint paper Jeff Bub and Itamar Pitowski argued that the quantum state represents `the credence function of a rational agent [...] who is updating probabilities on the basis of events that occur'. In the famous thought experiment designed by Wigner, Wigner's friend performs a measurement in an isolated laboratory which in turn is measured by Wigner. Here we consider Wigner's friend as a rational agent and ask what her `credence function' is. We find experimental situations in which the friend can convince herself that updating the probabilities on the basis of events that happen solely in…
Bell's theorem is a statement by which averages obtained from specific types of statistical distributions must conform to a family of inequalities. These models, in accordance with the EPR argument, provide for the simultaneous existence of quantum mechanically incompatible quantities. We first recall several contradictions arising between the assumption of a joint distribution for incompatible observables and the probability structure of quantum-mechanics, and conclude that Bell's theorem is not expected to be relevant to quantum phenomena described by non-commuting observables, irrespective …
We provide a detailed description of the EPR paradox (in the Bohm version) for a two qubit-state in the discrete Wigner function formalism. We compare the probability distributions for two qubit relevant to simultaneously-measurable observables (computed from the Wigner function) with the probability distributions representing two perfectly-correlated classic particles in a discrete phase-space. We write in both cases the updating formulae after a measure, thus obtaining a mathematical definition of \textit{classic collapse} and \textit{quantum collapse}. We study, with the EPR experiment, the…
We develop a technique to find simultaneous measurements for noisy quantum observables in finite-dimensional Hilbert spaces. We use the method to derive lower bounds for the noise needed to make incompatible measurements jointly measurable. Using our strategy together with recent devel- opments in the field of one-sided quantum information processing we show that the attained lower bounds are tight for various symmetric sets of quantum measurements. We use this characterisation to prove the existence of so called 4-Specker sets in the qubit case.
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…
The quantum measurement problem can be regarded as the tension between the two alternative dynamics prescribed by quantum mechanics: the unitary evolution of the wave function and the state-update rule (or "collapse") at the instant a measurement takes place. The notorious Wigner's friend gedankenexperiment constitutes the paradoxical scenario in which different observers (one of whom is observed by the other) describe one and the same interaction differently, one --the Friend-- via state-update and the other --Wigner-- unitarily. This can lead to Wigner and his friend assigning different prob…
We consider the question of characterising the incompatibility of sets of high-dimensional quantum measurements. We introduce the concept of measurement incompatibility in subspaces. That is, starting from a set of measurements that is incompatible, one considers the set of measurements obtained by projection onto any strict subspace of fixed dimension. We identify three possible forms of incompatibility in subspaces: (i) incompressible incompatibility: measurements that become compatible in every subspace, (ii) fully compressible incompatibility: measurements that remain incompatible in every…
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…
Joint qubit observables induced by indirect measurements in cavity QED
本研究针对量子力学中可观测量需加噪声才可联合测量的特性，探究腔QED场景下间接时间连续测量诱导的联合量子比特可观测量，发现可通过监测方案和腔初态调控其性质，且零差检测正交分量可得到最优有偏联合可测量可观测量对。
Kirkwood-Dirac Type Quasiprobabilities as Universal Identifiers of Nonclassical Quantum Resources
本文证明Kirkwood-Dirac型准概率分布可作为通用标识符揭示任意量子资源，通过不相容测量区分资源态与非资源态，其负性对应几何距离并关联反常弱值，揭示测量不相容性是量子优势的基础。
The existence of observables that are incompatible or not jointly measurable is a characteristic feature of quantum mechanics, which lies at the root of a number of nonclassical phenomena, such as uncertainty relations, wave--particle dual behavior, Bell-inequality violation, and contextuality. However, no intuitive criterion is available for determining the compatibility of even two (generalized) observables, despite the overarching importance of this problem and intensive efforts of many researchers. Here we introduce an information theoretic paradigm together with an intuitive geometric pic…
We formulate a new error-disturbance relation, which is free from explicit dependence upon variances in observables. This error-disturbance relation shows improvement over the one provided by the Branciard inequality and the Ozawa inequality for some initial states and for particular class of joint measurements under consideration. We also prove a modified form of Ozawa's error-disturbance relation. The later relation provides a tighter bound compared to the Ozawa and the Branciard inequalities for a small number of states.
The uncertainty relations, pioneered by Werner Heisenberg nearly 90 years ago, set a fundamental limitation on the joint measurability of complementary observables. This limitation has long been a subject of debate, which has been reignited recently due to new proposed forms of measurement uncertainty relations. The present work is associated with a new error trade-off relation for compatible observables approximating two incompatible observables, in keeping with the spirit of Heisenberg's original ideas of 1927. We report the first \textsl{direct} test and confirmation of the tight bounds pre…
We formulate entropic measurements uncertainty relations (MURs) for a spin-1/2 system. When incompatible observables are approximatively jointly measured, we use relative entropy to quantify the information lost in approximation and we prove positive lower bounds for such a loss: there is an unavoidable information loss. Firstly we allow only for covariant approximate joint measurements and we find state-dependent MURs for two or three orthogonal spin-1/2 components. Secondly we consider any possible approximate joint measurement and we find state-independent MURs for two or three spin-1/2 com…
Incompatible measurements in quantum theory always lead to Einstein-Podolsky-Rosen (EPR)-Schrödinger steering. Channel steering which is a generalized notion of EPR-Schrödinger steering, has been introduced recently. Here we establish a connection between lack of joint measureability and channel steering.
A coarse grained Wigner distribution p_{W}(x,u) obeying positivity derives out of information-theoretic considerations. Let p(x,u) be the unknown joint PDF (probability density function) on position- and momentum fluctuations x,u for a pure state particle. Suppose that the phase part Psi(x,z) of its Fourier transform F.T.[p(x,u)]=|Z(x,z)|exp[iPsi(x,z)] is constructed as a hologram. (Such a hologram is often used in heterodyne interferometry.) Consider a particle randomly illuminating this phase hologram. Let its two position coordinates be measured. Require that the measurements contain an ext…
The decoherence interpretation of quantum measurements is applied to Wigner's friend experiments. A framework in which all the experimental outcomes arise from unitary evolutions is proposed. Within it, a measurement is not completed until an uncontrolled environment monitorizes the state composed by the system, the apparatus and the observer. The (apparent) wave-function collapse and the corresponding randomness result from tracing out this environment; it is thus the ultimate responsible for the emergence of definite outcomes. Two main effects arise from this fact. First, external interferen…
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