Gunnar Bjork, Jonas Soderholm, Alexei Trifonov, Tedros Tsegaye, Anders Karlsson
We formulate a general complementarity relation starting from any Hermitian operator with discrete non-degenerate eigenvalues. We then elucidate the relationship between quantum complementarity and the Heisenberg-Robertson's uncertainty relation. We show that they are intimately connected. Finally we exemplify the general theory with some specific suggested experiments.
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For any ideal two-path interferometer it is shown that the wave-particle duality of quantum mechanics implies Heisenberg's uncertainty relation and vice versa. It is conjectured that complementarity and uncertainty are two aspects of the same general principle.
Since the uncertainty about an observable of a system prepared in a quantum state is usually described by its variance, when the state is mixed, the variance is a hybrid of quantum and classical uncertainties. Besides that, complementarity relations are saturated only for pure, single-quanton, quantum states. For mixed states, the wave-particle quantifiers never saturate the complementarity relation and can even reach zero for a maximally mixed state. So, to fully characterize a quanton it is not sufficient to consider its wave-particle aspect; one has also to regard its correlations with othe…
We present a scheme in which we investigate the two-slit experiment and we show that the principle of complementarity is more fundamental then the uncertainty principle.
Generalized uncertainty relations may depend not only on the commutator relation of two observables considered, but also on mutual correlations, in particular, on entanglement. The equivalence between the uncertainty relation and Bohr's complementarity thus holds in a much broader sense than anticipated.
Uncertainty relations and complementarity relations are core issues in quantum mechanics and quantum information theory. By use of the generalized Wigner-Yanase-Dyson (GWYD) skew information, we derive several uncertainty and complementarity relations with respect to mutually unbiased measurements (MUMs), and general symmetric informationally complete positive operator valued measurements (SIC-POVMs), respectively. Our results include some existing ones as particular cases. We also exemplify our results by providing a detailed example.
Complementarity-based complementarity: the choice of mutually unbiased observables shapes quantum uncertainty relations
针对MUBs导出的量子不确定性关系被认为与观测变量选择无关的问题，本文通过实验与数值方法证明5维系统中不同三MUB集合会导致不同的熵与方差互补界限。
We propose an uncertainty-complementarity balance relation and build quantitative connections among non-locality, complementarity, and uncertainty. Our balance relation, which is formulated in a theory-independent manner, states that for two measurements performed sequentially, the complementarity demonstrated in the first measurement in terms of disturbance is no greater than the uncertainty of the first measurement. Quantum theory respects our balance relation, from which the Tsirelson bound can be derived, up to an inessential assumption. In the simplest Bell scenario, we show that the boun…
The so-called preparation uncertainty can be understood in purely operational terms. Namely, it occurs when for some pair of observables, there is no preparation, for which they both exhibit deterministic statistics. However, the right-hand side of uncertainty relation is generally not operational as it depends on the quantum formalism. Also, while joint non-measurability of observables is an operational notion, the complementarity in Bohr sense (i.e. excess of information needed to describe the system) has not yet been expressed in purely operational terms. In this paper we propose a solution…
Reality of quantum observables, a feature of long-standing interest within foundations of quantum mechanics, has recently been quantified and deeply studied by means of entropic measures [Phys. Rev. A 97, 022107 (2018)]. However, there is no state-independent "reality trade-off" between noncommuting observables, as in certain systems all observables are real [Europhys. Lett. 112, 40005 (2015)]. We show that the entropic uncertainty relation in the presence of quantum memory [Nature Phys. 6, 659 (2010)] perfectly supplements the discussed notion of reality, rendering trade-offs between reality …
We establish a rigorous quantitative connection between (i) the interferometric duality relation for which-way information and fringe visibility and (ii) Heisenberg's uncertainty relation for position and modular momentum. We apply our theory to atom interferometry, wherein spontaneously emitted photons provide which way information, and unambiguously resolve the challenge posed by the metamaterial `perfect lens' to complementarity and to the Heisenberg-Bohr interpretation of the Heisenberg microscope thought experiment.
The two-slit experiment with quantum particles provides many insights into the behaviour of quantum mechanics, including Bohr's complementarity principle. Here we analyze Einstein's recoiling slit version of the experiment and show how the inevitable entanglement between the particle and the recoiling slit as a which-way detector is responsible for complementarity. We derive the Englert-Greenberger-Yasin duality from this entanglement, which can also be thought of as a consequence of sum-uncertainty relations between certain complementary observables of the recoiling slit. Thus, entanglement i…
A coherent account of the connections and contrasts between the principles of com- plementarity and uncertainty is developed starting from a survey of the various formalizations of these principles. The conceptual analysis is illustrated by means of a set of experimental schemes based on Mach-Zehnder interferometry. In particu- lar, path detection via entanglement with a probe system and (quantitative) quan- tum erasure are exhibited to constitute instances of joint unsharp measurements of complementary pairs of physical quantities, path and interference observables. The analysis uses the repr…
The fundamental principles of complementarity and uncertainty are shown to be related to the possibility of joint unsharp measurements of pairs of noncommuting quantum observables. A new joint measurement scheme for complementary observables is proposed. The measured observables are represented as positive operator valued measures (POVMs), whose intrinsic fuzziness parameters are found to satisfy an intriguing pay-off relation reflecting the complementarity. At the same time, this relation represents an instance of a Heisenberg uncertainty relation for measurement imprecisions. A model-indepen…
The overarching goal of this thesis is to demonstrate that complementarity is at the heart of quantum information theory, that it allows us to make (some) sense of just what information "quantum information" refers to, and that it is useful in understanding and constructing quantum information processing protocols. The detailed research results which form the basis of these claims are to be found in the included papers, and the aim here is to present an overview comprehensible to a more general audience.
We discuss complementarity and uncertainty in a gedanken Which-Way (Welcher-Weg) experiment in a Mach-Zehnder interferometer. Although a Welcher-Weg measurement can be performed with only a negligible amount of momentum change in the detector itself, the change in the interference pattern must correspond to a change in the momentum distribution of the particle. Entanglement between the Mach-Zehnder interferometer and the Welcher-Weg measurement introduces orthogonality that disables the interference of the particle, but simultaneously, it also generates a momentum transfer between the particle…
A model of the Einstein-Bohr double-slit experiment is formulated in a fully quantum theoretical setting. In this model, the state and dynamics of a movable wall that has the double slits in it, as well as the state of a particle incoming to the double slits, are described by quantum mechanics. Using this model, we analyzed complementarity between exhibiting the interference pattern and distinguishing the particle path. Comparing the Kennard-Robertson type and the Ozawa-type uncertainty relations, we conclude that the uncertainty relation involved in the double-slit experiment is not the Ozawa…
We revisit, in the framework of Mach-Zehnder interferometry, the connection between the complementarity and uncertainty principles of quantum mechanics. Specifically, we show that, for a pair of suitably chosen observables, the trade-off relation between the complementary path information and fringe visibility is equivalent to the uncertainty relation given by Schr\"odinger and Robertson, and to the one provided by Landau and Pollak as well. We also employ entropic uncertainty relations (based on Rényi entropic measures) and study their meaning for different values of the entropic parameter. …
The products of weak values of quantum observables are shown to be of value in deriving quantum uncertainty and complementarity relations, for both weak and strong measurement statistics. First, a 'product representation formula' allows the standard Heisenberg uncertainty relation to be derived from a classical uncertainty relation for complex random variables. We show this formula also leads to strong uncertainty relations for unitary operators, and underlies an interpretation of weak values as optimal (complex) estimates of quantum observables. Furthermore, we show that two incompatible obse…
We analyze Einstein's recoiling slit experiment and point out that the inevitable entanglement between the particle and the recoiling-slit was not part of Bohr's reply. We show that if this entanglement is taken into account, one can provided a simpler answer to Einstein. We also derive the Englert-Greenberger-Yasin duality relation from this entanglement. In addition, we show that the Englert-Greenberger-Yasin duality relation can also be thought of as a consequence of the sum uncertainty relation for certain observables of the recoiling slit. Thus, the uncertainty relations and entanglement …
Unveiling quantum steering by quantum-classical uncertainty complementarity
本文提出一种比熵不确定性关系更紧的量子-经典不确定性互补关系，并据此构造更高效的导引见证，揭示相干性与量子导引之间的定量操作联系。
A which-way measurement in Young's double-slit will destroy the interference pattern. Bohr claimed this complementarity between wave- and particle behaviour is enforced by Heisenberg's uncertainty principle: distinguishing two positions a distance s apart transfers a random momentum q \sim \hbar/s to the particle. This claim has been subject to debate: Scully et al. asserted that in some situations interference can be destroyed with no momentum transfer, while Storey et al. asserted that Bohr's stance is always valid. We address this issue using the experimental technique of weak measurement. …
We prove uncertainty relations that quantitatively express the impossibility of jointly sharp preparation of pre- and post-selected quantum states for measuring incompatible observables during the weak measurement. By defining a suitable operator whose average in the pre-selected quantum state gives the weak value, we show that one can have new uncertainty relations for variances of two such operators corresponding to two non-commuting observables. These generalize the recent stronger uncertainty relations that give non-trivial lower bounds for the sum of variances of two observables which ful…
Consideration of the von Neumann measurement process underlying interference experiments shows that the uncertainty in the incoming wave, responsible for its interference, translates during measurement into an uncertainty at the measuring apparatus. However, subsequent measurement on the apparatus does not reveal any new information about the interfering wave. This observation, in the context of recent advances in quantum information, suggests an argument for an information theoretic interpretation of quantum mechanics.
We present a unified view of the Berry phase of a quantum system and its entanglement with surroundings. The former reflects the nonseparability between a system and a classical environment as the latter for a quantum environment, and the concept of geometric time-energy uncertainty can be adopted as a signature of the nonseparability. Based on this viewpoint, we study their relationship in the quantum-classical transition of the environment, with the aid of a spin-half particle (qubit) model exposed to a quantum-classical hybrid field. In the quantum-classical transition, the Berry phase has …
In terms of operator, the two complementary quantities, the predictability and visibility, are reinvestigated in a two-way interferometer. One Hermitian operator and one non-Hermitian operator (composed of two Hermitian operators) are introduced for the predictability and visibility, respectively. The predictability and visibility can not be measured exactly simultaneously, due to the non-commutation between the two operators. The sum of the variances of the predictability and visibility (the total variance), is used to measure the uncertainty, which is linked to the complementarity relation t…
Relations connecting violation of any Bell inequalities and the complementarity between visibility and distinguishability in the interferometric experiments with different sources of decoherence are presented. A boundary of local-realistic explanation of the which-way complementarity is discussed in dependence on the choice of independent or collective tests of nonlocality.
We discuss complementarity relations in a bipartite continuous variable system. Building up from the work done on discrete d-dimensional systems, we prove that for symmetric two-mode states, quantum complementarity relations can be put in a simple relation with the elements of the variance matrix. When this condition is not satisfied, such a connection becomes non-trivial. Our investigation is the first step towards an operative characterization of the complementarity in a scenario that has not been investigated so far.
Bohr's complementarity and Schrödinger's entanglement are two prominent physical characters of quantum systems. In this letter, we formally connect them. It is known that complementarity relations for wave-particle duality are saturated only for pure, single-quanton, quantum states. For mixed states, the wave-particle quantifiers never saturate a complementarity relation and can even reach zero for a maximally mixed state. To fully characterize a quanton, it is not enough to consider its wave-particle aspect; we have also to regard its quantum correlations with other systems. Here we prove tha…
I argue that quantum optical experiments that purport to refute Bohr's principle of complementarity (BPC) fail in their aim. Some of these experiments try to refute complementarity by refuting the so called particle-wave duality relations, which evolved from the Wootters-Zureck reformulation of BPC (WZPC). I therefore consider it important for my forgoing arguments to first recall the essential tenets of BPC, and to clearly separate BPC from WZPC, which I will argue is a direct contradiction of BPC. This leads to a need to consider the meaning of particle-wave duality relations and to question…
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 complementarity relations in a multi-path interferometer with quantum memory
该研究针对带量子存储器的多路径干涉仪，以混合态为输入，建立可见度与路径可区分度的对偶关系，推导两类三重性关系，定量刻画纠缠作用，且所得互补性关系对双路径干涉仪完备。
The complementarity of two observables is often captured in uncertainty relations, which quantify an inevitable tradeoff in knowledge. Here we study complementarity in the context of an information processing task: we link the complementarity of two observables to their usefulness for superdense coding (SDC). In SDC, Alice sends two classical dits of information to Bob by sending a single qudit. However, we show that encoding with commuting unitaries prevents Alice from sending more than one dit per qudit, implying that complementarity is necessary for SDC to be advantagous over a classical st…
We develop an information theoretic interpretation of the number-phase complementarity in atomic systems, where phase is treated as a continuous positive operator valued measure (POVM). The relevant uncertainty principle is obtained as an upper bound on a sum of knowledge of these two observables for the case of two-level systems. A tighter bound characterizing the uncertainty relation is obtained numerically in terms of a weighted knowledge sum involving these variables. We point out that complementarity in these systems departs from mutual unbiasededness in two signalificant ways: first, the…
Quantum average correlations and complementarity relations via metric-adjusted skew information
本文基于度量调整的斜信息研究量子平均关联与互补关系，通过多种平均方法得出统一表达式，并建立波粒特征、量子熵与平均关联之间的互补关系。
The Einstein-Rupp experiments have been unduly neglected in the history of quantum mechanics. While this is to be explained by the fact that Emil Rupp was later exposed as a fraud and had fabricated the results, it is not justified, due to the importance attached to the experiments at the time. This paper discusses Rupp's fraud, the relation between Albert Einstein and Rupp, and the Einstein-Rupp experiments, and argues that these experiments were an influence on Niels Bohr's development of complementarity and Werner Heisenberg's formulation of the uncertainty relations.
We establish two complementarity relations for the relative entropy of coherence in quantum information processing, i.e., quantum dense coding and teleportation. We first give an uncertaintylike expression relating local quantum coherence to the capacity of optimal dense coding for bipartite system. The relation can also be applied to the case of dense coding by using unital memoryless noisy quantum channels. Further, the relation between local quantum coherence and teleportation fidelity for two-qubit system is given.
We present a method to derive Bell monogamy relations by connecting the complementarity principle with quantum non-locality. The resulting monogamy relations are stronger than those obtained from the no-signaling principle alone. In many cases, they yield tight quantum bounds on violation of single and multiple qubit correlation Bell inequalities. In contrast with the two-qubit case, a rich structure of possible violation patterns is shown to exist in the multipartite scenario.
Recently, a connection between quantum coherence and quantum steering was established and criteria for quantum steering or in other words, nonlocal advantage of quantum coherence (NAQC) were derived for two-qubit states. Here, we derive a set of complementarity relations between the steering or NAQC inequalities achieved by various criteria. We also extend the idea in the multi-partite scenario, specifically, in the three-qubit scenario, which can easily be generalized to the multi-partite scenario.
Quantum measurements necessarily disturb the state of physical system. Once we perform a complete measurement, the system undergoes decoherence and loses its coherence. If there is no disturbance, the state retains all of its coherence. It is therefore natural to ask if there is trade-off between disturbance caused to a state and its coherence. We present a coherence disturbance complementarity relation using the relative entropy of coherence. For bipartite states we prove a complementarity relation between the quantum coherence, entanglement and disturbance. Similar relation also holds for qu…
General framework of quantum complementarity from a measurement-based perspective
本文从测量视角建立以信息排斥关系表述量子互补性的一般框架，涵盖波粒二象性关系，并揭示其在纠缠目击及熵不确定性关系中的应用。
In the Copenhagen interpretation the Heisenberg uncertainty relation is interpreted as the mathematical expression of the concept of complementarity, quantifying the mutual disturbance necessarily taking place in a simultaneous or joint measurement of incompatible observables. This interpretation has already been criticized by Ballentine a long time ago, and has recently been challenged in an experimental way. These criticisms can be substantiated by using the generalized formalism of positive operator-valued measures, from which a new inequality can be derived, precisely illustrating the Cope…
Complementarity relations for wave-particle duality are saturated only for pure, single-quanton, quantum states. For a completely incoherent state, it is known that wave and particle quantifiers can reach zero, and hence no information about the wave and particle aspects of the system can be obtained. This means that the information is being shared with another systems, and quantum correlations can be seen as responsible for the loss of purity of the quanton, provided that the quanton is part of a multipartite pure quantum system. In this paper, by exploring the purity of bi- and tri-partite p…
Complementarity was originally introduced as a qualitative concept for the discussion of properties of quantum mechanical objects that are classically incompatible. More recently, complementarity has become a \emph{quantitative} relation between classically incompatible properties, such as visibility of interference fringes and "which-way" information, but also between purely quantum mechanical properties, such as measures of entanglement. We discuss different complementarity relations for systems of 2-, 3-, or \textit{n} qubits. Using nuclear magnetic resonance techniques, we have experimenta…
We derive two complementarity relations that constrain the individual and bipartite properties that may simultaneously exist in a multi-qubit system. The first expression, valid for an arbitrary pure state of n qubits, demonstrates that the degree to which single particle properties are possessed by an individual member of the system is limited by the bipartite entanglement that exists between that qubit and the remainder of the system. This result implies that the phenomenon of entanglement sharing is one specific consequence of complementarity. The second expression, which holds for an arbit…
We introduce a complete set of complementary quantities in bipartite, two-dimensional systems. Complementarity then relates the quantitative entanglement measure concurrence which is a bipartite property to the single-particle quantum properties predictability and visibility, for the most general quantum state of two qubits. Consequently, from an interferometric point of view, the usual wave-particle duality relation must be extended to a ``triality'' relation containing, in addition, the quantitative entanglement measure concurrence, which has no classical counterpart and manifests a genuine …
Validity of black hole complementarity in the context of generalized uncertainty principle
针对GUP背景下不可克隆定理有效性的争议，本文通过计算不同GUP形式下Schwarzschild黑洞的温度修正及复制信息所需能量，证明该能量超过黑洞质量，从而支持不可克隆定理依然成立。
It is well known that entanglement under Lorentz boosts is highly dependent on the boost scenario in question. For single particle states, a spin-momentum product state can be transformed into an entangled state. However, entanglement is just one of the aspects that completely characterizes a quantum system. The other two are known as the wave-particle duality. Although the entanglement entropy does not remain invariant under Lorentz boosts, and neither do the measures of predictability and coherence, we show here that these three measures taken together, in a complete complementarity relation…
Bohr's complementarity
本文回顾了互补性概念的历史演变，从玻尔的原始表述到现代量化，并探讨其与量子理论基本假设的深刻联系及相关发展。
We study number-phase uncertainty in a laser-driven, effectively four-level atomic system under electromagnetically induced transparency (EIT) and coherent population trapping (CPT). Uncertainty is described using (entropic) knowledge of the two complementary variables, namely, number and phase, where knowledge is defined as the relative entropy with respect to a uniform distribution. In the regime where the coupling and probe lasers are approximately of equal strength, and the atom exists in a CPT state, there is coherence between the ground states, and correspondingly large phase knowledge a…
'Complementarity' in paraxial and non-paraxial optical beams
本文通过将傍轴与非傍轴光束对应到qubit和qutrit系统，推导出相干性、可预测性与线性熵之间的互补关系，并揭示其与纠缠及复向量空间几何的联系。
We analyze an interferometric complementarity between one- and two-particle interference in the general case: $V_{i}^{2}+V_{12}^{2}\leq 1$ $(i=1$, $2)$, and further examine the relation among one-particle interference visibility $V_{i}$, two-particle interference visibility $V_{12}$ and the predication $P_{i}$ of the path of a single particle. An equality $V_{i}^{2}+V_{12}^{2}+P_{i}^{2}=1$ $(i=1$, $2)$ is achieved for any pure two-particle source, which implies the condition of the complementarity relation to reach the upper bound and its relation to another interferometric complementarity bet…
We propose a method to detect genuine quantum correlation for multi-qubit pure states. We then derive a complementarity relations for pure quantum states of N qubits. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. On the other hand, it is known that the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters is in general not true for multiqubit states. Inducing from the CKW equality, we find a proper form of entanglement monogamy equality for arbitrary quantum states. The total quantum correlation of qubit k with the remainin…
Bohr's complementarity principle is of fundamental historic and conceptual importance for Quantum Mechanics (QM), and states that, with a given experimental apparatus configuration, one can observe either the wave-like or the particle-like character of a quantum system, but not both. However, it was eventually realized that these dual behaviors can both manifest partially in the same experimental setup, and, using ad hoc proposed measures for the wave and particle aspects of the quanton, complementarity relations were proposed limiting how strong these manifestations can be. Recently, a formal…
We investigate the dynamics of information in isolated multi-qubit systems. It is shown that information is in not only local form but also nonlocal form. We apply a measure of local information based on fidelity, and demonstrate that nonlocal information can be directly related to some appropriate well defined entanglement measures. Under general unitary transformations, local and nonlocal information will exhibit unambiguous complementary behavior with the total information conserved.
The uncertainty relation for continuous variables due to Byalinicki-Birula and Mycielski expresses the complementarity between two $n$-uples of canonically conjugate variables $(x_1,x_2,\cdots x_n)$ and $(p_1,p_2,\cdots p_n)$ in terms of Shannon differential entropy. Here, we consider the generalization to variables that are not canonically conjugate and derive an entropic uncertainty relation expressing the balance between any two $n$-variable Gaussian projective measurements. The bound on entropies is expressed in terms of the determinant of a matrix of commutators between the measured varia…
Bohr's complementarity principle tested on a real quantum computer via interferometer experiments
本文通过在真实量子计算机上进行干涉实验，检验玻尔互补性原理，并利用量子态层析技术分析结果。
The ability that one system immediately affects another one by using local measurements is regarded as quantum steering, which can be detected by various steering criteria. Recently, Mondal et al. [Phys. Rev. A 98, 052330 (2018)] derived the complementarity relations of coherence steering criteria, and revealed that the quantum steering of system can be observed through the average coherence of subsystem. Here, we experimentally verify the complementarity relations between quantum steering criteria by employing two-photon Bell-like states and three Pauli operators. The results demonstrate that…
Clocks play a special role at the interface of general relativity and quantum mechanics. We analyze a clock-interferometry thought experiment and go on to theoretically derive and experimentally test a complementarity relation for quantum clocks in the context of the gravitational time lag. We study this relation in detail and discuss its application to various types of quantum clocks.
Wave-particle duality, an important and fundamental concept established upon pure quantum systems, is central to the complementarity principle in quantum mechanics. However, due to the environment effects or even the entanglement between the quanton and the which-way detector (WWD), the quanton should be described by a mixed quantum state but not a pure quantum state. Although there are some attempts to clarify the complementarity principle for mixed quantum systems, it is still unclear how the mixedness affects the complementary relation. Here, we give a ternary complementary relation (TCR) a…
Some recent works have introduced a quantum twist to the concept of complementarity, exemplified by a setup in which the which-way detector is in a superposition of being present and absent. It has been argued that such experiments allow measurement of particle-like and wave-like behavior at the same time. Here we derive an inequality which puts a bound on the visibility of interference and the amount of which-way information that one can obtain, in the context of such modified experiments. As the wave-aspect can only be revealed by an ensemble of detections, we argue that in such experiments,…
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