Abstract Bohr’s complementarity and Schrödinger’s entanglement are two prominent physical characters of quantum systems. In this article, 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 that for any complete complementarity relation involving predictability and visibility measures that satisfy the criteria established in the literature, the corresponding quantum correlations are entanglement monotones. Therefore, we formally connect entanglement monotones with complementarity relations without appealing to a particular measure.
Abstract Bohr’s complementarity and Schrödinger’s entanglement are two prominent physical characters of quantum systems. In this article, 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 that for any complete complementarity relation involving predictability and visibility measures that satisfy the criteria established in the literature, the corresponding quantum correlations are entanglement monotones. Therefore, we formally connect entanglement monotones with complementarity relations without appealing to a particular measure.
Abstract 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 other 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 pure quantum states, we show that it is possible to obtain complete complementarity relations. This procedure allows us to create a general framework for obtaining complete complementarity relations for a subsystem that belongs to an arbitrary multi-partite quantum system in a pure state. By some simple examples, we show that if the predictability measure is changed then the correlation measure must also be changed in order to obtain complete complementarity relations for pure cases.
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 formalism was developed and quantifiers for the particleness and waveness of a quantum system were derived from the mathematical structure of QM entailed in the density matrix's basic properties ($\ensuremath{\rho}\ensuremath{\ge}0, \mathrm{Tr}\ensuremath{\rho}=1$). In this article, using IBM Quantum Experience quantum computers, we perform experimental tests of these complementarity relations applied to a particular class of one-qubit quantum states and also for random quantum states of one, two, and three qubits.
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) among wave, particle and mixedness aspects for arbitrary multi-state systems, which are respectively quantified by the l1 measure for quantum coherence, the which-path predictability, and the linear entropy. In particular, we show how a WWD can transform entropy into predictability and coherence. Through modifying the POVM (positive-operator valued measure) measurement on WWD, our TCR can be simplified as the wave-mixedness and particle-mixedness duality relations. Beyond enclosing the wave-particle duality relation [PRL 116, 160406 (2016)], our TCR relates to the wave-particle-entanglement complementarity relation [PRL 98, 250501 (2008); Opt. Commun. 283, 827 (2010)].
Bohr’s complementarity principle has long been a fundamental concept in quantum mechanics, positing that, within a given experimental set-up, a quantum system (or quanton) can exhibit either its wave-like character, denoted as W , or its particle-like character, denoted as P , but not both simultaneously. Modern interpretations of Bohr’s complementarity principle acknowledge the coexistence of these aspects in the same experiment while introducing the constraint W + P ≤ α . Notably, estimations of W or P frequently rely on indirect retrodiction methods, a practice that has led to the claim of the violation of Bohr’s complementarity principle. By taking a different route, recent advancements demonstrate that quantum complementarity relations can be rigorously derived from the axioms of quantum mechanics. To reconcile these observations and eliminate potential paradoxes or violations, we propose an updated formulation for the quantum complementarity principle, which is stated as follows: For a given quantum state preparation ρ t at a specific instant of time t , the wave and particle behaviours of a quanton are constrained by a complementarity relation 𝔚 ( ρ t ) + 𝔓 ( ρ t ) ≤ α ( d ) , which is derived directly from the axioms of quantum mechanics.
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 quantum aspect of bipartite systems. A generalized duality relation, that also governs possible violations of the Bell's inequality, arises between single- and bipartite properties.
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 (CCR), are Lorentz invariant. Peres et al. (Peres et al. 2002 Phys. Rev. Lett. 88 , 230402. ( doi:10.1103/PhysRevLett.88.230402 )) realized that even though it is possible to formally define spin in any Lorentz frame, there is no relationship between the observable expectation values in different Lorentz frames. Analogously, one can, in principle, define complementary relations in any Lorentz frame, but there is no obvious transformation law relating complementary relations in different frames. However, our result shows that the CCRs have the same value in any Lorentz frame, i.e. there is a transformation law connecting the CCRs. In addition, we explore relativistic scenarios for single and two-particle states, which helps in understanding the exchange of different aspects of a quantum system under Lorentz boosts.