A bipartite quantum state (for two systems in any dimensions) can be decomposed as a superposition of many components. For a superposition of more than two components we prove that there is a bound of the entanglement of the superposition state which can be expressed according to entanglements of its component states. Especially, if the component states are mutually bi-orthogonal, the entanglement of the superposition state can be exactly given in terms of the entanglements of the states being superposed.
We investigate upper and lower bounds on the entropy of entanglement of a superposition of bipartite states as a function of the individual states in the superposition. In particular, we extend the results in [G. Gour, arxiv.org:0704.1521 (2007)] to superpositions of several states rather than just two. We then investigate the entanglement in a subspace as a function of its basis states: we find upper bounds for the largest entanglement in a subspace and demonstrate that no such lower bound for the smallest entanglement exists. Finally, we consider entanglement of superpositions using measures…
We find tight lower and upper bounds on the entanglement of a superposition of two bipartite states in terms of the entanglement of the two states constituting the superposition. Our upper bound is dramatically tighter than the one presented in Phys. Rev. Lett 97, 100502 (2006) and our lower bound can be used to provide lower bounds on different measures of entanglement such as the entanglement of formation and the entanglement of subspaces. We also find that in the case in which the two states are one-sided orthogonal, the entanglement of the superposition state can be expressed explicitly in…
Given a bipartite quantum state (in arbitrary dimension) and a decomposition of it as a superposition of two others, we find bounds on the entanglement of the superposition state in terms of the entanglement of the states being superposed. In the case that the two states being superposed are bi-orthogonal, the answer is simple, and, for example, the entanglement of the superposition cannot be more than one e-bit more than the average of the entanglement of the two states being superposed. However for more general states, the situation is very different.
Entanglement characteristics of a pair coherent state is studied using entanglement of superposition. It is demonstrated only few states in the expansion of a pair coherent state, in a harmonic oscillator basis, contribute significantly to the entanglement. We also study a case of two superposed pair coherent states
We derive the lower and upper bounds on the entanglement of a given multipartite superposition state in terms of the entanglement of the states being superposed. The first entanglement measure we use is the geometric measure, and the second is the q-squashed entanglement. These bounds allow us to estimate the amount of the multipartite entanglement of superpositions. We also show that two states of high fidelity to one another do not necessarily have nearly the same q-squashed entanglement.
The entanglement of superpositions [Phys. Rev. Lett. 97, 100502 (2006)] is generalized to the multipartite scenario: an upper bound to the multipartite entanglement of a superposition is given in terms of the entanglement of the superposed states and the superposition coefficients. This bound is proven to be tight for a class of states composed by an arbitrary number of qubits. We also extend the result to a large family of quantifiers which includes the negativity, the robustness of entanglement, and the best separable approximation measure.
Superposition states are at the origin of many paradoxes in quantum mechanics. By unraveling the von Neumann equation for density matrices, we develop a superposition-free formulation of quantum mechanics. Stochastic quantum jumps are a key feature of this approach, in blatant contrast with the continuity of the deterministic Schrödinger equation. We explain how quantum entanglement arises. Our superposition-free formulation results offers a new perspective on quantum mechanics.
We show relations between superposition of macroscopically distinct states and entanglement. These relations lead to the important conclusion that if a state contains superposition of macroscopically distinct states, the state also contains large multipartite entanglement in terms of several measures. Such multipartite entanglement property also suggests that if a state contains superposition of macroscopically distinct states, a measurement on a single particle drastically changes the state of macroscopically many other particles, as in the case of the N-qubit GHZ state.
This work discuss the entanglement and quantum polarization of superpositions of two-mode coherent states. We use the concurrence to measure their entanglements and the quantum Stokes parameters and the Q function in order to analyze their polarization and degree of polarization.
We propose a scheme for generating the superposition and the entanglement of squeezed vacuum states of electromagnetical fields. The scheme involves single-photon source, single photon detector, and cross Kerr nonlinearity. The Kerr nonlinearity required for generating the superposition of squeezed vacuum states is 1/2 of that required for generating the superposition of coherent states. The proposal can also be extended to generate the entanglement states between two coherent states and that between one coherent state and one squeezed vacuum state.
We propose a simple scheme to generate deterministic entanglement between two movable end mirrors in a Fabry-Perot cavity using a single photon superposition state. We derive analytically the expressions of the generated entangled states and the degree of entanglement for each state. We show that strong entanglement can be obtained either in the single-photon strong coupling regime deterministically or in the single-photon weak coupling regime conditionally.
We investigate how the genuine multipartite entanglement is distributed among the components of superposed states. Analytical lower and upper bounds for the usual multipartite negativity and the genuine multipartite entanglement negativity are derived. These bounds are shown to be tight by detailed examples.
We investigate the multiparticle quantum superposition and the persistence of multipartite entanglement of the quantum superposition generated by the quantum injected high-gain optical parametric amplification of a single photon. The physical configuration based on the optimal universal quantum cloning has been adopted to investigate how the entanglement and the quantum coherence of the system persists for large values of the nonlinear parametric gain g.
This article discusses the important primitives of Superposition and Entanglement in Quantum Information Processing from physics point of view. System of spin-1/2 particles has been considered which presents itself as a logical and conceptual candidate to understand these concepts. The article is intended as a review of these important concepts and hopes to bring forth a conceptual framework in this regard.
A new entangled quantum state is introduced by applying local coherent superposition (ra^+ +ta) of photon subtraction and addition to each mode of even entangled coherent state (EECS) and the properties of entanglement are investigated. It is found that the Shchukin-Vogel inseparability, the degree of entanglement and the average fidelity of quantum teleportation of the EECS can be improved due to the coherent superposition operation. The effects of improvement by coherent superposition operation are better than those by single (a^+) and two-photon (a^+ b^+) addition operations under a small r…
A scheme for preparation of coherent superposition of Fock states of electromagnetic field is constructed. The superposition state is created inside the cavity via a strong interaction of a four-level atom with quantum field of the cavity and classic laser fields. We demonstrate the possibility to create desired arbitrary superposition of the cavity Fock states just by changing the relative delay of the fields. Then, as another application of our model, we study a means of creating entanglement of neutral four-level atoms using different sequences of interactions with the cavity and laser fiel…
The abstract framework of quantum mechanics (QM) causes the well-known weirdness, which leads to the field of foundation of QM. We constructed the new concept, i.e., scope, to lay the foundation of quantum coherence and openness, also the principles of superposition and entanglement. We studied analytically and quantitatively the quantum correlations and information, also we discussed the physical essence of the existed entanglement measures. We compared with several other approaches to the foundation of QM, and we stated that the concept of scope is unique and has not been demonstrated before…
We investigate entanglement properties of a recently introduced class of macroscopic quantum superpositions in two-mode mixed states. One of the tools we use in order to infer the entanglement in this non-Gaussian class of states is the power to entangle a qubit system. Our study reveals features which are hidden in a standard approach to entanglement investigation based on the uncertainty principle of the quadrature variables. We briefly describe the experimental setup corresponding to our theoretical scenario and a suitable modification of the protocol which makes our proposal realizable wit…
Density matrices of pure multiphoton Fock polarization states and of arising from them reduced density matrices of mixed states are expressed in similar ways in terms of matrices of correlators defined as averaged products of equal numbers of creation and annihilation operators. Degree of entanglement of considered states is evaluated for various combinations of parameters of states and character of their reduction.
A multiparticle quantum superposition state has been generated by a novel phase-selective parametric amplifier of an entangled two-photon state. This realization is expected to open a new field of investigations on the persistence of the validity of the standard quantum theory for systems of increasing complexity, in a quasi decoherence-free environment. Because of its nonlocal structure the new system is expected to play a relevant role in the modern endeavor on quantum information and in the basic physics of entanglement.
Measures are introduced to quantify the degree of superposition in mixed states with respect to orthogonal decompositions of the Hilbert space of a quantum system. These superposition measures can be regarded as analogues to entanglement measures, but can also be put in a more direct relation to the latter. By a second quantization of the system it is possible to induce superposition measures from entanglement measures. We consider the measures induced from relative entropy of entanglement and entanglement of formation. We furthermore introduce a class of measures with an operational interpret…
Two quantum Macro-states and their Macroscopic Quantum Superpositions (MQS) localized in two far apart, space - like separated sites can be non-locally correlated by any entangled couple of single-particles having interacted in the past. This novel Macro - Macro paradigm is investigated on the basis of a recent study on an entangled Micro-Macro system involving N=10^5 particles. Crucial experimental issues as the violation of Bell's inequalities by the Macro - Macro system are considered.
We calculate the analytic expression for geometric measure of entanglement for arbitrary superposition of two $N$-qubit canonical orthonormal Greenberger-Horne-Zeilinger ($GHZ$) states and the same for two $W$ states. In course of characterizing all kinds of nonclassical correlations, explicit formula for quantum discord (via relative entropy) for the former class of states has been presented. Contrary to the $GHZ$ state, the closest separable state to the $W$ state is not classical. Therefore, in this case the discord is different from the relative entropy of entanglement. We conjecture that …
We consider the case of a pair of particles initially in a superposition state corresponding to a separated pair of wave packets. We calculate \emph{exactly} the time development of this non-Gaussian state due to interaction with an \emph{arbitrary} heat bath. We find that coherence decays continuously, as expected. We then investigate entanglement and find that at a finite time the system becomes separable (not entangled). Thus, we see that entanglement sudden death is also prevalent in continuous variable systems which should raise concern for the designers of entangled systems.
Quantum superpositions and entanglement are at the heart of the quantum information science. There have been only a few investigations of these phenomena at the mesoscopic level, despite the fact that these systems are promising for quantum state storage and processing. Here we present two novel experiments with surface plasmons propagating on cm-long metallic stripe waveguides. We demonstrate that two plasmons can be entangled at remote places. In addition, we create a single plasmon in a temporal superposition state: it exists in a superposition of two widely separated moments. These quantum…
We predict the occurrence of single-photon-induced entanglement and quantum superposition in a hybrid quantum model, introducing an optomechanical coupling into the Rabi model. Originally, it comes from the photon-dependent quantum property of ground state featured by the proposed hybrid model. It associates with a single-photon-induced quantum phase transition, and is immune to the $A^2$ term of the spin-field interaction. Moreover, the obtained quantum superposition state is actual a squeezed cat state, which can give a significant precision enhancement in quantum metrology. This work offers…
Entanglement of superpositions in zero-energy subspace: insights from stabilizer theory
Faculty of Physics, M.V. Lomonosov Moscow State University（莫斯科国立大学物理系）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Artificial Intelligence Research Institute (AIRI)（人工智能研究院）
In this paper we study the evolution of the two two-level atoms interacting with a single-mode quantized radiation field, namely, two-atom multiphoton Jaynes-Cummings model when the radiation field and atoms are initially prepared in the superpostion of displaced number states and excited atomic states, respectively. For this system we investigate the atomic inversion, Wigner function, phase distribution and entanglement.
The multiphoton quantum superposition generated by a quantum-injected optical parametric amplifier (QI-OPA) seeded by a single-photon belonging to an EPR entangled pair is made to interact with a Mirror-BEC shaped as a Bragg interference structure. The effect can be greatly enhanced if the mirror is part of a cavity structure or is used within an optical interferometer. The overall process will realize an Entangled Macroscopic Quantum Superposition involving a ''microscopic'' singe-photon state of polarization and the coherent ''macroscopic'' displacement of the BEC structure acting in space-l…
Quantum entanglement and coherence are two fundamental features of nature, arising from the superposition principle of quantum mechanics. While considered as puzzling phenomena in the early days of quantum theory, it is only very recently that entanglement and coherence have been recognized as resources for the emerging quantum technologies, including quantum metrology, quantum communication, and quantum computing. In this work we study the limitations for the interconversion between coherence and entanglement. We prove a fundamental no-go theorem, stating that a general resource theory of sup…
We propose the use of hybrid entanglement in an entanglement swapping protocol, as means of distributing a Bell state with high fidelity to two parties, Alice and Bob. The hybrid entanglement used in this work is described as a discrete variable (Fock state) and a continuous variable (cat state superposition) entangled state. We model equal and unequal levels of photonic loss between the two propagating continuous variable modes, before detecting these states via a projective vacuum-one-photon measurement, and the other mode via balanced homodyne detection. We investigate homodyne measurement …
The quasidistributions corresponding to the diagonal representation of quantum states are discussed within the framework of operator-symbol construction. The tomographic-probability distribution describing the quantum state in the probability representation of quantum mechanics is reviewed. The connection of the diagonal and probability representations is discussed. The superposition rule is considered in terms of the density-operator symbols. The separability and entanglement properties of multipartite quantum systems are formulated as the properties of the density-operator symbols of the sys…
We study characteristics of superpositions and entanglement of thermal states at high temperatures and discuss their applications to quantum information processing. We introduce thermal-state qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states without photon number resolving detection but with Kerr nonlinear interactions and two single-photon detectors. This enables one to perform quantum teleportation and gate operations for quantum computation with …
We investigate how the entanglement properties of a two-mode state can be improved by performing a coherent superposition operation of photon subtraction and addition, proposed by Lee and Nha [Phys. Rev. A 82, 053812 (2010)], on each mode. We show that the degree of entanglement, the EPR-type correlation, and the performance of quantum teleportation can be all enhanced for the output state when the coherent operation is applied to a two-mode squeezed state. The effects of the coherent operation are more prominent than those of the mere photon subtraction and the addition particularly in the sm…
Entanglement can modify the interference patterns of multi-particle systems. We analyse, using the path integral formalism, a novel example of multi-particle interference and some unexplored aspects of this phenomenon by considering the two-slit arrangement with two distinguishable particles in a superposition state. The two components of entanglement, multiplicity of multi-particle terms and their coherence, can be studied separately by comparing the patterns of product, mixed and superposition states. This suggests a scheme to detect multi-particle superposition in some favourable cases. Mor…
Monogamy and polygamy relations characterize the distributions of entanglement in multipartite systems. We investigate the monogamy and polygamy relations with respect to any partitions for a superposition of the generalized $W$-class state and vacuum in terms of the Tsallis-$q$ entanglement and the Rényi-$α$ entanglement. By using the Hamming weight of the binary vectors related to the partitions of the subsystems, new classes of monogamy and polygamy inequalities are derived, which are shown to be tighter than the existing ones. Detailed examples are presented to illustrate the finer charact…
Quantum entanglement is the quintessence of quantum information processing mostly limited to the microscopic regime governed by Heisenberg uncertainty principle. For practical applications, however, macroscopic entanglement gives great benefits in both photon loss and sensitivity. Recently, a novel method of macroscopic entanglement generation has been proposed and demonstrated in a coupled interferometric system using classical laser light, where superposition between binary bases in each interferometric system plays a key role. Here, the function of path superposition applied to independent …
We describe a set of measurement protocols for performing non-classicality tests and the verification of entangled superposition states of macroscopic continuous variable systems, such as nanomechanical resonators. Following earlier works, we first consider a setup where a two-level system is used to indirectly probe the motion of the mechanical system via Ramsey measurements and discuss the application of this methods for detecting non-classical mechanical states. We then show that the generalization of this techniques to multiple resonator modes allows the conditioned preparation and the det…
The entanglement quantified by negativity of pure bipartite superposed states is studied. We show that if the entanglement is quantified by the concurrence two pure states of high fidelity to one another still have nearly the same entanglement. Furthermore this conclusion can be guaranteed by our obtained inequality, and the concurrence is shown to be a continuous function even in infinite dimensions. The bounds on the negativity of superposed states in terms of those of the states being superposed are obtained. These bounds can find useful applications in estimating the amount of the entangle…
Pairwise correlation is really an important property for multi-qubit states. For the two-qubit X states extracted from Dicke states and their superposition states, we obtain a compact expression of the quantum discord by numerical check. We then apply the expression to discuss the quantum correlation of the reduced two-qubit states of Dicke states and their superpositions, and the results are compared with those obtained by entanglement of formation, which is a quantum entanglement measure.
We present a new interpretation of the terms superposition, entanglement, and measurement that appear in quantum mechanics. We hypothesize that the structure of the wave function for a quantum system at the sub-Planck scale has a deterministic cyclic structure. Each cycle comprises a sequential succession of the eigenstates that comprise a given wave function. Between unitary operations or measurements on the wave function, the sequential arrangement of the current eigenstates chosen by the system is immaterial, but once chosen it remains fixed until another unitary operation or measurement ch…
Characterizing Entanglement in Combinations of Bell States through Superposition and Mixing: An Increase in Entanglement on Introducing Depolarizing, Phase Damping, and Amplitude Damping Noise
University of Southern Mississippi（南密西西比大学）；Brigham Young University（杨百翰大学）
Quantum entanglement and quantum nonlocality of $N$-photon entangled states $|ψ_{N m}> =\mathcal{N}_{m}[\cosγ|N-m>_{1}|m>_{2} +e^{iθ_{m}}\sinγ|m>_{1}|N-m>_{2}]$ and their superpositions are studied. We indicate that the relative phase $θ_{m}$ affects quantum nonlocality but not quantum entanglement for the state $|ψ_{N m}>$. We show that quantum nonlocality can be controlled and manipulated by adjusting the state parameters of $|ψ_{N m}>$, superposition coefficients, and the azimuthal angles of the Bell operator. We also show that the violation of the Bell inequality can reach its maximal valu…
In this paper we re-investigate the core of Schroedinger's 'cat paradox'. We argue that one has to distinguish clearly between superpositions of macroscopic cat states and superpositions of entangled states which comprise both the state of the cat. It is shown, that in the first instance recurrence to decoherence or other mechanisms is not necessary in this special case in order to explain the absence of macroscopic superpositions. Additionally, we present modified versions of two quantum optical experiments as experimenta crucis. Applied rigorously, quantum mechanical formalism reduces the pr…
The extended Bloch representation of quantum mechanics was recently derived to offer a (hidden-measurement) solution to the measurement problem. In this article we use it to investigate the geometry of superposition and entangled states, explaining the interference effects, and the entanglement correlations, in terms of the different orientations that a state-vector can take within the generalized Bloch sphere. We also introduce a tensorial determination of the generators of SU(N), particularly suitable to describe multipartite systems, from the viewpoint of the sub-entities. We then use it to…
The question whether Quantum Mechanics describes all aspects of both the micro-world of elementary particles and living organisms adequately is still open. A Bell inequality test without locality and detection loopholes will deliver the answer when applied to macroscopic systems. Here we show that an experimentally feasible loophole-free Bell test is possible using only multi-photon polarization entanglement and linear optics. We apply a preselection protocol which produces genuine macroscopic superpositions and entanglement, thus eliminating postselection and dark counts. Such superpositions …
Quantum mechanics predicts microscopic phenomena with undeniable success. Nevertheless, current theoretical and experimental efforts still do not yield conclusive evidence that there is, or not, a fundamental limitation on the possibility to observe quantum phenomena at the macroscopic scale. This question prompted several experimental efforts producing quantum superpositions of large quantum states in light or matter. Here we report on the observation of entanglement between a single photon and an atomic ensemble. The certified entanglement stems from a light-matter micro-macro entangled stat…
We show that a cavity field can evolve from an initial displaced mixed thermal state to a macroscopic superpositions of displaced thermal states via resonant interaction with a two-level atom. As a macroscopic system (meter) is really in a mixed state before coupling with the microscopic system at some temperature, our result is important for studying the quantum measurement problem and decoherence under real conditions. For the two-mode case, entanglement of displaced thermal states between the modes can be obtained.
We motivate and introduce a class of "hierarchical" quantum superposition states of $N$ coupled quantum oscillators. Unlike other well-known multimode photonic Schrödinger cat states such as entangled coherent states, the hierarchical superposition states are characterized as two-branch superpositions of tensor products of single-mode Schrödinger cat states. In addition to analyzing the photon statistics and quasiprobability distributions of prominent examples of these nonclassical states, we consider their usefulness for high-precision quantum metrology of nonlinear optical Hamiltonians and q…
The probability of simultaneous tunnelling of two particles is modified when the system is in a non-separable state, either entangled or symmetrised. We compare both effects in the rectangular potential barrier by evaluating the transmission rates in superposition states. For large momenta, their simultaneous presence greatly changes the form of the transmission rates. The joint effects are much larger than the superposition ones. Moreover, there are significant differences between bosons and fermions. We present an unified view of the combined effects as a quantum interference phenomenon. The…
We proposed two linear optics based entanglement concentration protocols (ECPs) to obtain maximally entangled 4-mode Cluster-type entangled coherent state (ECS) from less (partially) entangled Cluster-type ECS. The first ECP is designed using a superposition of single-mode coherent state with two unknown parameters, whereas the second ECP is obtained using a superposition of single-mode coherent state and a superposition of two-mode coherent state with four unknown parameters. The success probabilities have been calculated for both the ECPs. Necessary quantum circuits enabling future experimen…
An entangled two-mode coherent state is studied within the framework of $2\times 2$ dimensional Hilbert space. An entanglement concentration scheme based on joint Bell-state measurements is worked out. When the entangled coherent state is embedded in vacuum environment, its entanglement is degraded but not totally lost. It is found that the larger the initial coherent amplitude, the faster entanglement decreases. We investigate a scheme to teleport a coherent superposition state while considering a mixed quantum channel. We find that the decohered entangled coherent state may be useless for qu…
