We show an example of benign non-separability in an apparently separable system consisting of $n$ free non-correlated quantum particles, solitonic solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The non-separability manifests itself in the wave function of a single particle being influenced by the very presence of other particles. In the simplest case of identical particles, it is the number of particles that affects the wave function of each particle and, in particular, the width of its Gaussian probability density. As a result, this width, a local property, is directly linked to the mass of the entire Universe in a very Machian manner. In the realistic limit of large $n$ if the width in question is to be microscopic, the coupling constant must be very small resulting in an ``almost linear'' theory. This provides a model explanation of why the linearity of quantum mechanics can be accepted with such a high degree of certainty even if the more fundamental underlying theory could be nonlinear. We also demonstrate that when such non-correlated solitons are coupled to harmonic oscilators they lead to a faster-than-light nonlocal telegraph since changing the frequency of one oscillator affects instantaneously the probability density of particles associated with other oscillators. This effect can be alleviated by fine-tuning the parameters of the solution. Exclusion rules of a novel kind that we term supersuperselection rules also emerge from these solutions. They are similar to the mass and the univalence superselection rules in linear quantum mechanics. The effects in question and the exclusion rules do not appear if a weakly separable extension to $n$-particles is employed.
We derive N-particle Bell-type inequalities under the assumption of partial separability, i.e. that the N-particle system is composed of subsystems which may be correlated in any way (e.g. entangled) but which are uncorrelated with respect to each other. These inequalities provide, upon violation, experimentally accessible sufficient conditions for full N-particle entanglement, i.e. for N-particle entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. The inequalities are shown to be maximally violated by the N-particle Greenberger-Horne…
A new interpretation offers a consistent conceptual basis for nonrelativistic quantum mechanics. The violation of Bell's inequality is explained by maintaining realism, inductive inference and Einstein separability.
Careful analysis of cluster separability opens a way to a completely new understanding of preparation and registration procedures for microsystems: they are changes of separation status. An important observation is that quantum mechanics does not specify what happens to systems when they change their separation status. New rules that close this gap can therefore be added without disturbing the logic of quantum mechanics. Another important observation is that registration apparatuses for microsystems must contain detectors and that their readings are signals from detectors. This leads to furthe…
We give a necessary and sufficient condition for a mixed quantum mechanical state to be separable. The criterion is formulated as a boundedness condition in terms of the greatest cross norm on the tensor product of trace class operators.
A modified Beltrametti-Cassinelli-Lahti model of measurement apparatus that satisfies both the probability reproducibility condition and the objectification requirement is constructed. Only measurements on microsystems are considered. The cluster separability forms a basis for the first working hypothesis: the current version of quantum mechanics leaves open what happens to systems when they change their separation status. New rules that close this gap can therefore be added without disturbing the logic of quantum mechanics. The second working hypothesis is that registration apparatuses for mi…
We analyze the structure of correlations among more than two quantum systems. We introduce a classification of correlations based on the concept of non-separability, which is different {\em a priori} from the concept of entanglement. Generalizing a result of Svetlichny [Phys. Rev. D {\bf 35} (1987) 3066] on 3-particle correlations, we find an inequality for $n$-particle correlations that holds under the most general separability condition and that is violated by some quantum-mechanical states.
A finite dimensional quantum mechanical system is modeled by a density rho, a trace one, positive semi-definite matrix on a suitable tensor product space H[N] . For the system to demonstrate experimentally certain non-classical behavior, rho cannot be in S, a closed convex set of densities whose extreme points have a specificed tensor product form. Two mathematical problems in the quantum computing literature arise from this context: (1) the determination whether a given rho is in S and (2) a measure of the ``entanglement'' of such a rho in terms of its distance from S. In this paper we descri…
What ontology does realism about the quantum state suggest? The main extant view in contemporary philosophy of physics is wave-function realism. We elaborate the sense in which wave-function realism does provide an ontological picture; and defend it from certain objections that have been raised against it. However, there are good reasons to be dissatisfied with wave-function realism, as we go on to elaborate. This motivates the development of an opposing picture: what we call spacetime state realism; a view which takes the states associated to spacetime regions as fundamental. This approach en…
The paper describes a solution to the problem of quantum measurement that has been proposed recently. The literal understanding of the basic rule of quantum mechanics on identical particles violates the cluster separation principle and so leads to difficulties. A proposal due to Peres of how such difficulties could be removed is reformulated and extended. Cluster separability leads to a locality requirement on observables and to the key notion of separation status. Separation status of a microsystem is shown to change in preparation and registration processes. The indispensability of detectors…
Recently, Braunstein et al. [1] introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs whose normalized Laplacian matrix is multipartite entangled under any vertex labeling. Furthermore, we give conditions on the vertex degrees such that there is a vertex labeling under which the normalized …
The separability problem is one of the basic and emergent problems in the present and future quantum information processing. The latter focuses on information and computing based on quantum mechanics and uses quantum bits as its basic information units. In this paper, we present an overview of the progress in the separability problem in bipartite systems, more specifically in two quantum bits systems from the criterion based on the inequalities of Bell in $1964$ to the recent criteria of separability in 2018.
Let C be the set of all possible quantum states. We study the convex subsets of C with attention focused on the lattice theoretical structure of these convex subsets and, as a result, find a framework capable of unifying several aspects of quantum mechanics, including entanglement and Jaynes' Max-Ent principle. We also encounter links with entanglement witnesses, which leads to a new separability criteria expressed in lattice language. We also provide an extension of a separability criteria based on convex polytopes to the infinite dimensional case and show that it reveals interesting facets c…
Recent frameworks describing quantum mechanics in the absence of a global causal order admit the existence of causally indefinite processes, where it is impossible to ascribe causal order for events A and B. These frameworks even allow for processes that violate the so-called causal inequalities, which are analogous to Bell's inequalities. However, the physicality of these exotic processes is, in the general case, still under debate, bringing into question their foundational relevance. While it is known that causally indefinite processes can be probabilistically realised by means of a quantum …
The formalism of classical and quantum mechanics on phase space leads to symplectic and Heisenberg group representations, respectively. The Wigner functions give a representation of the quantum system using classical variables. The correspondence between the classical and the quantum criterion of separability for the system is obtained in terms of these functions. Entanglement is generic and separability is special. Some applications are discussed in commonly occuring examples and possibly in exotic systems.
Quantum mechanics, in principle, allows for processes with indefinite causal order. However, most of these causal anomalies have not yet been detected experimentally. We show that every such process can be simulated experimentally by means of non-Markovian dynamics with a measurement on additional degrees of freedom. Explicitly, we provide a constructive scheme to implement arbitrary acausal processes. Furthermore, we give necessary and sufficient conditions for open system dynamics with measurement to yield processes that respect causality locally, and find that tripartite entanglement and no…
We show that the Bub-Clifton uniqueness theorem for 'no collapse' interpretations of quantum mechanics (Studies in the History and Philosophy of Modern Physics 27, 181-219 (1996)) can be proved without the 'weak separability' assumption.
Schroedinger equation on a Hilbert space ${\cal H}$, represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space $P {\cal H}$. Separable states of a bipartite quantum system form a special submanifold of $P {\cal H}$. We analyze the Hamiltonian dynamics that corresponds to the quantum system constrained on the manifold of separable states, using as an important example the system of two interacting qubits. The constraints introduce nonlinearities which render the dynamics nontrivial. We show that the qualitative properties of the constrain…
Clarifying the relation between the whole and its parts is crucial for many problems in science. In quantum mechanics, this question manifests itself in the quantum marginal problem, which asks whether there is a global pure quantum state for some given marginals. This problem arises in many contexts, ranging from quantum chemistry to entanglement theory and quantum error correcting codes. In this paper, we prove a correspondence of the marginal problem to the separability problem. Based on this, we describe a sequence of semidefinite programs which can decide whether some given marginals are …
Quantum mechanics is already 100 years old, but remains alive and full of challenging open problems. On one hand, the problems encountered at the frontiers of modern theoretical physics like Quantum Gravity, String Theories, etc. concern Quantum Theory, and are at the same time related to open problems of modern mathematics. But even within non-relativistic quantum mechanics itself there are fundamental unresolved problems that can be formulated in elementary terms. These problems are also related to challenging open questions of modern mathematics; linear algebra and functional analysis in pa…
The purpose of this short article is to build on the work of Ghirardi, Marinatto and Weber (Ghirardi, Marinatto & Weber 2002; Ghirardi & Marinatto 2003, 2004, 2005) and Ladyman, Linnebo and Bigaj (2013), in supporting a redefinition of entanglement for "indistinguishable" systems, particularly fermions. According to the proposal, non-separability of the joint state is insufficient for entanglement. The re-definition is justified by its physical significance, as enshrined in three biconditionals whose analogues hold of "distinguishable" systems.
We have studied the emergence of classical states in the perturbative interaction model. The states which interact with many other degrees of freedom, such as the center of mass of a macro-object, play important role. Although the random phase mechanism is effective as same as Zurek's strong correlation model, there are enormous states, each of which independently developes due to the orthogonality of the environmental states. In these privileged states, the subsystem picture with the separability is stable.
A general method for extending a non-dissipative nonlinear Schrödinger and Liouville-von Neumann 1-particle dynamics to an arbitrary number of particles is described. It is shown at a general level that the dynamics so obtained is completely separable, which is the strongest condition one can impose on dynamics of composite systems. It requires that for all initial states (entangled or not) a subsystem not only cannot be influenced by any action undertaken by an observer in a separated system (strong separability), but additionally that the self-consistency condition $Tr_2\circ ϕ^t_{1+2}=ϕ^t_{…
This paper addresses arguments that "separability" is an assumption of Bell's theorem, and that abandoning this assumption in our interpretation of quantum mechanics (a position sometimes referred to as "holism") will allow us to restore a satisfying locality principle. Separability here means that all events associated to the union of some set of disjoint regions are combinations of events associated to each region taken separately. In this article, it is shown that: (a) localised events can be consistently defined without implying separability; (b) the definition of Bell's locality condition…
Recently normalized Laplacian matrices of graphs are studied as density matrices in quantum mechanics. Separability and entanglement of density matrices are important properties as they determine the nonclassical behavior in quantum systems. In this note we look at the graphs whose normalized Laplacian matrices are separable or entangled. In particular, we show that the number of such graphs is related to the number of 0-1 matrices that are line sum symmetric and to the number of graphs with at least one vertex of degree 1.
We address the question of description of qubit system in a formalism based on the nilpotent commuting variables. In this formalism qubits exhibit properties of composite objects being subject of the Pauli exclusion principle, but otherwise behaving boson-like. They are not fundamental particles. In such an approach the classical limit yields the nilpotent mechanics. Using the space of $η$-wavefunctions, generalized Schrödinger equation etc. we study properties of pure qubit systems and also properties of some composed, hybrid models: fermion-qubit, boson-qubit. The fermion-qubit system can be…
It is shown that any quantum operation that perfectly clones the entanglement of all maximally-entangled qubit pairs cannot preserve separability. This ``entanglement no-cloning'' principle naturally suggests that some approximate cloning of entanglement is nevertheless allowed by quantum mechanics. We investigate a separability-preserving optimal cloning machine that duplicates all maximally-entangled states of two qubits, resulting in 0.285 bits of entanglement per clone, while a local cloning machine only yields 0.060 bits of entanglement per clone.
Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are responsible for many nonintuitive physical phenomena. The entanglement property of normalized Laplacian matrices is in general not invariant under graph isomorphism. In recent papers, graphs were identified wh…
In this paper we review a proposed geometrical formulation of quantum mechanics. We argue that this geometrization makes available mathematical methods from classical mechanics to the quantum frame work. We apply this formulation to the study of separability and entanglement for states of composite quantum systems.
Recently, [{arXiv:0810.3134}] is accepted and published. We show that local realistic theories violate separability of $N$-qubit quantum states by a factor of $3^N$. Thus we have a violation factor of 3 when N=1. Realistic theories violate a proposition of a single qubit (a state with two-dimensional space) system, the Bloch sphere machinery of quantum mechanics. Our method uses the ratio of each of the scalar products. The maximal possible value of the scalar product in local realistic theories grows exponentially larger than one in $N$-qubit separable states when the number of qubits increas…
Quantumness and separability criteria for continuous variable systems are discussed for the case of a noncommutative (NC) phase-space. In particular, the quantum nature and the entanglement configuration of NC two-mode Gaussian states are examined. Two families of covariance matrices describing standard quantum mechanics (QM) separable states are deformed into a NC QM configuration and then investigated through the positive partial transpose criterium for identifying quantum entanglement. It is shown that the entanglement of Gaussian states may be exclusively induced by switching on the NC def…
Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5. Keywords: non-linear and linear entanglement …
Some fundamental aspects related with the construction of Robertson-Schrödinger like uncertainty principle inequalities are reported in order to provide an overall description of quantumness, separability and nonlocality of quantum systems in the noncommutative phase-space. Some consequences of the deformed noncommutative algebra are also considered in physical systems of interest.
Symmetry plays an important role in the field of quantum mechanics. In this paper, we consider a subclass of symmetric quantum states in the multipartite system $N^{\otimes d}$, namely, the completely symmetric states, which are invariant under any index permutation. It was conjectured by L. Qian and D. Chu [arXiv:1810.03125 [quant-ph]] that the completely symmetric states are separable if and only if it is a convex combination of symmetric pure product states. In this paper, we proved that this conjecture is true for both bipartite and multipartite cases. And we proved the completely symmetri…
A general method for testing essential nonlocality of nonlinear modifications of quantum mechanics is presented and applied to show the inconsistency of I. Bialynicki-Birula's and J. Mycielski's nonlinear quantum theory.
The Schroedinger equation with the nonlinear term is derived by the natural generalization of the hydrodynamical model of quantum mechanics. The nonlinear term appears to be logically necessary because it enables explanation of the classical limit of the wave function, the collaps of the wave function and solves the Schroedinger cat paradox.
We present a general description of separable states in Quantum Mechanics. In particular, our result gives an easy proof that inseparabitity (or entanglement) is a pure quantum (noncommutative) notion. This implies that distinction between separability and inseparabitity has sense only for composite systems consisting of pure quantum subsystems. Moreover, we provide the unified characterization of pure-state entanglement and mixed-state entanglement.
Bell suggested that a new perspective on quantum mechanics was needed. We propose a solution of the measurement problem based on a reconsideration of the nature of particles. The solution is presented with an idealized model involving non-locality or non-separability, identified in 1927 by Einstein and implicit in the standard interpretation of single slit (or hole) diffraction. Considering particles as being localizable entities leads to an `induced collapse model', a parameter-free alternative to spontaneous collapse models, that affords a new perspective on, \emph{inter alia}, nuclear decay…
The groupoids description of Schwinger's picture of quantum mechanics is continued by discussing the closely related notions of composition of systems, subsystems, and their independence. Physical subsystems have a neat algebraic description as subgroupoids of the Schwinger's groupoid of the system. The groupoids picture offers two natural notions of composition of systems: Direct and free products of groupoids, that will be analyzed in depth as well as their universal character. Finally, the notion of independence of subsystems will be reviewed, finding that the usual notion of independence, …
A framework for a quantum mechanical information theory is introduced that is based entirely on density operators, and gives rise to a unified description of classical correlation and quantum entanglement. Unlike in classical (Shannon) information theory, quantum (von Neumann) conditional entropies can be negative when considering quantum entangled systems, a fact related to quantum non-separability. The possibility that negative (virtual) information can be carried by entangled particles suggests a consistent interpretation of quantum informational processes.
If quantum states exhibit small nonlinearities during time evolution, then quantum computers can be used to solve NP-complete problems in polynomial time. We provide algorithms that solve NP-complete and #P oracle problems by exploiting nonlinear quantum logic gates. It is argued that virtually any deterministic nonlinear quantum theory will include such gates, and the method is explicitly demonstrated using the Weinberg model of nonlinear quantum mechanics.
To help resolve stubborn aporias in quantum mechanics, Kastner, Kauffman and Epperson have recently proposed a new interpretation of Heisenbergs ontological duality of res extensa and res potentia. In related work, Khrennikov and Aerts and their respective associates have explored the intermediate domain of mesoscopic phenomena in which quantum-like behavior can be observed and formalized. We generalize this work to define a third domain, one of macroscopic, interactive processes to which quantum concepts, but not quantum formalism apply. We show that they follow the non-standard logic propose…
Due to its connection to the diffeomorphism group, nonlinear quantum mechanics may play an important role in quantum geometry. The Doebner-Goldin nonlinearity (arising from representations of the diffeomorphism group) amplifies nonlocal signaling effects under extreme localization, suggesting that even if greatly suppressed at low energies, such effects may be significant at the Planck scale. This offers new perspectives on Planck-scale physics.
A model of reality is called separable if the state of a composite system is equal to the union of the states of its parts, located in different regions of space. Spekkens has argued that it is trivial to reproduce the predictions of quantum mechanics using a separable ontological model, provided one allows for arbitrary violations of `dynamical locality'. However, since dynamical locality is strictly weaker than local causality, this leaves open the question of whether an ontological model for quantum mechanics can be both separable and dynamically local. We answer this question in the affirm…
We show how one can compute multiple-time multi-particle correlation functions in nonlinear quantum mechanics in a way which guarantees locality of the formalism.
It is shown that, in order to avoid unacceptable nonlocal effects, the free parameters of the general Doebner-Goldin equation have to be chosen such that this nonlinear Schrödinger equation becomes Galilean covariant.
The possibility of teleportation is by sure the most interesting consequence of quantum non-separability. So far, however, teleportation schemes have been formulated by use of state vectors and considering individual entities only. In the present article the feasibility of teleportation is examined on the basis of the rigorous ensemble interpretation of quantum mechanics (not to be confused with a mere treatment of noisy EPR pairs) leading to results which are unexpected from the usual point of view.
Raman-type laser excitation of a trapped atom allows one to realize the quantum mechanical counterpart of phenomena of nonlinear optics, such as Kerr-type nonlinearities, parametric amplification, and multi-mode mixing. Additionally, huge nonlinearities emerge from the interference of the atomic wave function with the laser waves. They lead to a partitioning of the phase space accompanied by a significantly different action of the time evolution in neighboring phase-space zones. For example, a nonlinearly modified coherent "displacement" of the motional quantum state may induce strong amplitud…
We propose a technique to couple the position operator of a nano mechanical resonator to a SQUID device by modulating its magnetic flux bias. By tuning the magnetic field properly, either linear or quadratic couplings can be realized, with a discretely adjustable coupling strength. This provides a way to realize coherent nonlinear effects in a nano mechanical resonator by coupling it to a Josephson quantum circuit. As an example, we show how squeezing of the nano mechanical resonator state can be realized with this technique. We also propose a simple method to measure the uncertainty in the po…
In view of the recent quest for well-behaved nonlinear extensions of the traditional Schroedinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, in this paper, together with a review of the general features of the nonlinear quantum (thermo)dynamics I proposed in a series of papers [see references in G.P. Beretta, Found.Phys. 17, 365 (1987)], I show its exact equivalence with the maximal-entropy-production variational-principle formulation recently derived in S. Gheorghiu-Svirschevsk…
