We show that quantum entanglement and the Heisenberg uncertainty principle are inextricably connected. Toward this end, a complete set of commuting observables (CSCO) criterion for the entanglement is developed. Assuming (A1,A2,...) and (B1,B2,...) being two CSCO's for a given system, and C being the matrix, Cij = i [Bi,Aj], for each given row i (i=1,2,...) if at least one matrix element Cij (j=1,2,...) is nonzero, then for the simultaneous eigenstates |ψ)=|A1',A2',...) of (A1,A2,...), the simultaneous measurements of (B1,B2,...) are, in general,entangled. The only exception is when all the simultaneous eigenstates |ψ)= A1', A2',...), (ψ|C|ψ)=0. This CSCO criterion may be considered as an extension of the Heisenberg uncertainty principle to quantum systems with either two (or more) particles or multi-degrees of freedom (MDF).
We show that quantum entanglement and the Heisenberg uncertainty principle are inextricably connected. Toward this end, a complete set of commuting observables (CSCO) criterion for the entanglement is developed. Assuming (A1,A2,...) and (B1,B2,...) being two CSCO's for a given system, and C being the matrix, Cij = i [Bi,Aj], for each given row i (i=1,2,...) if at least one matrix element Cij (j=1,2,...) is nonzero, then for the simultaneous eigenstates |ψ)=|A1',A2',...) of (A1,A2,...), the simultaneous measurements of (B1,B2,...) are, in general,entangled. The only exception is when all the simultaneous eigenstates |ψ)= A1', A2',...), (ψ|C|ψ)=0. This CSCO criterion may be considered as an extension of the Heisenberg uncertainty principle to quantum systems with either two (or more) particles or multi-degrees of freedom (MDF).
Abstract A criterion for entanglement detection based on covariance matrices for an arbitrary set of observables is formulated. The method generalizes the covariance matrix entanglement criterion by Simon (2000 Phys. Rev. Lett. 84 2726) to a more general set of operators using the positive partial transpose test for the covariance matrix. The relation is found by starting from the generalized uncertainty relation for multiple operators, and taking the partial transpose on the bipartition. The method is highly efficient and versatile in the sense that the set of measurement operators can be freely chosen, and there is no constraint on the commutation relations. The main restriction on the chosen set of measurement operators is that the correlators and expectation values of the partially transposed observable operators can be evaluated. The method is particularly suited for systems with higher dimensionality since the computations do not scale with the dimension of the Hilbert space—rather they scale with the number of chosen observables. We illustrate the approach by examining the entanglement between two spin ensembles, and show that it detects entanglement in a basis independent way.
refer to the joint measurement problem for noncommuting quantum observables: “One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations).” The object of the present paper is to consider what approximate solution one may hope to find to the joint measurement problem. Our main conclusion, based on a rigorous notion of unsharp observable, is that one can reasonably expect to find approximate joint measurements for any pair of noncommuting sharp observables, provided a trade-off in the sense of a Heisenberg uncertainty relation for measurement inaccuracies is taken into account. In addition, it will be seen that as a consequence of the noncom-mutativity of the given observables, the approximating observables must have degrees of unsharpness that obey yet another trade-off relation in the spirit of Heisenberg’s principle. These connections will be
Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.
Abstract. Quantum physics against intuition. Since Galileo, deeper understanding of physical reality often contradicts everyday’s intuition. This trend has become dramatic a century ago, rendering modern physics abstruse to the laymen. The scientist can exert his intuition and his imagination within an elaborate mathematical language, but it is impossible to translate his ideas faithfully into ordinary language; this difficulty may be one of the causes of the present tarnish of the image of science. A first shock was brought in by relativity, which revolutionized our conceptions of space and time, but quantum physics presents even more counter-intuitive features. One has been led to represent physical quantities pertaining to a system, such as the position or the velocity of a particle, no longer by ordinary numbers but by “observables”, mathematical objects which belong to a non-commutative algebra. Namely, the product of two such observables is not a unique new observable but may depend on their order: the products AB and BA may differ.The values that a physical quantity can take constitute a set of ordinary numbers, the eigenvalues a( n) of the observable A. This set may be discrete as a consequence of the algebra. The discreteness of the eigenvalues of the energy of an atom explains the spectrum of its radiation. Any component of the intrinsic angular momentum of an electron can take only two opposite values, baffling our geometric intuition.If two observables commute (i.e., AB = BA), they are compatible: both may take well-defined values chosen among the eigenvalues. If they do not commute (i.e., AB ≠ BA), which occurs for the position and velocity of a particle, for the electric and magnetic field, or for the components of an angular momentum, they are incompatible: their values cannot be measured nor even specified simultaneously. Each of them presents statistical fluctuations which must be larger than some lower bound (Heisenberg’s inequality). The state of a system, prepared under some definite conditions, should be identified as the set of expectation values of all observables.Thus, quantum physics, our most fundamental description of nature, is a theory involving irreducible probabilities and unavoidable uncertainties. Its predictions do not refer to a single system, but to a statistical ensemble, and the observer cannot be completely disregarded. Moreover it may involve aspects which are apparently contradictory (Bohr’s complementarity). An electron, an atom or a molecule may behave in limiting cases either as a particle or as a wave; under intermediate conditions, it behaves as both, but each of the descriptions is blurred with probabilities. As a wave, it can interfere with itself. Likewise, an electromagnetic field has the double nature of a set of photons and of a wave. At the macroscopic scale, relative fluctuations are small, so that determinism emerges.Quantum probabilities differ from standard ones. In particular, quantum correlations involving four observables pair wise violate Bell’s inequalities, so that microscopic quantities are intrinsically indeterminate and depend on the observation process.The GHZ paradox, due to Greenberger, Horne and Zeilinger, exemplifies the most troublesome conceptual feature of quantum physics. Indeed, it baffles not only intuition, but also current logics. It deals with a physical system made of three correlated particles. Various observables, A1, A2, A3, B1, B2 and B3 ≡ B1 B2, which we do not specify here, can be measured on such a composite system. Each of them may take discrete values a1 , a2, a3, b1, b2 equal either to +1 or to –1. Since the three observables A1 B1, A2 B2 and A3 B3 are compatible, they can be simultaneously controlled. Consider then a statistical ensemble of systems, all prepared identically in such a way that A1 B1 = A2 B2 = A3 B3 = +1. If we measure A1 and B1, we shall find either a1 = b1 = +1 or a1 = b1 = –1, that is always a1 = b1; likewise for a2 = b2 = ±1 and a3 = b1 b2 = ±1. We are tempted to infer that a simultaneous determination of the values a1, a2 and a3 of A1, A2 and A3 (which are compatible) should verify a3 = a1 a2, that is, a1 a2 a3 = 1. However both theory and experiment provide in all cases a1 a2 a3 = –1. In fact the four statements a1 = b1, a2 = b2, a3 = b1 b2 and a1 a2 a3 = –1 which seem contradictory are separately “true”, in the sense that each one can be verified experimentally; but such experiments cannot be performed on a single system. We are not allowed to make a logical reasoning based on a simultaneous consideration of the values of B1 and A2 because these observables do not commute. No calculation, no experiment can simultaneously provide values for the numbers b1 and a2. Hence, we cannot regard as true together the statements a1 = b1 and a2 = b2, although they can be checked by separate experiments. Each one is true, but their union is meaningless.Thus, if theory does not allow us to imagine thought experiments in which several statements on a system might simultaneously be checked, we cannot draw conclusions by putting such statements together. We are led to regard the “truth” of a quantum statement as contextual. The values of physical quantities are not attached only to the system in itself, but also refer to the context in which their relations may be verified. Quantum mechanics does not deal with objects per se, but with what we know and what we can infer about them. It does not provide a direct image of the reality of the object since it involves its observation. Although it is a perfect tool for letting us make statistical predictions, and although it appears as our most basic theory owing to its precision and generality, it violates our daily intuition, our elementary mathematical intuition and our feelings about reality. Its probabilities and even its logics obey rather shocking new rules.
Information is not in the carrier; it is in the definition. Communication does not transmit content; it only transmits a trigger. The Same-Origin World System is a carrier-independent formal theory. Its core quantity is the same-origin degree, its structure is the same-origin matrix, and its evolution is row-stochastic dynamics. Core chain: 1 = N → same-origin degree → same-origin matrix → dynamics → convergence → one controls all. 1 = N: The 1 selected from N has an internal structure that is the N of the next level. This recursion can continue indefinitely. Same-origin degree: D_same(i,j) = exp(-D_fine × Σ w_k × d_k²), describing the similarity between attributes. Same-origin matrix: The matrix of all pairwise same-origin degrees, normalized to be row-stochastic. Dynamics: x(t+1) = D × x(t). The spectral radius of a row-stochastic matrix is ρ(D)=1, and the system converges to the principal eigenvector. One controls all: Changing one attribute changes all attributes. Four-copy local verification: End A has A1 + B2, End B has B1 + A2. Rigid binding A1=A2, B1=B2. Local comparison equals cross-end comparison. No communication is needed to verify consistency. Pulsar triggering: Both ends observe the same pulsar. The N-th pulse triggers both ends simultaneously. Event occurrence + index selection. Non-local connection, carrier-independent. Quantum function correspondences: Superposition (the same-origin matrix contains all states simultaneously), interference (phase θ_ij), entanglement (shared trigger source), non-locality (pulsar triggering), measurement (triggering), collapse (state change after triggering), many-worlds (decision point selects matrix). Information geometry correspondences: Probability distribution (difference distribution), Fisher metric (relational distance d = -ln D_same), divergence (same-origin degree), e-connection (internal weight), m-connection (external weight), geodesic (minimum-cost path), curvature (non-uniformity of difference distribution), natural gradient (same-origin dynamics). Carrier independence: The core is mathematics, independent of the speed of light, Planck constant, or electron charge. It can be implemented in optics, acoustics, electronics, or quantum systems. 1 = N. Control one = control all. Information is definition, communication is trigger, verification is local, synchronization is pulsar. Carrier-independent; the core is mathematics.
We propose a unifying approach to the separability problem using covariance matrices of locally measurable observables. From a practical point of view, our approach leads to strong entanglement criteria that allow us to detect the entanglement of many bound entangled states in higher dimensions and which are at the same time necessary and sufficient for two qubits. From a fundamental perspective, our approach leads to insights into the relations between several known entanglement criteria--such as the computable cross-norm and local uncertainty criteria--as well as their limitations.