The Experts below are selected from a list of 7674 Experts worldwide ranked by ideXlab platform
Frederic Holweck - One of the best experts on this subject based on the ideXlab platform.
-
embedding qubits into fermionic fock space peculiarities of the four qubit case
Physical Review D, 2015Co-Authors: Peter Levay, Frederic HolweckAbstract:We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physically sound meaning of some of the direct sum structures showing up in the context of the so-called Black-Hole/Qubit Correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of $Spin(16,\mathbb{C})$ invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with 8 modes. Here we find an algebraically independent generating set of the generalized SLOCC invariants and calculate their restriction to the Dense Orbit. We point out the special role the largest exceptional group $E_8$ is playing in these considerations.
-
embedding qubits into fermionic fock space peculiarities of the four qubit case
Physical Review D, 2015Co-Authors: Peter Levay, Frederic HolweckAbstract:We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physical sound meaning of some of the direct sum structures showing up in the context of the so-called black-hole/qubit correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self-conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of $\text{Spin}(16,\mathbb{C})$ invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with eight modes. Here we find an algebraically independent generating set of the generalized stochastic local operations and classical communication invariants and calculate their restriction to the Dense Orbit. We point out the special role the largest exceptional group ${E}_{8}$ is playing in these considerations.
Peter Levay - One of the best experts on this subject based on the ideXlab platform.
-
embedding qubits into fermionic fock space peculiarities of the four qubit case
Physical Review D, 2015Co-Authors: Peter Levay, Frederic HolweckAbstract:We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physically sound meaning of some of the direct sum structures showing up in the context of the so-called Black-Hole/Qubit Correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of $Spin(16,\mathbb{C})$ invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with 8 modes. Here we find an algebraically independent generating set of the generalized SLOCC invariants and calculate their restriction to the Dense Orbit. We point out the special role the largest exceptional group $E_8$ is playing in these considerations.
-
embedding qubits into fermionic fock space peculiarities of the four qubit case
Physical Review D, 2015Co-Authors: Peter Levay, Frederic HolweckAbstract:We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physical sound meaning of some of the direct sum structures showing up in the context of the so-called black-hole/qubit correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self-conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of $\text{Spin}(16,\mathbb{C})$ invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with eight modes. Here we find an algebraically independent generating set of the generalized stochastic local operations and classical communication invariants and calculate their restriction to the Dense Orbit. We point out the special role the largest exceptional group ${E}_{8}$ is playing in these considerations.
Víctor Jiménez López - One of the best experts on this subject based on the ideXlab platform.
-
On the Lebesgue measure of Li-Yorke pairs for interval maps
Communications in Mathematical Physics, 2010Co-Authors: Henk Bruin, Víctor Jiménez LópezAbstract:We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a Dense Orbit for $f \times f$. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where $f$ has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.
Xie Junyi - One of the best experts on this subject based on the ideXlab platform.
-
The existence of Zariski Dense Orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker)
2021Co-Authors: Xie JunyiAbstract:In this paper we prove the following theorem. Let $f$ be a dominant endomorphism of a smooth projective surface over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then there exists a closed point whose Orbit under $f$ is Zariski Dense. This result gives us a positive answer to the Zariski Dense Orbit conjecture proposed by Medvedev and Scanlon, by Amerik, Bogomolov and Rovinsky, and by Zhang, for endomorphisms of smooth projective surfaces. Moreover, we define a new canonical topology on varieties over an algebraically closed field which has finite transcendence degree over $\mathbb{Q}$. We call it the adelic topology. The adelic topology is stronger than the Zariski topology and an irreducible variety is still irreducible in this topology. Using the adelic topology, we propose an adelic verison of the Zariski Dense Orbit conjecture. This version is stronger then the original one and it quantifies how many such Orbits there are. We also proved this adelic version for endomorphisms of smooth projective surfaces. Moreover, we proved the adelic verison of the Zariski Dense Orbit conjecture for endomorphisms of abelian varieties and split polynomial maps. This yields new proofs for the original version in this two cases. In Appendix A, we study the endomorphisms on the $k$-affinoid spaces. We show that for certain endomorphism $f$ on a $k$-affinoid space $X$, the attractor $Y$ of $f$ is a Zariski closed subset and the dynamics of $f$ semi-conjugates to its restriction on $Y.$ A special case of this result is used in the proof of the main theorem. In Appendix B, written in collaboration with Thomas Tucker, we prove the Zariski Dense Orbit conjecture for endomorphisms of $(\mathbb{P}^1)^N.$Comment: 66 pages. We modified the definition of the adelic topolog
-
Remarks on algebraic dynamics in positive characteristic
2021Co-Authors: Xie JunyiAbstract:In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. We generalise some basic facts on arithmetic degree and canonical height in positive characteristic. As applications, we prove the dynamical Mordell-Lang conjecture for automorphisms of projective surfaces of positive entropy, the Zariski Dense Orbit conjecture for automorphisms of projective surfaces and for endomorphisms of projective varieties with large first dynamical degree. We also study ergodic theory for constructible topology. For example, we prove the equidistribution of backward Orbits for finite flat endomorphisms with large topological degree. As applications, we give a simple proof for weak dynamical Mordell-Lang and prove a counting result for backward Orbits without multiplicities. This gives some applications for equidistributions on Berkovich spaces.Comment: 34 page
-
Endomorphisms of quasi-projective varieties -- towards Zariski Dense Orbit and Kawaguchi-Silverman conjectures
2021Co-Authors: Jia Jia, Xie Junyi, Shibata Takahiro, Deqi ZhangAbstract:Let $X$ be a quasi-projective variety and $f\colon X\to X$ a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree $d_1(f)$ of $f$ equals the arithmetic degree $\alpha_f(P)$ at a point $P$ having Zariski Dense $f$-forward Orbit. Assuming $X$ is a smooth affine surface, such that the log Kodaira dimension $\bar{\kappa}(X)$ is non-negative (resp. the \'etale fundamental group $\pi_1^{\text{\'et}}(X)$ is infinite), we confirm AZO, (hence) ZDO, and KSC (when $\operatorname{deg}(f)\geq 2$) (resp. AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every surjective endomorphism on any projective variety with ''larger'' first dynamical degree (resp. every dominant endomorphism of any semiabelian variety).Comment: 35 pages; comments are welcome
Henk Bruin - One of the best experts on this subject based on the ideXlab platform.
-
On the Lebesgue measure of Li-Yorke pairs for interval maps
Communications in Mathematical Physics, 2010Co-Authors: Henk Bruin, Víctor Jiménez LópezAbstract:We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a Dense Orbit for $f \times f$. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where $f$ has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.