There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

Journal of Statistical Mechanics: theory and experiment

We give a general introduction to quantum phase transitions in strongly-correlated electron systems. These transitions which occur at zero temperature when a non-thermal parameter $g$ like pressure, chemical composition or magnetic field is tuned to a critical value are characterized by a dynamic exponent $z$ related to the energy and length scales $\Delta$ and $\xi$. Simple arguments based on an expansion to first order in the effective interaction allow to define an upper-critical dimension $D_{C}=4$ (where $D=d+z$ and $d$ is the spatial dimension) below which mean-field description is no longer valid. We emphasize the role of pertubative renormalization group (RG) approaches and self-consistent renormalized spin fluctuation (SCR-SF) theories to understand the quantum-classical crossover in the vicinity of the quantum critical point with generalization to the Kondo effect in heavy-fermion systems. Finally we quote some recent inelastic neutron scattering experiments performed on heavy-fermions which lead to unusual scaling law in $\omega /T$ for the dynamical spin susceptibility revealing critical local modes beyond the itinerant magnetism scheme and mention new attempts to describe this local quantum critical point.

https://arxiv.org/pdf/cond-mat/0102119.pdf

Quantum Phase Transitions

Entangled quantum systems have properties that have fundamentally overthrown the classical worldview. Increasing the complexity of entangled states by expanding their dimensionality allows the implementation of novel fundamental tests of nature, and moreover also enables genuinely new protocols for quantum information processing. Here we present the creation of a (100x100)-dimensional entangled quantum system, using spatial modes of photons. For its verification we develop a novel nonlinear criterion which infers entanglement dimensionality of a global state by using only information about its subspace correlations. This allows very practical experimental implementation as well as highly efficient extraction of entanglement dimensionality information. Applications in quantum cryptography and other protocols are very promising.

Studies of Quantum Entanglement in 100 Dimensions

Unraveling multipartite entanglement for an important family of mixed states

Entanglement in high-dimensional many-body systems plays an increasingly vital role in the foundations and applications of quantum physics. In the present paper, we introduce a theoretical concept which allows to categorize multipartite states by the number of degrees of freedom being entangled. In this regard, we derive computable and experimentally friendly criteria for arbitrary multipartite qudit systems that enable to examine in how many degrees of freedom a mixed state is genuine multipartite entangled.

Examining the dimensionality of genuine multipartite entanglement

The distribution of quantum entanglement between distant parties is a key challenge in the pursuit of a worldwide quantum network. Quantum repeaters and optical satellite links have both been proposed to overcome the distance limitations of ﬁber-based quantum networks. In order to test the viability of a world spanning quantum satellite network, several proof-of-concept studies have already demonstrated high-ﬁdelity transmission of photonic entanglement via terrestrial long-distance free-space links. With the recent launch of the Micius quantum satellite by the Chinese Academy of Sciences, the technological challenge of implementing this scheme in situ was overcome. All of these free-space experiments utilized entanglement in a two-dimensional state space, which in most cases was encoded in the polarization of photons. However, high-dimensional entanglement can yield signiﬁcant beneﬁts in quantum key distribution by increasing the secure key rate and enhancing the resilience to eavesdropping attacks. 

Energy-time entanglement is an established photonic degree of freedom (DOF); it is routinely used for the distribution of entanglement in ﬁber-based quantum cryptography networks but has only recently been considered as a viable option for atmospheric free-space quantum communications. The dimensionality can be further increased by exploiting simultaneous entanglement in several DOF. This so-called hyperentanglement has previously been used in various quantum protocols, but not yet been demonstrated outside a protected laboratory environment. 

For our proof-of-concept experiment (doi:10.1038/ncomms15971), we use energy-time and polarization hyperentanglement to distribute, for the first time, 4-dimensional entanglement via a 1.2-km-long intra-city free-space link. A source of hyperentangled photons and a detection module (Alice) were located in our lab and a receiver station (Bob) was located at a different university building. The source produced polarization entangled photon pairs via type-0 down-conversion in a Sagnac loop conﬁguration. The emission time of a photon pair is uncertain within the signiﬁcantly longer coherence time of the pump laser, thus resulting in an energy-time and polarization hyperentangled state. One photon was guided to a local measurement module and the partner photon was guided to a transmitter telescope on the roof of the institute. After transmission over the 1.2-km-long free-space link, the photons were received by a telescope and detected in Bob’s measurement module. The measurement modules for Alice and Bob each featured a polarization and an energy-time analyzer.

We observe high-visibility two-photon interference in both polarization and energy-time subspaces. The measured visibilities certify entanglement in both subspaces individually. Additionally, they establish a lower bound on the Bell-state ﬁdelity of the hyperentangled state of 0.94, thus certifying genuine 4-dimensional entanglement and 1.47 ebits of entanglement of formation.

We have thus successfully distributed hyperentangled photon pairs via an intra-city free-space link under conditions of strong atmospheric turbulence. The transmission of quantum information embedded in a genuine high-dimensional state space under real-world link conditions is a ﬁrst important step towards real-world implementations of advanced quantum information processing protocols in the future. In particular, it enables the implementation of high-dimensional QKD protocols over long-distance free-space links, and, ultimately, over satellite links with only minor changes to existing mission proposals.

Towards high-dimensional quantum key distribution over long-distance free-space links (Conference Presentation)

To determine whether a given multipartite quantum state is separable with respect to some partition, we construct a family of entanglement measures {Rm(ρ)}. This is done utilizing generalized concurrences as building blocks which are defined by flipping of M constituents and indicate states that are separable with regard to bipartitions when vanishing. Furthermore, we provide an analytically computable lower bound for {Rm(ρ)} via a simple ordering relation of the convex roof extension. Using the derived lower bound, we illustrate the effect of the isotropic noise on a family of four-qubit mixed states for each intermediate separability.

Marcus Huber

Papers

Studies of Quantum Entanglement in 100 Dimensions

Unraveling multipartite entanglement for an important family of mixed states

Examining the dimensionality of genuine multipartite entanglement

Towards high-dimensional quantum key distribution over long-distance free-space links (Conference Presentation)

Mixed state entanglement measures for intermediate separability