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Although the path-integral formalism is known to be equivalent to conventional quantum mechanics, it is not generally obvious how to implement path-based calculations for multi-qubit entangled states. Whether one takes the formal view of entangled states as entities in a high-dimensional Hilbert space, or the intuitive view of these states as a connection between distant spatial configurations, it may not even be obvious that a path-based calculation can be achieved using only paths in ordinary space and time. Previous work has shown how to do this for certain special states; this paper extends those results to all pure two-qubit states, where each qubit can be measured in an arbitrary basis. Certain three-qubit states are also developed, and path integrals again reproduce the usual correlations. These results should allow for a substantial amount of conventional quantum analysis to be translated over into a path-integral perspective, simplifying certain calculations, and more generally informing research in quantum foundations.
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Device-independent quantum key distribution (DI-QKD) represents one of the most fascinating challenges in quantum communication, exploiting concepts of fundamental physics, namely Bell tests of nonlocality, to ensure the security of a communication link. This requires the loophole-free violation of a Bell inequality, which is intrinsically difficult due to losses in fibre optic transmission channels. Heralded photon amplification is a teleportation-based protocol that has been proposed as a means to overcome transmission loss for DI-QKD. Here we demonstrate heralded photon amplification for path entangled states and characterise the entanglement before and after loss by exploiting a recently developed displacement-based detection scheme. We demonstrate that by exploiting heralded photon amplification we are able to reliably maintain high fidelity entangled states over loss-equivalent distances of more than 50~km.
Multiphoton entanglement, as a quantum resource, plays an essential role in linear optical quantum information processing. Krenn et al. (Phys. Rev. Lett. 118, 080401 2017) proposed an innovative scheme that generating entanglement by path identity, in which two-photon interference (called Hong-Ou-Mandel effect) is not necessary in experiment. However, the experiments in this scheme have strict requirements in stability and scalability, which is difficult to be realized in bulk optics. To solve this problem, in this paper we first propose an on-chip scheme to generate multi-photon polarization entangled states, including Greenberger-Horne-Zeilinger (GHZ) states and W states. Moreover, we also present a class of generalized graphs for W states (odd-number-photon) by path identity in theory. The on-chip scheme can be implemented in existing integrated optical technology which is meaningful for multi-party entanglement distribution in quantum communication networks.
We experimentally show that two-photon path-entangled states can be coherently manipulated by multi-mode interference in multi-mode waveguides. By measuring the output two-photon spatial correlation function versus the phase of the input state, we show that multi-mode waveguides perform as nearly-ideal multi-port beam splitters at the quantum level, creating a large variety of entangled and separable multi-path two-photon states.
The roles of Lie groups in Feynmans path integrals in non-relativistic quantum mechanics are discussed. Dynamical as well as geometrical symmetries are found useful for path integral quantization. Two examples having the symmetry of a non-compact Lie group are considered. The first is the free quantum motion of a particle on a space of constant negative curvature. The system has a group SO(d,1) associated with the geometrical structure, to which the technique of harmonic analysis on a homogeneous space is applied. As an example of a system having a non-compact dynamical symmetry, the d-dimensional harmonic oscillator is chosen, which has the non-compact dynamical group SU(1,1) besides its geometrical symmetry SO(d). The radial path integral is seen as a convolution of the matrix functions of a compact group element of SU(1,1) on the continuous basis.
The extension of many-body quantum dynamics to the non-unitary domain has led to a series of exciting developments, including new out-of-equilibrium entanglement phases and phase transitions. We show how a duality transformation between space and time on one hand, and unitarity and non-unitarity on the other, can be used to realize steady state phases of non-unitary dynamics that exhibit a rich variety of behavior in their entanglement scaling with subsystem size -- from logarithmic to extensive to emph{fractal}. We show how these outcomes in non-unitary circuits (that are spacetime-dual to unitary circuits) relate to the growth of entanglement in time in the corresponding unitary circuits, and how they differ, through an exact mapping to a problem of unitary evolution with boundary decoherence, in which information gets radiated away from one edge of the system. In spacetime-duals of chaotic unitary circuits, this mapping allows us to uncover a non-thermal volume-law entangled phase with a logarithmic correction to the entropy distinct from other known examples. Most notably, we also find novel steady state phases with emph{fractal} entanglement scaling, $S(ell) sim ell^{alpha}$ with tunable $0 < alpha < 1$ for subsystems of size $ell$ in one dimension. These fractally entangled states add a qualitatively new entry to the families of many-body quantum states that have been studied as energy eigenstates or dynamical steady states, whose entropy almost always displays either area-law, volume-law or logarithmic scaling. We also present an experimental protocol for preparing these novel steady states with only a very limited amount of postselection via a type of teleportation between spacelike and timelike slices of quantum circuits.
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