Entanglement Depth
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quantum physics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, entanglement depth characterizes the strength of multiparticle entanglement. An entanglement depth k means that the quantum state of a particle ensemble cannot be described under the assumption that particles interacted with each other only in groups having fewer than k particles. It has been used to characterize the quantum states created in experiments with cold gases.


Definition

Entanglement depth appeared in the context of spin squeezing. It turned out that to achieve larger and larger spin squeezing, and thus larger and larger precision in parameter estimation, a larger and larger entanglement depth is needed. Later it was formalized in terms of
convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
s of
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
s, independent of spin squeezing as follows. Let us consider a
pure state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
that is the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of multi-particle quantum states , \Psi\rangle=, \phi_1\rangle\otimes, \phi_2\rangle\otimes ... \otimes, \phi_n\rangle. The pure state , \Psi\rangle is said to be k-producible if all \phi_i are states of at most k particles. A mixed state is called k-producible, if it is a mixture of pure states that are all at most k-producible. The k-producible mixed states form a convex set. A quantum state contains at least multiparticle entanglement of k+1 particles, if it is not k-producible. A N-particle state with N-entanglement is called genuine multipartite entangled. Finally, a quantum state has an entanglement depth k, if it is k-producible, but not (k-1)-producible. It was possible to detect the entanglement depth close to states different from spin-squeezed states. Since there is not a general method to detect multipartite entanglement, these methods had to be tailored to experiments with various relevant quantum states. Thus, entanglement criteria has been developed to detect entanglement close to symmetric Dicke states with \langle J_z\rangle=0. They are very different from spin-squeezed states, since they do not have a large spin polarization. They can provide Heisenberg limited metrology, while they are more robust to particle loss than Greenberger-Horne-Zeilinger (GHZ) states. There are also criteria for detecting the entanglement depth in ''planar-squeezed state''s. Planar squeezed states are quantum states that can be used to estimate a rotation angle that is not expected to be small. Finally, multipartite entanglement can be detected based on the metrological usefulness of the quantum state. The criteria applied are based on bounds on the
quantum Fisher information The quantum Fisher information is a central quantity in quantum metrology and is the quantum analogue of the classical Fisher information. It is one of the central quantities used to qualify the utility of an input state, especially in Mach–Zehnd ...
.


Experiments

The entanglement criterion in Ref. has been used in many experiments with cold gases in spin-squeezed states. There have also been experiments in cold gases for detecting multipartite entanglement in symmetric Dicke states. There have been also experiments with Dicke states that detected entanglement based on metrological usefulness in cold gases and in photons.


References

{{Reflist Quantum information science Quantum optics Optical quantities