quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...

, 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 first in the context of cold gases, together with entanglement criteria that made it possible to bound it from below based on measured quantities. We will now present a general definition based on

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

s of

quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...

s. First, we will define

k

-producibility. Let us consider a

pure state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...

that is the

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...

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 genuine multiparticle entanglement of

k+1

particles, if it is not

k

-producible. Finally, a quantum state has an entanglement depth

k

, if it is

k

-producible, but not

(k-1)

-producible.

References

Quantum information science Quantum optics {{quantum-stub