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We investigate the features of the entanglement spectrum (distribution of the eigenvalues of the reduced density matrix) of a large quantum system in a pure state. We consider all Renyi entropies and recover purity and von Neumann entropy as particular cases. We construct the phase diagram of the theory and unveil the presence of two critical lines.
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The most general quantum object that can be shared between two distant parties is a bipartite channel, as it is the basic element to construct all quantum circuits. In general, bipartite channels can produce entangled states, and can be used to simulate quantum operations that are not local. While much effort over the last two decades has been devoted to the study of entanglement of bipartite states, very little is known about the entanglement of bipartite channels. In this work, we rigorously study the entanglement of bipartite channels as a resource theory of quantum processes. We present an infinite and complete family of measures of dynamical entanglement, which gives necessary and sufficient conditions for convertibility under local operations and classical communication. Then we focus on the dynamical resource theory where free operations are positive partial transpose (PPT) superchannels, but we do not assume that they are realized by PPT pre- and post-processing. This leads to a greater mathematical simplicity that allows us to express all resource protocols and the relevant resource measures in terms of semi-definite programs. Along the way, we generalize the negativity from states to channels, and introduce the max-logarithmic negativity, which has an operational interpretation as the exact asymptotic entanglement cost of a bipartite channel. Finally, we use the non-positive partial transpose (NPT) resource theory to derive a no-go result: it is impossible to distill entanglement out of bipartite PPT channels under any sets of free superchannels that can be used in entanglement theory. This allows us to generalize one of the long-standing open problems in quantum information - the NPT bound entanglement problem - from bipartite states to bipartite channels. It further leads us to the discovery of bound entangled POVMs.
We consider a non-interacting bipartite quantum system $mathcal H_S^Aotimesmathcal H_S^B$ undergoing repeated quantum interactions with an environment modeled by a chain of independant quantum systems interacting one after the other with the bipartite system. The interactions are made so that the pieces of environment interact first with $mathcal H_S^A$ and then with $mathcal H_S^B$. Even though the bipartite systems are not interacting, the interactions with the environment create an entanglement. We show that, in the limit of short interaction times, the environment creates an effective interaction Hamiltonian between the two systems. This interaction Hamiltonian is explicitly computed and we show that it keeps track of the order of the successive interactions with $mathcal H_S^A$ and $mathcal H_S^B$. Particular physical models are studied, where the evolution of the entanglement can be explicitly computed. We also show the property of return of equilibrium and thermalization for a family of examples.
We study the behavior of bipartite entanglement at fixed von Neumann entropy. We look at the distribution of the entanglement spectrum, that is the eigenvalues of the reduced density matrix of a quantum system in a pure state. We report the presence of two continuous phase transitions, characterized by different entanglement spectra, which are deformations of classical eigenvalue distributions.
We first show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter $r$ of a Bell state $|Phi^{+} rangle$ and the completely incoherent state, $hat{rho}_r = r |Phi^{+} rangle langle Phi^{+}| + (1-r)frac{hat{mathbb{I}}}{N^2}$: we find that for $r_0 < r < 1$, where $r_0$ is obtained as a function of the dimensionality $N$, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.
We propose an explicit protocol for the deterministic transformations of bipartite pure states in any dimension using deterministic transformations in lower dimensions. As an example, explicit solutions for the deterministic transformations of $3otimes 3$ pure states by a single measurement are obtained, and an explicit protocol for the deterministic transformations of $notimes n$ pure states by three-outcome measurements is presented.
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