Abstract Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state
∣ψ⟩ | ψ ⟩ . Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform
∣ψ⟩ | ψ ⟩ into another state
∣ϕ⟩ | ϕ ⟩ . Famously, Nielsen’s theorem [28] provides a necessary and sufficient algebraic criterion for such a transformation to be possible (namely, the entanglement spectrum of
∣ϕ⟩ | ϕ ⟩ should majorise the entanglement spectrum of
∣ψ⟩ | ψ ⟩ ). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states
∣ψ⟩,∣ϕ⟩ | ψ ⟩ , | ϕ ⟩ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states
∣ψ⟩,∣ϕ⟩ | ψ ⟩ , | ϕ ⟩ are entangled in fundamentally different ways, that cannot be converted to each other via local operations and classical communication. Via Nielsen’s theorem, this conjecture can be equivalently stated as a conjecture about majorisation of spectra of random matrices from the so-called trace-normalised complex Wishart–Laguerre ensemble . Concretely, let X and Y be independent
n×m n × m random matrices whose entries are i.i.d. standard complex Gaussians; then Nielsen’s conjecture says that the probability that the spectrum of
XX†/tr(XX†) X X † / tr ( X X † ) majorises the spectrum of
YY†/tr(YY†) Y Y † / tr ( Y Y † ) tends to zero as both n and m grow large. We prove this conjecture, and we also confirm some related predictions of Cunden et al. [12].
Suppose Alice and Bob try to transform an entangled state shared between them into another one by local operations and classical communications. Then in general a certain amount of entanglement contained in the initial state will decrease in the process of transformation. However, an interesting phenomenon called partial entanglement recovery shows that it is possible to recover some amount of entanglement by adding another entangled state and transforming the two entangled states collectively. In this paper, we are mainly concerned with the feasibility of partial entanglement recovery. The basic problem we address is whether a given state is useful in recovering entanglement lost in a specified transformation. In the case where the source and target states of the original transformation satisfy the strict majorization relation, a necessary and sufficient condition for partial entanglement recovery is obtained. For the general case we give two sufficient conditions. We also give an efficient algorithm for the feasibility of partial entanglement recovery in polynomial time. As applications, we establish some interesting connections between partial entanglement recovery and the generation of maximally entangled states, quantum catalysis, mutual catalysis, and multiple-copy entanglement transformation.
We analyze the conditions under which local operations and classical communication enable entanglement transformations within the set of bipartite pure Gaussian states. A set of necessary and sufficient conditions had been found in [Quant. Inf. Comp. 3, 211 (2003)] for the interconversion between such states that is restricted to Gaussian local operations and classical communication. Here, we exploit majorization theory in order to derive more general (sufficient) conditions for the interconversion between bipartite pure Gaussian states that goes beyond Gaussian local operations. While our technique is applicable to an arbitrary number of modes for each party, it allows us to exhibit surprisingly simple examples of 2 x 2 Gaussian states that necessarily require non-Gaussian local operations to be transformed into each other.
We present a theory of entanglement transformations of Gaussian pure states with local Gaussian operations and classical communication. This is the experimentally accessible set of operations that can be realized with optical elements such as beam splitters, phase shifts and squeezers, together with homodyne measurements. We provide a simple necessary and sufficient condition for the possibility to transform a pure bipartite Gaussian state into another one. We contrast our criterion with what is possible if general local operations are available.