Ordering physical states is the key to quantifying some physical property of the states uniquely. Bipartite pure entangled states are totally ordered under local operations and classical communication (LOCC) in the asymptotic limit and uniquely quantified by the well-known entropy of entanglement. However, we show that mixed entangled states are partially ordered under LOCC even in the asymptotic limit. Therefore, non-uniqueness of entanglement measure is understood on the basis of an operational notion of asymptotic convertibility.
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].
Without additional resources, it is often impossible to transform one entangled quantum state into another with local quantum operations and classical communication. Jonathan and Plenio (Phys. Rev. Lett., vol. 83, p. 3566, 1999) presented an interesting example showing that the presence of another state, called a catalyst, enables such a transformation without changing the catalyst. They also pointed out that in general it is very hard to find an analytical condition under which a catalyst exists. In this paper, we study the existence of catalysts for two incomparable quantum states. For the simplest case of 2/spl times/2 catalysts for transformations from one 4/spl times/4 state to another, a necessary and sufficient condition for existence is found. For the general case, we give an efficient polynomial time algorithm to decide whether a k/spl times/k catalyst exists for two n/spl times/n incomparable states, where k is treated as a constant.
We show that entanglement-assisted transformations of bipartite entangled states can be more efficient than catalysis [Jonathan and Plenio, Phys. Rev. Lett. 83, 3566 (1999)], i.e., given two incomparable bipartite states not only can the transformation be enabled by performing collective operations with an auxiliary entangled state, but the entanglement of the auxiliary state itself can be enhanced. We refer to this phenomenon as supercatalysis. We provide results on the properties of supercatalysis and its relationship with catalysis. In particular, we obtain a useful necessary and sufficient condition for catalysis, and provide several sufficient conditions for supercatalysis and study the extent to which entanglement of the auxiliary state can be enhanced via supercatalysis.
Quantum entanglement is a phenomenon which has shown great potential use in modern technical implementations, but there is still much development needed in the field. One major problem is how to measure the amount of entanglement present in a given entangled state. There are numerous different entanglement measures suggested, all satisfying some conditions being of either operational, or more abstract, mathematical nature. However, in contradiction to what one might expect, the measures show discrepancies in the ordering of entangled states. Concretely this means that with respect to one measure, a state can be more entangled than another state, but the ordering may be opposite for the same states using another measure. In this thesis we take a closer look at some of the most commonly occurring entanglement measures, and find examples of states showing inequivalent entanglement ordering for the different measures.