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].
It is shown that the order property of pure bipartite states under stochastic local operations and classical communications (SLOCC) changes radically when dimensionality shifts from finite to infinite. In contrast to finite-dimensional systems where there is no pure incomparable state, the existence of infinitely many mutually SLOCC incomparable states is shown for infinite-dimensional systems even under the bounded energy and finite information exchange condition. These results show that the effect of the infinite dimensionality of Hilbert space, the ``infinite workspace'' property, remains even in physically relevant infinite-dimensional systems.
Incomparability of pure bipartite entangled states under deterministic LOCC is a very strange phenomena. We find two possible ways of getting our desired pure entangled state which is incomparable with the given input state, by collective LOCC with certainty. The first one is by providing some pure entanglement through the lower dimensional maximally-entangled states or using further less amount of entanglement and the next one is by collective operation on two pairs which are individually incomparable. It is quite surprising that we are able to achieve maximally entangled states of any Schmidt rank from a finite number of 2x2 pure entangled states only by deterministic LOCC. We provide general theory for the case of 3x3 system of incomparable states by the above processes where incomparability seems to be the most hardest one.
The conditions for transforming pure entangled states under local operations and classical communication (LOCC) are well understood. A natural question then arises: Can we determine the transformation conditions for mixed entangled states under LOCC based on the properties of their pure-state ensembles? While much effort has been devoted to this issue, in this paper, we rule out this possibility. Our findings address several open questions, including: (i) The conditions
Efcr(ρ)≥Efcr(σ) for all convex roof entanglement measures
Efcr is insufficient to guarantee the existence of an LOCC transformation
ΛL(⋅) from
ρ to
σ; and (ii) The inequalities
∑jpjE(φj)≥∑lqlE(ψl) for all entanglement monotones
E are not sufficient to ensure the existence of an LOCC transformation from
{pj,∣φj⟩} to
{ql,∣ψl⟩}.
Abstract We advance on the conversion of bipartite quantum states via local operations and classical communication (LOCC) for infinite-dimensional systems. We introduce δ-LOCC convertibility based on the observation that any pure state can be approximated by a state with finite-support Schmidt coefficients. We show that δ-LOCC convertibility of bipartite states is fully characterized by a majorization relation between the sequences of squared Schmidt coefficients, providing a novel extension of Nielsen’s theorem for infinite-dimensional systems. Hence, our definition is equivalent to the one of ε-LOCC convertibility (Owari et al 2008 Quantum Inf. Comput. 8 0030), but deals with states having finitely supported sequences of Schmidt coefficients. Additionally, we discuss the notions of optimal common resource and optimal common product in this scenario. The optimal common product always exists, whereas the optimal common resource depends on the existence of a common resource. This highlights a distinction between the resource-theoretic aspects of finite versus infinite-dimensional systems. Our results rely on the order-theoretic properties of majorization for infinite sequences, applicable beyond the LOCC convertibility problem.
Based on set theoretic ordering properties, a general formulation for constructing a pair of convertibility monotones, which are generalizations of distillable entanglement and entanglement cost, is presented. The new pair of monotones do not always coincide for pure bipartite infinite dimensional states under SLOCC (stochastic local operations and classical communications), demonstrating the existence of SLOCC incomparable pure bipartite states, a new property of entanglement in infinite dimensional systems, with no counterpart in the corresponding finite dimensional systems.