Structural change in multipartite entanglement sharing: a random matrix approach G. Gennaro, 1 ∗ S. Campbell, 2 ∗ M. Paternostro, 2 and G. M. Palma, 3 $^{1}$Dipartimento di Scienze Fisiche ed Astronomiche, Universita’ degli Studi di Palermo, via Archirafi 36, I-90123 Palermo, Italy 2 School of Mathematics and Physics. Queen’s University Belfast, BT7 1NN, United Kingdom 3 NEST-CNR-INFM & Dipartimento di Scienze Fisiche ed Astronomiche, Universita’ degli Studi di Palermo, via Archirafi 36, I-90123 Palermo, Italy (Dated: November 16, 2018) We study the typical entanglement properties of a system comprising two independent qubit envi- ronments interacting via a shuttling ancilla. The initial preparation of the environments is modelled using random-matrix techniques. The entanglement measure used in our study is then averaged over many histories of randomly prepared environmental states. Under a Heisenberg interaction model, the average entanglement between the ancilla and one of the environments remains con- stant, regardless of the preparation of the latter and the details of the interaction. We also show that, upon suitable kinematic and dynamical changes in the ancilla-environment subsystems, the entanglement-sharing structure undergoes abrupt modifications associated with a change in the mul- tipartite entanglement class of the overall system’s state. These results are invariant with respect to the randomized initial state of the environments. PACS numbers: 03.65.Yz, 03.67.Mn,03.67.-a Open-system dynamics involving environments com- prising only a finite number of elements have been proven a valuable arena for the study of interesting physical phenomena such as quantum chaos [1], quantum ther- modynamics [2], entanglement and relaxation [3, 4]. In most of these studies, the use of random-matrix theory has proven technically very advantageous in modelling random collisions between parts of the overall system at hand. Random matrices have been helpful in deal- ing with many tasks of quantum information processing, including quantum data hiding, quantum state distinc- tion, superdense coding and noise estimation [5]. Very recently, random matrices have found extensive applica- tion in the characterization of Markovian decoherence [6]. In this paper we unveil a further interesting situation where the theory of random matrices finds fertile appli- cations: We study the typical amount of entanglement that can be set in a multipartite system comprising two environments of arbitrary (finite) size and a shuttling two-level ancilla that bridges their cross-talking (Fig. 1). Differently from Refs. [3, 4, 6, 7], random matrices are used in order to model the initial preparation of the en- vironments. These interact with the shuttling ancilla via a Hamiltonian model having pre-determined interaction strength. This allows us to investigate on the typical ancilla-environment as well as all-environment degree of entanglement simply by averaging the values correspond- ing to many hystories of random initial preparations. We point out the sensitivity of the entanglement-sharing structure on the kinematic and dynamical aspects of the interactions and its independence from the random preparation of the environments. In particular, we show that upon tuning of the coupling Hamiltonian regulating the ancilla-environment interactions, an abrupt transi- tion between two inequivalent classes of multipartite en- tanglement is achieved. The dimensions of the environ- ments and the number of interactions enter preponder- antly into the determination of the entanglement-sharing structure, as we quantitatively reveal. It is important to remark that our model does not al- low for phenomena of thermalization or homogeneization typical of effectively Markovian reservoirs such as those considered in Refs. [8]. In fact, the evolution here at hand is profoundly different from a forgetful open dynam- ics comprising environments which are weakly perturbed by the interactions with the ancilla. Memory effects are clearly seen in our analysis, which reveals how the state of each environment is deeply affected by its coupling to the shuttling ancilla. As we do not impose restrictions to the strength of each interaction, also the Born approxi- mation does not hold in our analysis. The remainder of this paper is organized as follows. In Sec. I we address the case of a single environment interacting with the ancilla A . Through numerical sim- ulations supported by a clear analytical study we show that when the ancilla interacts repeatedly with the same environmental qubit, the degree of entanglement is in- variant to both the random initial preparation and di- mension of the environment. Sec. II extends our study to the case of two multi-qubit environments mutually connected by the bouncing ancilla. A repeatedly inter- acts ( collides , as we often say in this paper) with two specific qubits, each beloning to the respective environ- ment. In this case, the amount of entanglement that can be set is a delicate trade-off between both the di- mensions of the two environments and the order of the ancilla-environment interaction. This scenario allows us to address the main points of our study. First we show that, by tuning the form of the coupling Hamiltonian, the system undergoes an entanglement-sharing transi- tion between two non-equivalent classes of multipartite entanglement. As no time is explicitly involved, we re- fer to this effect as a “kinematic” transition. We then study how repeated sequential collisions are able to af- arXiv:1001.0705v2 [quant-ph] 7 Jan 2010