Entanglement has long been regarded a cornerstone of quantum information science, distinguishing quantum mechanics from classical theories and serving as a pivotal resource for quantum technologies [1]. Since the advent of the stabilizer formalism, it has been clear that entanglement is not enough to provide computational advantage [2]. Such a formalism identifies a set of states, called stabilizer states, which have the peculiar feature of being efficiently simulable using classical computational resources despite being possibly highly entangled [2, 3]. States away from the set of stabilizer states are a fundamental resource for universal quantum computation. Indeed, the distance from the set of the stabilizer states [4] defines the non-stabilizer resource, which also plays an important role in characterizing the complexity of quantum states and processes [5–12]. Entanglement and magic are thus distinct but interrelated resources for understanding the structure and behavior of quantum states. Previous investigations into this interplay have yielded several key insights. Notably, entanglement can be computed exactly for stabilizer states [13, 14], establishing a foundational link between entanglement and classical simulability. The probability distribution of entanglement in random stabilizer states [15] establishes another connection between entanglement and the free stabilizer resources. A series of works show that this kind of entanglement has a simple pattern [16] and that entanglement complexity arises when enough non-stabilizer resources (also known as magic) are injected [16–18], also by measurement in monitored quantum circuits [19, 20]. Furthermore, a connection between non-stabilizerness and the entanglement response of quantum systems, i.e. anti-flatness of the reduced density operator, has been identified [21], highlighting how these resources interact under system dynamics. Additionally, a computational phase separation has been observed, categorizing quantum states into two distinct regimes: entanglement-dominated and magic-dominated phases [22]. This work aims at establishing some exact results in the interplay between entanglement and nonstabilizerness in random pure quantum states. We will utilize Stabilizer Entropy (SE) [8] as the unique computable monotone for non-stabilizerness [23]. We start with a simple consideration that has, though, profound consequences: the separable state with maximal SE is much less resourceful than the average pure state drawn from the Haar measure. Haar-random states are typically highly entangled so we see that most entangled states possess a (much) higher SE than the maximum-SE separable state. This means that entanglement makes room for nonstabilizerness and that the two resources must feature a rich interplay. In this paper, we give a quantitative and analytical analysis of the dependence of these two quantities in random pure states. A numerical analysis with similar scope was recently presented in [24]. The main result of this work is the surprising fact that magic and entanglement are exactly uncorrelated (in their linear versions), yet dependent, and a way to picture this intricate dependence is via the foliation of the Hilbert space of bipartite pure states using the concept of Schmidt orbits.
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