Aspects of Generic Entanglement

By Patrick M. Hayden, Debbie W. Leung, Andreas J. Winter. 2004

438 citations recorded by the source corpus.

We study entanglement and other correlation properties of random states in high-dimensional bipartite systems. These correlations are quantified by parameters that are subject to the ``concentration of measure'' phenomenon, meaning that on a large-probability set these parameters are close to their expectation. For the entropy of entanglement, this has the counterintuitive consequence that there exist large subspaces in which all pure states are close to maximally entangled. This, in turn, implies the existence of mixed states with entanglement of formation near that of a maximally entangled state, but with negligible quantum mutual information and, therefore, negligible distillable entanglement, secret key, and common randomness. It also implies a very strong locking effect for the entanglement of formation: its value can jump from maximal to near zero by tracing over a number of qubits negligible compared to the size of the total system. Furthermore, such properties are generic. Similar phenomena are observed for random multiparty states, leading us to speculate on the possibility that the theory of entanglement is much simplified when restricted to asymptotically generic states. Further consequences of our results include a complete derandomization of the protocol for universal superdense coding of quantum states.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/64733d32e7e6537950c4718a04c2aafe5579c59c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Atomic and Molecular Physics, and Optics

Citations in the corpus

Schur–Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations (2017, background): “ctions. Another motivation arises from randomized constructions. Unitaries and states drawn from the Haar measure appear in many situations, including in quantum cryptography, coding, and data hiding [HLW06]. While randomized constructions are often nearoptimal and frequently out-perform all known d”
Random matrix techniques in quantum information theory (2015, methodology): “[ASW11] to show violations at p= 1. Later, Fukuda provided a simpler proof of violation [Fuk14], using this time "-net arguments and Levy’s lemma, the techniques used also in the pioneering work [HLW06]. In [BCN12,BCN13], the authors use free probability theory to compute exactly the minimum ou”
Quantum coherence and geometric quantum discord (2017, background): “As is known, some measures of quantumness manifest concentration effect, e.g., the random bipartite pure states sampled from the uniform Haar measure are typically maximally entangled (Hayden et al., 2006).”
Geometry of Quantum States: Index (2007, background): “35), while the probability of finding a state with the entropy S smaller by one than the average 〈S〉N,N is exponentially small [418].”
Faithful Squashed Entanglement (2010, background): “The antisymmetric state of sufficiently high dimension is an example of a data hiding state [49], as are random mixed states with high probability [ 50 ] (given an appropriate choice of the dimensions and the rank of the state).”
Multiboundary wormholes and holographic entanglement (2014, background): “ies of the state in question using only LOCC [56]. In general, E F E D and the gap can be very large: the entanglement of formation can be near maximal even when the mutual information is very small [57]. So in the example above, it is possible that E F(B 1 : B 2) remains large even after the HRT ph”
Local random quantum circuits are approximate polynomial-designs: numerical results (2012, neutral): “[47]) Pr U [ |〈α|U |β〉| ≥ γ d ]”
Many-Body Quantum Magic (2020, background): “[30] Patrick Hayden, Debbie W.”
Shadow tomography of quantum states (2017, background): “Lemma 18 ([20, Lemma III.5]) Let N be even, and let S be a subspace of C N chosen uniformly at random subject to dim ( S ) = N 2 .”
Adiabatic Quantum Computing (2016, background): “While generic quantum states do not obey an area law (Hayden et al., 2006), and there are 1D systems for which there is exponentially more entanglement than suggested by the area law (Movassagh and Shor, 2016), a sweeping conjecture in condensed matter physics is that in a gapped system the…”
Exponential Decay of Correlations Implies Area Law (2012, background): “This is not only a pathological case among quantum states, but it is actually the general rule: The overwhelming majority of quantum states will have such peculiar type of correlations, whenever the regions X and Y have approximately equal sizes [27].”
Unbounded Violation of Tripartite Bell Inequalities (2007, background): “This Theorem shows once more that random states exhibit unexpected extremal properties [1, 14, 34, 48].”
Multiple shocks (2013, background): “In a typical ensemble, the distribution of entanglement is very sharply peaked, and deficits are highly suppressed in the measure [36].”
Sample-Optimal Tomography of Quantum States (2015, background): “[50] does not explicitly cover the z > 1 case for the ﬁrst inequality, though it implicitly covered in their proof.”
Entanglement and the foundations of statistical mechanics (2005, methodology): “It has already been used in quantum information theory to study entanglement and other correlation properties of typical states in large bipartite system”
The Theory of Quantum Information (2018, background): “51 is a variant of a theorem due to Hayden, Leung, and Winter [97].”
The Church of the Symmetric Subspace (2013, methodology): “Thus our results will be similar in many ways to those of [24], which used the more conventional methods of (cid:15) -nets and Levy’s Lemma (which is based on Gaussian concentration, which in turn can be derived from moment-generating functions).”