Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0605126 · Physical Review Letters

Generic Entanglement Can Be Generated Efficiently

Roberto I. Oliveira, Oscar Dahlsten, Martin Bodo Plenio

Published Mar 30, 2007• 125 citations
arXivHTMLPDF

Abstract

We find that generic entanglement is physical, in the sense that it can be generated in polynomial time from two-qubit gates picked at random. We prove as the main result that such a process generates the average entanglement of the uniform (unitarily invariant) measure in at most O(N3) steps for N qubits. This is despite an exponentially growing number of such gates being necessary for generating that measure fully on the state space. Numerics furthermore show a variation cutoff allowing one to associate a specific time with the achievement of the uniform measure entanglement distribution. Various extensions of this work are discussed. The results are relevant to entanglement theory and to protocols that assume generic entanglement can be achieved efficiently.

DOI
10.1103/physrevlett.98.130502

Related papers

Optimal two-qubit gate for generation of random bipartite entanglement

Marko Žnidarič

We numerically study protocols consisting of repeated applications of two qubit gates used for generating random pure states. A necessary number of steps needed in order to generate states displaying bipartite entanglement typical of random states is considered. Numerics indicates that for a generic two qubit entangling gate the decay of purity is exponential with the decay time scaling as $\ensuremath{\sim}n$, implying that of order $\ensuremath{\sim}{n}^{2}$ steps are needed to reach random bipartite entanglement. We also numerically identify the optimal two qubit gate for which the convergence is the fastest. Perhaps surprisingly, applying the same good two qubit gate in addition to a random single qubit rotations at each step leads to a faster generation of entanglement than applying a random two qubit transformation at each step.

arXiv:quant-ph/0702240•Physical Review A•Jul 17, 2007•38 citations

The emergence of typical entanglement in two-party random processes

Oscar Dahlsten, Roberto I. Oliveira, Martin Bodo Plenio

We investigate the entanglement within a system undergoing a random, local process. We find that there is initially a phase of very fast generation and spread of entanglement. At the end of this phase the entanglement is typically maximal. In Oliveira et al (2007 Phys. Rev. Lett. 98 130502) we proved that the maximal entanglement is reached to a fixed arbitrary accuracy within O ( N 3 ) steps, where N is the total number of qubits. Here we provide a detailed and more pedagogical proof. We demonstrate that one can use the so-called stabilizer gates to simulate this process efficiently on a classical computer. Furthermore, we discuss three ways of identifying the transition from the phase of rapid spread of entanglement to the stationary phase: (i) the time when saturation of the maximal entanglement is achieved, (ii) the cutoff moment, when the entanglement probability distribution is practically stationary, and (iii) the moment block entanglement exhibits volume scaling. We furthermore investigate the mixed state and multipartite setting. Numerically, we find that the mutual information appears to behave similarly to the quantum correlations and that there is a well-behaved phase-space flow of entanglement properties towards an equilibrium. We describe how the emergence of typical entanglement can be used to create a much simpler tripartite entanglement description. The results form a bridge between certain abstract results concerning typical (also known as generic) entanglement relative to an unbiased distribution on pure states and the more physical picture of distributions emerging from random local interactions.

arXiv:quant-ph/0701125•Journal of Physics A Mathematical and Theoretical•Jun 27, 2007•71 citations

Simulating typical entanglement with many-body Hamiltonian dynamics

Yoshifumi Nakata, Mio Murao

We study the time evolution of the amount of entanglement generated by one-dimensional spin-1/2 Ising-type Hamiltonians composed of many-body interactions. We investigate sets of states randomly selected during the time evolution generated by several types of time-independent Hamiltonians by analyzing the distributions of the amount of entanglement of the sets. We compare such entanglement distributions with that of typical entanglement, entanglement of a set of states randomly selected from a Hilbert space with respect to the unitarily invariant measure. We show that the entanglement distribution obtained by a time-independent Hamiltonian can simulate the average and standard deviation of the typical entanglement, if the Hamiltonian contains suitable many-body interactions. We also show that the time required to achieve such a distribution is polynomial in the system size for certain types of Hamiltonians.

arXiv:1108.0485•Physical Review A•Nov 21, 2011•1 citation

Efficient generation of random multipartite entangled states using time-optimal unitary operations

Antoni Borràs, A. P. Majtey, M. Casas

We review the generation of random pure states using a protocol of repeated two-qubit gates. We study the dependence of the convergence to states with Haar multipartite entanglement distribution. We investigate the optimal generation of such states in terms of the physical (real) time needed to apply the protocol, instead of the gate complexity point of view used in other works. This physical time can be obtained, for a given Hamiltonian, within the theoretical framework offered by the quantum brachistochrone formalism, the quantum analogue to the brachistochrone problem in classical mechanics [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006)]. Using an anisotropic Heisenberg Hamiltonian as an example, we find that different optimal quantum gates arise according to the optimality point of view used in each case. We also study how the convergence to random entangled states depends on different entanglement measures.

arXiv:0805.2070•Physical Review A•Aug 19, 2008•4 citations

Entanglement of Three-Qubit Random Pure States

Marco Enríquez, Francisco Delgado, Karol Życzkowski

We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.

arXiv:1809.00642•Entropy•Sep 29, 2018•35 citations

Entanglement, randomness and chaos

Giuliano Benenti

Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in n_q. In particular, quantum chaotic maps are efficient generators of multipartite entanglement among the qubits, close to that expected for random states. This review discusses several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer; (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics; (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment.

arXiv:0807.4364•Jul 28, 2008•3 citations
We find that generic entanglement is physical, in the sense that it can be generated in polynomial time from two-qubit gates picked at random. We prove as the main result that such a process generates the average entanglement of the uniform (unitarily invariant) measure in at most O(N3) steps for N qubits. This is despite an exponentially growing number of such gates being necessary for generating that measure fully on the state space. Numerics furthermore show a variation cutoff allowing one to associate a specific time with the achievement of the uniform measure entanglement distribution. Various extensions of this work are discussed. The results are relevant to entanglement theory and to protocols that assume generic entanglement can be achieved efficiently.
We numerically study protocols consisting of repeated applications of two qubit gates used for generating random pure states. A necessary number of steps needed in order to generate states displaying bipartite entanglement typical of random states is considered. Numerics indicates that for a generic two qubit entangling gate the decay of purity is exponential with the decay time scaling as $\ensuremath{\sim}n$, implying that of order $\ensuremath{\sim}{n}^{2}$ steps are needed to reach random bipartite entanglement. We also numerically identify the optimal two qubit gate for which the convergence is the fastest. Perhaps surprisingly, applying the same good two qubit gate in addition to a random single qubit rotations at each step leads to a faster generation of entanglement than applying a random two qubit transformation at each step.
We investigate the entanglement within a system undergoing a random, local process. We find that there is initially a phase of very fast generation and spread of entanglement. At the end of this phase the entanglement is typically maximal. In Oliveira et al (2007 Phys. Rev. Lett. 98 130502) we proved that the maximal entanglement is reached to a fixed arbitrary accuracy within O ( N 3 ) steps, where N is the total number of qubits. Here we provide a detailed and more pedagogical proof. We demonstrate that one can use the so-called stabilizer gates to simulate this process efficiently on a classical computer. Furthermore, we discuss three ways of identifying the transition from the phase of rapid spread of entanglement to the stationary phase: (i) the time when saturation of the maximal entanglement is achieved, (ii) the cutoff moment, when the entanglement probability distribution is practically stationary, and (iii) the moment block entanglement exhibits volume scaling. We furthermore investigate the mixed state and multipartite setting. Numerically, we find that the mutual information appears to behave similarly to the quantum correlations and that there is a well-behaved phase-space flow of entanglement properties towards an equilibrium. We describe how the emergence of typical entanglement can be used to create a much simpler tripartite entanglement description. The results form a bridge between certain abstract results concerning typical (also known as generic) entanglement relative to an unbiased distribution on pure states and the more physical picture of distributions emerging from random local interactions.
We study the time evolution of the amount of entanglement generated by one-dimensional spin-1/2 Ising-type Hamiltonians composed of many-body interactions. We investigate sets of states randomly selected during the time evolution generated by several types of time-independent Hamiltonians by analyzing the distributions of the amount of entanglement of the sets. We compare such entanglement distributions with that of typical entanglement, entanglement of a set of states randomly selected from a Hilbert space with respect to the unitarily invariant measure. We show that the entanglement distribution obtained by a time-independent Hamiltonian can simulate the average and standard deviation of the typical entanglement, if the Hamiltonian contains suitable many-body interactions. We also show that the time required to achieve such a distribution is polynomial in the system size for certain types of Hamiltonians.
We review the generation of random pure states using a protocol of repeated two-qubit gates. We study the dependence of the convergence to states with Haar multipartite entanglement distribution. We investigate the optimal generation of such states in terms of the physical (real) time needed to apply the protocol, instead of the gate complexity point of view used in other works. This physical time can be obtained, for a given Hamiltonian, within the theoretical framework offered by the quantum brachistochrone formalism, the quantum analogue to the brachistochrone problem in classical mechanics [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006)]. Using an anisotropic Heisenberg Hamiltonian as an example, we find that different optimal quantum gates arise according to the optimality point of view used in each case. We also study how the convergence to random entangled states depends on different entanglement measures.
We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.
Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in n_q. In particular, quantum chaotic maps are efficient generators of multipartite entanglement among the qubits, close to that expected for random states. This review discusses several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer; (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics; (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment.