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Quantum Information and Computation

Entanglement probability distribution of bi-partite randomised stabilizer states

Oscar Dahlsten, Martin Bodo Plenio

Published Sep 1, 2006• 17 citations
Publisher

Abstract

We study the entanglement properties of random pure stabilizer states in spin-1/2 particles. We obtain a compact and exact expression for the probability distribution of the entanglement values across any bipartite cut. This allows for exact derivations of the average entanglement and the degree of concentration of measure around this average. We also give simple bounds on these quantities. We find that for large systems the average entanglement is near maximal and the measure is concentrated around it.

DOI
10.26421/qic6.6-5

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We study the entanglement properties of random pure stabilizer states in spin-1/2 particles. We obtain a compact and exact expression for the probability distribution of the entanglement values across any bipartite cut. This allows for exact derivations of the average entanglement and the degree of concentration of measure around this average. We also give simple bounds on these quantities. We find that for large systems the average entanglement is near maximal and the measure is concentrated around it.
We study the entanglement properties of random pure stabilizer states in spin-1/2 particles. For two contiguous groups of spins of arbitrary size we obtain a compact and exact expression for the probability distribution for the entanglement values between the two sets. This allows for exact derivations of the average entanglement and the degree of concentration of measure around this average. We also give simple bounds on these quantities. We find that for large systems the average entanglement is near maximal and the measure is concentrated around it.
The average entanglement of random pure states of an $N\ifmmode\times\else\texttimes\fi{}N$ composite system is analyzed. We compute the average value of the determinant DDD of the reduced state, which forms an entanglement monotone. Calculating higher moments of the determinant, we characterize the probability distribution P(D)P(D)P(D). Similar results are obtained for the rescaled NthN\text{th}Nth root of the determinant, called the GGG concurrence. We show that in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ this quantity becomes concentrated at a single point ${G}_{\ensuremath{\star}}=1∕e$. The position of the concentration point changes if one consider an arbitrary $N\ifmmode\times\else\texttimes\fi{}K$ bipartite system, in the joint limit $N,K\ensuremath{\rightarrow}\ensuremath{\infty}$, with K∕NK∕NK∕N fixed.
How entangled is a randomly chosen bipartite stabilizer state? We show that if the number of qubits each party holds is large, the state will be close to maximally entangled with probability exponentially close to 1. We provide a similar tight characterization of the entanglement present in the maximally mixed state of a randomly chosen stabilizer code. Finally, we show that typically very few Greenberger-Horne-Zeilinger states can be extracted from a random multipartite stabilizer state via local unitary operations. Our main tool is a concentration inequality which bounds deviations from the mean of random variables which are naturally defined on the Clifford group.
In this thesis we study the behavior of bipartite entanglement of a large quantum system, by analyzing the distribution of the Schmidt coefficients of the reduced density matrix. Applying the general methods of classical statistical mechanics, we develop a canonical approach for the study of the distribution of these coefficients for a fixed value of the average entanglement. We introduce a partition function depending on a fictitious temperature, which localizes the measure on the set of states with higher and lower entanglement, if compared to typical (random) states with respect to the Haar measure. The purity of one subsystem, which is our entanglement measure/indicator, plays the role of energy in the partition function. This thesis consists of two parts. In the first part, we completely characterize the distribution of the purity and of the eigenvalues for pure states. The global picture unveils several locally stable solutions exchange stabilities, through the presence of first and second order phase transitions. We also detect the presence of metastable states. In the second part, we focus on mixed states. Through the same statistical approach, we determine the exact expression of the first two moments of the local purity and the high temperature expansion of the first moment. We also bridge our problem with the theory of quantum channels, more precisely we exploit the symmetry properties of the twirling transformations in order to compute the exact expression for the first moment of the local purity.
Average entanglement of random pure states of anN×N composite system is analyzed. We compute the average value of the determinantD of the reduced state, which forms an entanglement monotone. Calculating higher moments of the determinant we characterize the probability distributionP(D). Similar results are obtained for the rescaledN th root of the determinant, called G‐concurrence. We show that in the limitN→∞ this quantity becomes concentrated at a single pointG ⋆ = 1/e. The position of the concentration point changes if one consider an arbitraryN×K bipartite system, in the joint limitN,K→∞,K/N fixed.
Entanglement properties of random multipartite quantum states which are invariant under global SU(d)\textnormal{SU}(d)SU(d) action are investigated. The random states live in the tensor power of an irreducible representation of SU(d)\textnormal{SU}(d)SU(d). We calculate and analyze the expectation and fluctuation of the second-order R\'enyi entanglement measure of the random invariant and near-invariant states in high dimension, and reveal the phenomenon of concentration of measure the random states exhibit. We show that with high probability a random SU(ddd)-invariant state is close to being maximally entangled with respect to any bipartite cut as the dimension of individual system goes to infinity. We also show that this generic entanglement property of random SU(2)-invariant state is robust to arbitrarily finite disturbation.