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UCL, 23 Feb 2006 Entanglement Probability Distribution of Random Stabilizer States Oscar C.O. Dahlsten Martin B. Plenio. Explaining the Title. The title is ‘Entanglement Probability Distribution of Random Stabilizer States’

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Explaining the title

UCL, 23 Feb 2006Entanglement Probability Distribution of Random Stabilizer StatesOscar C.O. DahlstenMartin B. Plenio

Explaining the title

Explaining the Title

  • The title is ‘Entanglement Probability Distribution of Random Stabilizer States’

  • Entanglement is the amount of quantum correlations, here taken between two parties sharing a pure state.

  • By Entanglement Probability Distribution we mean P(E), the likelihood of having entanglement of value E.

  • Stabilizer states are an important discrete subset of general states.

  • By random stabilizer states, me mean that we are sampling

    at random and without bias from states that are restricted to be stabilizer states.

Talk structure

Talk Structure

This talk aims to explain the paper:

Exact Entanglement Probability Distribution in

Randomised Bipartite Stabilizer States.

[Dahlsten, Plenio, quant-ph/0511119]

1. Introduction, aim of work

2. Entanglement Probability Distribution

3. Properties of Distribution

4. Summary and Outlook



  • Entanglement is a fundamental resource in quantum information tasks.

  • We can classify and quantify entanglement between two parties quite well, but there is a plethora of classes for more than two parties.

  • Here we consider two simplifications to the problem:

    A. Restrict the states to be ‘stabilizer states’, a discrete subset of all possible quantum states.

    B. Restrict entanglement types to those that are ‘typical’.

A only stabilizer states

A. Only Stabilizer States

  • Stabilizer states are an important discrete subset of all possible states [Gottesman, Caltech PhD thesis].

  • Called stabilizer states as the state is defined by listing the Pauli Matrices that ’stabilize it’, i.e.

  • They can be parametrised efficiently, yet form a rich variety of states: and etc.

  • Bipartite Entanglement in stabilizer states comes in integer values, E=0,1,2,…Emax

    [Audenaert, Plenio, quant-ph/0505036 ]

    [Fattall et al. quant-ph/0406168]

B only typical entanglement

B. Only Typical Entanglement

  • Second simplification: Consider only the typical entanglement in a completely randomised system. [Hayden et al., quant-ph/0407049]

  • Physical setting: imagine two-level atoms in a gas colliding at random, causing entanglement between energy levels.

  • Asymptotically the system is completely randomised.

  • Alice and Bob E-Entangled with probability P(E).





Typical entanglement cont d

Typical Entanglement cont’d.

  • In general states it is known that the average typical/generic entanglement is near maximal.

    (Page’s conjecture).

  • Here typical is defined relative to the uniform distribution on states, given by the ’Haar measure’ on unitaries.

  • There is a concentration of the distribution around this average with increasing N -’concentration of measure’.

  • Is the above still true under the restriction of stabilizer states?

Exact objective find p e

Exact Objective: find P(E)

  • The first question in this line of enquiry is: what is the typical bipartite entanglement in randomised stabilizer states?

  • To answer this we need the probability distribution P(E).

  • Entanglement value E is typical if P(E) significant, atypical if P(E) insignificant.

  • Hence theobjective is to find and study P(E) for randomised bipartite stabilizer states.



1.(Done) Introduction, aim of work

-Simplify entanglement classification by

restricting classes to those that are typical in

stabilizer states.

-Therefore aim to find P(E) of randomised stabilizer

states, where E isbipartite entanglement .


2. Entanglement Probability Distribution

We derive an expression for P(E).

3. Properties of Distribution

4. Summary and Outlook

P e theorem statement

P(E) Theorem Statement

  • Notation: The N qubits are grouped such that NA belong to Alice(the smaller party) and NB to Bob.

  • The total state is pure and N=NA+NB.

  • The state is restricted to be a stabilizer state, but any such state is equally likely.

  • Then P(E), the probability of E entanglement between Alice and Bob is:

Proof outline i

Proof Outline (i)

  • Take probability distribution on stabilizer states as flat. Then p(state)=1/ntot where ntot is the total number of states for the given N.

  • Entanglement E is an integer,

  • So P(E)=nE/ntot where nE(N,NA) is the number of possible stabilizer states with entanglement E.

  • Simplest example: N=2, NA=1 whereby

    Then an explicit count gives ntot=60, n0=36, n1=24.

    Thus P(0) =36/60 and




All ntot states

Proof outline ii

Proof Outline (ii)

  • Finding nE(N, NA) for any N and NA is tricky. Use three lemmas:

  • Lemma 1: The total number of states is known to be

    [Gottesmann, Aaronson quant-ph/052328]

    [Gross, quant-ph/0602001]

  • Lemma 2: The number of unentangled states n0is

  • Lemma 3: There is an invariant ratio (proof complicated)

  • The lemmas together give an iterative expression for nE. This gives P(E) as P(E)=nE/ntot



1.(Done) Introduction, aim of work

2. (Done)Entanglement Probability Distribution



3. Properties of Distribution

-Distribution is ‘Gaussianish’

-Average is nearly maximal

-Concentration around average

-Similar to general states

4. Summary and Outlook

Distribution is gaussian ish

Distribution is ‘Gaussian-ish’

  • An entirely equivalent form of the distribution is

  • Where is messy but comparatively small

  • Therefore P(E) is roughly the side of a Gaussian curve, centred on N/2

Example of p e

Example of P(E)

  • An example of P(E), for N=12, NA=5.

Average is nearly maximal

Average is Nearly Maximal

  • Recall maximal entanglement possible is NA, the number of qubits in the smallest of the two groups.

  • By the main P(E) theorem, one sees the average entanglement, , is nearly maximal for large N.

  • Therefore if we pick stabilizer states at random we expect to get near maximal entanglement on average.

Concentration at average

Concentration at Average

  • Distribution squeezes up around the average with increasing N.

  • Typical entanglement for large N is thus nearly maximal.

  • Animation to the right shows P(E) with fixed NA but N increasing.

Similar to general states

Similar to General States

  • The average entanglement in general states is also near maximal [‘Page’s conjecture’]. The figure below compares the averages for N=10 and varying NA.

  • There isconcentration around the average for general states too [Hayden et al., quant-ph/0407049].



  • We give the Probability Distribution of Entanglement in randomised stabilizer states.

  • It shows the typical entanglement is near maximal.

  • Surprisingly this is very similar to the case for general states.

  • Note: [Smith&Leung, quant-ph/0510232] also interesting.


  • Is there a stabilizer-general state similarity for other quantities than entanglement?

  • What about multipartite entanglement?

  • What happens during the process of randomisation?

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