Transcript Presentation() - D

UCL, 23 Feb 2006
Entanglement Probability
Distribution of Random
Stabilizer States
Oscar C.O. Dahlsten
Martin B. Plenio
UCL, 23 Feb 2006
Explaining the Title
UCL, 23 Feb 2006
• 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
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
UCL, 23 Feb 2006
Motivation
UCL, 23 Feb 2006
• 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
UCL, 23 Feb 2006
• 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: 00  11 and 000  111 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
UCL, 23 Feb 2006
• 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.
Alice
Bob
t=0
t=1
• Asymptotically the system is completely randomised.
• Alice and Bob E-Entangled with probability P(E).
Typical Entanglement cont’d.
UCL, 23 Feb 2006
• 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)
UCL, 23 Feb 2006
• 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 the objective is to find and study P(E) for
randomised bipartite stabilizer states.
Overview
UCL, 23 Feb 2006
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 is bipartite entanglement .
Next
2. Entanglement Probability Distribution
We derive an expression for P(E).
3. Properties of Distribution
4. Summary and Outlook
P(E) Theorem Statement
UCL, 23 Feb 2006
• 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:
 2  1
N
i
P( E ) 
i 1
N
 2
k  N  N A 1
k
E

1

j 1
2
N  N A 1 j

 1 2 N A  j  2 2 j 1
22 j  1

Proof Outline (i)
UCL, 23 Feb 2006
• 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, E 0,1,2,...N A 
• 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 E 0,1
Then an explicit count gives ntot=60, n0=36, n1=24.
Thus P(0) =36/60 and
p(1)=24/60
n
n
0
All ntot states
1
Proof Outline (ii)
UCL, 23 Feb 2006
• 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
ntot ( N )  2
N
 2  1
N
i
i 1
[Gottesmann, Aaronson quant-ph/052328]
[Gross, quant-ph/0602001]
• Lemma 2: The number of unentangled states n0 is
n0  ntot ( N A )ntot ( N B )
• Lemma 3: There is an invariant ratio (proof complicated)
nENa 1 N 
 f E , N A 
Na
nE N  1
• The lemmas together give an iterative expression for nE.
This gives P(E) as P(E)=nE/ntot
Overview
UCL, 23 Feb 2006
1.(Done) Introduction, aim of work
2. (Done)Entanglement Probability Distribution
N
Derived
i
2
 1 E N  NA1 j

NA  j
2 j 1
2

1
2

2
P( E )  i 1N

2j
2
1
k
j 1
2

1






k  N  NA 1
Next
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’
UCL, 23 Feb 2006
• An entirely equivalent form of the distribution is
P E   2
 N A  N B 2  N / 2  E 2  
4
1 2
• Where 1   2 is messy but comparatively small
• Therefore P(E) is roughly the side of a Gaussian
curve, centred on N/2
Example of P(E)
• An example of P(E), for N=12, NA=5.
UCL, 23 Feb 2006
Average is Nearly Maximal
UCL, 23 Feb 2006
• 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, E 
EP( E ) , is nearly maximal for
large N.
E

• Therefore if we pick stabilizer states at random we expect
to get near maximal entanglement on 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.
UCL, 23 Feb 2006
Similar to General States
UCL, 23 Feb 2006
• 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 is concentration around the average for general states
too [Hayden et al., quant-ph/0407049].
Summary
UCL, 23 Feb 2006
• 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.
Outlook
• Is there a stabilizer-general state similarity for other
quantities than entanglement?
• What about multipartite entanglement?
• What happens during the process of randomisation?