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UCL, 23 Feb 2006 UCL, 23 Feb 2006 Entanglement Probability Distribution of Random Stabilizer States Oscar C.O. Dahlsten Martin B. Plenio

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UCL, 23 Feb 2006 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.

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UCL, 23 Feb 2006 Talk Structure This talk aims to explain the paper: Exact Entanglement Probability Distribution in Randomised Bipartite Stabilizer States. [Dahlsten, Plenio, quant-ph/ ] 1. Introduction, aim of work 2. Entanglement Probability Distribution 3. Properties of Distribution 4. Summary and Outlook

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UCL, 23 Feb 2006 Motivation 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’.

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UCL, 23 Feb 2006 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,…E max [Audenaert, Plenio, quant-ph/ ] [Fattall et al. quant-ph/ ]

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UCL, 23 Feb 2006 B. Only Typical Entanglement Second simplification: Consider only the typical entanglement in a completely randomised system. [Hayden et al., quant-ph/ ] 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). t=0t=1 Alice Bob

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UCL, 23 Feb 2006 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?

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UCL, 23 Feb 2006 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 the objective is to find and study P(E) for randomised bipartite stabilizer states.

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UCL, 23 Feb 2006 Overview 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

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UCL, 23 Feb 2006 P(E) Theorem Statement Notation: The N qubits are grouped such that N A belong to Alice(the smaller party) and N B to Bob. The total state is pure and N=N A +N B. 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:

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UCL, 23 Feb 2006 Proof Outline (i) Take probability distribution on stabilizer states as flat. Then p(state)=1/n tot where n tot is the total number of states for the given N. Entanglement E is an integer, So P(E)=n E /n tot where n E (N,NA) is the number of possible stabilizer states with entanglement E. Simplest example: N=2, N A =1 whereby Then an explicit count gives n tot =60, n 0 =36, n 1 =24. Thus P(0) =36/60 and p(1)=24/60 n0n0 n1n1 All n tot states

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UCL, 23 Feb 2006 Proof Outline (ii) Finding n E (N, N A ) for any N and N A is tricky. Use three lemmas: Lemma 1: The total number of states is known to be [Gottesmann, Aaronson quant-ph/052328] [Gross, quant-ph/ ] Lemma 2: The number of unentangled states n 0 is Lemma 3: There is an invariant ratio (proof complicated) The lemmas together give an iterative expression for n E. This gives P(E) as P(E)=n E /n tot

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UCL, 23 Feb 2006 Overview 1.(Done) Introduction, aim of work 2. (Done)Entanglement Probability Distribution Derived Next 3. Properties of Distribution -Distribution is ‘Gaussianish’ -Average is nearly maximal -Concentration around average -Similar to general states 4. Summary and Outlook

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UCL, 23 Feb 2006 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

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UCL, 23 Feb 2006 Example of P(E) An example of P(E), for N=12, N A =5.

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UCL, 23 Feb 2006 Average is Nearly Maximal Recall maximal entanglement possible is N A, 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.

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UCL, 23 Feb 2006 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 N A but N increasing.

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UCL, 23 Feb 2006 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 N A. There is concentration around the average for general states too [Hayden et al., quant-ph/ ].

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UCL, 23 Feb 2006 Summary 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/ ] 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?

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