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Entanglement between Collective Operators in a Linear Harmonic Chain

Johannes Kofler1, Vlatko Vedral2, Myungshik S. Kim3, Časlav Brukner1,4

1University of Vienna, Austria 2University of Leeds, United Kingdom 3Queen’s University Belfast, United Kingdom 4Austrian Academy of Sciences, Austria

Quantum Information Theory & Technology Summer School Belfast, United Kingdom

September 2, 2005

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Motivation

• Aim: Detect entanglement between macroscopic parts of a system by looking at collective operators (macroscopic observables) for different regions

• Collective operators: Sum over (average of) individual operators

• Why: (i) Experimentally approachable

(ii) Fundamentally interesting: “Can collective operators be entangled?”

• Up to now: entanglement between single particles or between “mathematical” blocks

• Here: Entanglement between two “physical” blocks

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The Linear Harmonic Chain

Coupling:

Hamiltonian

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Upgrading position and momentum to operators and expansion into modes

Pseudo-momentumDispersion relation

Ground state

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Two-point Vacuum Correlation Functions

A. Botero, B. Reznik, Phys. Rev. A 70, 052329 (2004)

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Collective Blocks of Oscillators

Collective Operators

same block

different blocks

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Notation

• Usual: “Mathematical” Block

- Every measurement in the block is allowed in principle

• Here: “Physical” Block

- Characterized by collective operators

- Measurement couples only to the block as a whole

- Information loss

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Microscopic Mesoscopic Macroscopic

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Entanglement between two blocks of equal size

Matrix of second moments

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Degree of Entanglement between Blocks

Peres–Horodecki–Simon–Kim

Negativity

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Trade-off

• Microscopic case: p = 0

- Measurement apparatus has to be accurate to resolve individual oscillators

- Absolute error in the measured numbers can be large ( 1)

• Macroscopic case: p = 1

- Measurement apparatus can be coarse

- Absolute accuracy has to scale with n–2

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Neighbouring Blocks (of equal size n)

• Entanglement for all coupling parameters and block sizes n (macroscopic parts)

• Strength of entanglement decreases with decreasing coupling and increasing block size

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Separated Blocks (of equal size n)

• No entanglement between separated individual oscillators (n = 1)

• Genuine multi-particle entanglement for n = 2, 3 and 4

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Entanglement Detection by Uncertainty Relations

then the two blocks are entangled (Duan et al)

if

• For nA or nB > 1 this witness is weaker than the entanglement degree

• Entanglement not for all neighbouring blocks

• No multi-particle entanglement detected

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Scalar Quantum Field Theory

Collective operators

etc.

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Conclusions

• Entanglement between collective operators (macroscopic parts) of a system is demonstrated and quantified

• Neighbouring blocks in the harmonic chain are entangled for all block sizes and coupling constants

• Genuine multi-particle entanglement between (small) separated blocks where no individual pair is entangled

arXiv:quant-ph/0506236