Foray into Relativistic Quantum Information Science:

Wigner Rotations and Bell States

Chopin Soo

Laboratory for Quantum Information Science (LQIS)

(http://www.phys.ncku.edu.tw/~QIS/)

Physics Dept., NCKU

& Foray into Microsoft Powerpoint presentation

ref : quant-ph/0307107 seminar: Inst. of Phys. Acad. Sinica (Sept. 26, 2003)

Apology:

Motivations for investigating Relativistic(Lorentz Invariant) QIS:Applications: e.g.quantum cryptography, entanglement-enhanced communication, high precision clock synchronization based upon shared entanglement, quantum-enhanced positioning, quantum teleportation,…

Need: careful analysis of properties of entangled particles under Lorentz transformations, & construction of meaningful measures of entanglement (key concept and primary resource in QIS)

Issues: Lorentz invariance of entanglement (?)Possible modifications to Bell Inequality violations => alter efficiency of eavesdropper detection, compromise security of quantum protocols.Quantum teleportation: Realizable, and compatible with QFT ?

Conceptual/consistency issues: e.g.

LOCC (local operation and classical communication) is often invoked (e.g. in quantum teleportation) in non-relativistic QIS, but quantum-classical interface not sharply defined.

Bell Inequality violation: =>

Not compatible with local, non-superluminal hidden variable theory.

“Compatible” with QM, and no faster-than-light communication.

But non-rel. QM not fully consistent (!) with Lorentz invariance and causal structure of spacetime.

OR (a better formulation(?))

violation is consequence of, and fully compatible with, quantum theory which is local, Lorentz invariant & causal => (QFT).

x2

x1

In Non-Relativistic Quantum Mechanics ([x,p] =1):

<x2|exp[-iH(x2- x1)0/] | x1> 0(x2)0 > (x1)0

Even if (s)2 = [(x2- x1)0]2 - [(x2- x1)].[(x2- x1)] < 0 (space-like) “faster-than-light”

If (s)2 < 0, Lorentz trans. : (x2’)0 < (x1’)0 (reversal of temporal order)

In Quantum Field Theory microcausality is ensured as

[i(x2 ),k(x1)]± = 0 (s)2 < 0

Quantum Mechanics:Wavefunction (“state”) does not transform unitarily under Lorentz trans.

Quantum Field Theory: = field operatorPhysical states |> are unitary (albeit infinite-dimensional) representation spaces of Lorentz group

Lorentz group: non-compact, no finite-dimensional unitary rep.

=> Questions regarding the validity of“fundamental 2-state qubit” of non-rel QIS (?) and “fundamental entangled(Bell) spin-up spin-down states” Of non-rel QIS with 1-ebit (?)

Book : Quantum Theory of Fields, Vol. I.

Steven Weinberg

Preface:

:

L = Pure Lorentz Boost

(Eq. A)

To evaluate:

Massive classified by momentum and spin

Wigner Transformation :

(W. k = k)

D[W] is a unitary representation of the Little Group of k

=> Little Group of k = SO(3) (Wigner Rotation)

Note:

consistently produces no rotation in spin space (c.f. Eq. A) for this special case

Infinitesimal Wigner angle:

In absence of boost: Wigner rotations = ordinary rotations

Explicit Unitary Representation:

Writing

Lie Algebra of Lorentz Grp :

Note: Explicit infinite-dimensional unitary representation with Hermitian generators for non-compact Lorentz group!

Finite Wigner rotations:

=>

=>Not as easy to write finite expression in closed form using infinite products of infinitesimal transformations

for generic Lorentz trans =>

Complete Wigner rotation :

For spin ½ particles: Specialize to

&

Under Lorentz trans.:

Two-particle states: n1,2 = species label

Notes:

=> :

But=>

Hence

suggests combining rotational “singlet”(1) and “triplet”(3) Bell states as the 4 .

c.f. Conventional assignment (see e.g. Nielsen and Chuang)

Under arbitrary Lorentz transformations:

=> Complete behaviour of Bell states under Lorentz trans. is :

Under pure rotations

Reduced Density Matrices and Identical Particles

Reduced ( ) density matrices

=> Reduced Density Matrices are therefore defined as partial traces of higher particle no. matrices

equivalent to Yang’s definition

m-particle operator

Lorentz Invariance of von Neumann Entropies of Reduced Density Matrices

=>

von Neumann entropy

=>

Lorentz Invariant!

Worked example: System of two identical fermions

“Diagonalization” :

1-particle reduced density matrix:

Note: for total system

But => Entropy of reduced density matrix

Maximizing and minimizing, subject to

=> (c.f. for bosons)

e.g. “Unentangled” 2-particle state :

“entanglement entropy” (lowest value)

Consider “Entangled” Bell state:

=>

Results are Lorentz invariant!

than lowest value

True for

Entropy: In general, divergent in QFT

Generalized Zeta Function

Von Neumann Entropy

=>

e.g.

=>

Alternative and generalization:

Summary:

Modest results/observations from our foray:

1. Computation of explicit Wigner rotations for massive particles2. Explicit unitary rep. of Lorentz group and its generators3. Definition, and behaviour of Bell States under arbitrary Lorentz trans.4. Definition, and applications of Lorentz covariant reduced density matrices to identical particle systems.5. Lorentz-invariant characterization of entanglement.6. Relation betn. von Neumann entropy and generalized zeta function

=> towards Relativistic(Lorentz invariant) QIS <=> (founded upon QFT) => towards General Relativistic QIS <=> QG(?)

Real Life

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Glimm’s vector

Physics

Mathematics

Engineering

Truth

QIS & QC ?

The End.

That’s all folks!