Foray into Relativistic Quantum Information Science:

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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)
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Apology
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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 gt
alter efficiency of eavesdropper detection,
compromise security of quantum protocols. Quantum
teleportation Realizable, and compatible with
QFT ?
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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 gt 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 gt (QFT).
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In Non-Relativistic Quantum Mechanics (x,p
1) ltx2exp-iH(x2- x1)0/? x1gt ? 0 (x2)0 gt
(x1)0 Even if (?s)2 (x2- x1)02 - (x2-
x1).(x2- x1) lt 0 (space-like)
faster-than-light
x2
x1
If (?s)2 lt 0, ? Lorentz trans. (x2)0 lt
(x1)0 (reversal of temporal order)
In Quantum Field Theory microcausality is ensured
as ?i(x2 ),?k(x1) 0 ? (?s)2 lt 0
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Quantum Mechanics Wavefunction (state) ? does
not transform unitarily under Lorentz trans.

• Quantum Field Theory
• ? field operator
• Physical states ?gt are unitary (albeit
  infinite-dimensional)
• representation spaces of Lorentz group
• Lorentz group non-compact, no finite-dimensional
  unitary rep.
• gt 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 (?)

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Book Quantum Theory of Fields, Vol. I. Steven
Weinberg Preface
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Massive
classified by momentum and
spin
To evaluate


L Pure Lorentz Boost
(Eq. A)
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Wigner Transformation
(W. k k)
DW is a unitary representation of the Little
Group of k
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gt Little Group of k SO(3) (Wigner Rotation)
Note
consistently produces no rotation in spin space
(c.f. Eq. A) for this special case
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Infinitesimal Wigner angle
In absence of boost Wigner rotations ordinary
rotations
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Explicit Unitary Representation
Writing
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Lie Algebra of Lorentz Grp
Note Explicit infinite-dimensional unitary
representation with Hermitian
generators for non-compact Lorentz group!
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Finite Wigner rotations
gt
gtNot as easy to write finite expression in
closed form using infinite products of
infinitesimal transformations
for generic Lorentz trans gt
Complete Wigner rotation
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For spin ½ particles Specialize to

Under Lorentz trans.
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Two-particle states
n1,2 species label
Notes
But
gt

gt
Hence
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suggests combining rotational singlet(1) and
triplet(3) Bell states as the 4 .
c.f. Conventional assignment (see e.g. Nielsen
and Chuang)
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Under arbitrary Lorentz transformations
gt Complete behaviour of Bell states under
Lorentz trans. is
Under pure rotations
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Reduced Density Matrices and Identical Particles
Reduced ( )
density matrices
m-particle operator
equivalent to Yangs definition
gt Reduced Density Matrices are therefore defined
as partial traces of higher particle no.
matrices
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Lorentz Invariance of von Neumann Entropies of
Reduced Density Matrices
gt
von Neumann entropy
Lorentz Invariant!
gt
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Worked example System of two identical fermions
Diagonalization
1-particle reduced density matrix
Note for total system
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But gt Entropy of reduced density matrix
Maximizing and minimizing, subject to
gt
(c.f. for bosons)
e.g. Unentangled 2-particle state
entanglement entropy
(lowest value)
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Consider Entangled Bell state
gt
than lowest value
True for
Results are Lorentz invariant!
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Entropy In general, divergent in QFT
e.g.
Von Neumann Entropy
gt
Generalized Zeta Function
gt
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Alternative and generalization
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Summary Modest results/observations from our
foray 1. Computation of explicit Wigner
rotations for massive particles 2. Explicit
unitary rep. of Lorentz group and its
generators 3. 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 gt towards Relativistic(Lorentz
invariant) QIS ltgt (founded upon QFT) gt towards
General Relativistic QIS ltgt QG(?)
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• Real Life

• Give an example or real life anecdote

• Add a strong statement that summarizes how you
  feel or think about this topic

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Glimms vector
Physics
QIS QC
?
Truth
Mathematics
Engineering
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The End.Thats all folks!