Transcript Forays into Relativistic Quantum Information Science:

Foray into Relativistic
Quantum Information Science:
Wigner Rotations and Bell States
& Foray into Microsoft Powerpoint presentation
Chopin Soo
Laboratory for Quantum Information Science (LQIS)
(http://www.phys.ncku.edu.tw/~QIS/)
Physics Dept., NCKU
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).
In Non-Relativistic Quantum Mechanics ([x,p] =1):
x2
x1
<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 operator
Physical 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:
Massive
classified by momentum and spin
To evaluate:
:
L = Pure Lorentz Boost
(Eq. A)
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
m-particle operator
equivalent to Yang’s definition
=> Reduced Density Matrices are therefore defined as partial traces of
higher particle no. matrices
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.
e.g. “Unentangled” 2-particle state :
“entanglement entropy”
for bosons)
(lowest value)
Consider “Entangled” Bell state:
=>
than lowest value
True for
Results are Lorentz invariant!
Entropy: In general, divergent in QFT
e.g.
Von Neumann Entropy
=>
Generalized Zeta Function
=>
Alternative and generalization:
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
=> towards Relativistic(Lorentz invariant) QIS <=> (founded upon QFT)
=> towards General Relativistic QIS <=> QG(?)
Real Life

Give an example or real life anecdote

Add a strong statement that
summarizes how you feel or think
about this topic
Glimm’s vector
Physics
QIS & QC
Truth
?
Mathematics
Engineering
The End.
That’s all folks!