PHYSICAL REVIEW A 70, 034102 (2004)
Special relativity and reduced spin density matrices
Cezary Gonera, Piotr Kosiński, and Paweĺ Maślanka
Department of Theoretical Physics II, University of Łódź, Pomorska 149/153, 90-236 Łódź, Poland
(Received 28 October 2003; published 22 September 2004)
We derive the general formula for Lorentz-transformed spin density matrix. It is shown that an appropriate
Lorentz transformation can produce totally unpolarized state out of pure one. Further properties, as depurification by an arbitrary Lorentz boost and its relation to the localization properties are also discussed.
DOI: 10.1103/PhysRevA.70.034102
PACS number(s): 03.65.Pm, 03.65.Ud, 03.30.⫹p
It has been shown that special relativity imposes severe
restrictions on quantum information processing (for a review,
see Ref. [1]). In fact, the number of properties important
from this point of view should be reexamined if relativistic
corrections become important. This can be easily seen by
studying the simplest case of free relativistic particle with
spin. It appeared that the reduced density matrix for its spin
is not covariant under Lorentz transformations and spin entropy ceases to be the relativistic scalar [2,3]. The decrease in
spin purity caused by Lorentz transformations is related to
the spatial localization of the wave packet observed in the
rest frame and any localized pure state with separate spin and
momentum in the rest frame becomes mixed when observed
from moving inertial frame [4].
The modifications enforced by special relativity are even
more drastic for two- and many-particle systems [5–7], both
in massive and massless cases. An important property here is
that the entaglement between momenta can be transformed
by Lorentz transformation to spin and vice-versa; only the
joint entaglement is a Lorentz invariant notion [8].
The properties of relativistic transformations suggest that
the protocol for quantum communication should be appropriately changed. In this paper we consider again the single
particle with spin one-half. The general formula for Lorentztransformed Bloch vector is derived. Then, three applications
are presented. First, we rederive the following result [4]: any
boost applied to the pure spin state gives a mixed state. Second, the explicit example of general relation between depurification and localization [4] is given. Finally, we prove that
the pure state can be transformed to the state arbitrarily close
to the totally depolarized one.
Consider quantum relativistic particle of positive mass m.
The space of states carries an unitary irreducible representation of Poincare group and is spanned by 兩p , ␴典- the common
eigenvectors of four momentum and fixed (say, third) component of spin in the rest frame. Their scalar product reads
具p⬘, ␴⬘兩p, ␴典 = 2p0␦共3兲共p − p⬘兲␦␴␴⬘
兺␴
冕
具 ␺ ⬘兩 ␺ 典 =
U共A,0兲兩p, ␴典 =
d 3p
a⬘共p, ␴兲a共p, ␴兲.
2p0
共4兲
兺 D␴⬘␴„w共p,A兲…兩⌳p, ␴⬘典;
共5兲
␴⬘
here ⌳ = ⌳共A兲 is the Lorentz matrix corresponding to A,
D-unitary irreducible representation of the SU共2兲 group
while w共p , A兲 苸 SU共2兲 is the so-called Winger matrix defined
by
w共p,A兲 = b−1共⌳p兲Ab共p兲.
共6兲
Moreover, b共p兲 is the standard boost [9]:
b共p兲 =
m + p0 + p · ␴
冑2m共m + p0兲 .
共7兲
The counterpart of Eq. (5) for the wave functions reads
a⬘共p, ␴兲 =
兺 D␴␴⬘„w共⌳−1p,A兲…a共⌳−1p, ␴⬘兲.
共8兲
␴⬘
Note that for the case of spin one-half, D(w共p , A兲) = w共p , A兲.
The Wigner matrix has the following properties [9]:
(i) if A 苸 SU共2兲 , A+ = A−1, the corresponding Lorentz
transformation is simply a rotation and
共9兲
w共p,A兲 = A,
(ii) if A 苸 SL共2 , C兲 represents pure boost corresponding
to the four velocity u␮ = 共cosh␤ , sinh␤n兲, i.e.:
冉
A = A+ = exp −
冊
␤
n·␴ ,
2
共10兲
the relevant Wigner element can be computed to be
共2兲
冉
冊 冉冊 冉冊
⍀
⍀
⍀
+ i sin
e · ␴,
w共p,A兲 = exp i e · ␴ = cos
2
2
2
e⬅
where
1050-2947/2004/70(3)/034102(3)/$22.50
冕
兺␴
Unitary irreducible representation of the Poincare group
[more precisely—its universal covering ISL共2 , C兲] is given
by 关A 苸 SL共2 , C兲兴:
3
dp
a共p, ␴兲兩p, ␴典,
2p0
共3兲
is the corresponding wave function. The scalar product reads
共1兲
and p2 = m2. An arbitrary state can be expanded in terms of
basic vectors as follows:
兩␺典 =
a共p, ␴兲 ⬅ 具p, ␴兩␺典
70 034102-1
pÃn
,
兩p à n兩
©2004 The American Physical Society
PHYSICAL REVIEW A 70, 034102 (2004)
BRIEF REPORTS
sin⍀ =
2关共1 + u0兲共p0 + m兲 − u · p兴兩p à u兩
,
关共1 + u0兲共p0 + m兲 − u · p兴2 + 兩p à u兩2
cos⍀ =
关共1 + u0兲共p0 + m兲 − u · p兴2 − 兩p à u兩2
.
关共1 + u0兲共p0 + m兲 − u · p兴2 + 兩p à u兩2
共11兲
entz boost causes depurification [4]. Assume the reduced
spin density matrix in the rest frame describes pure spin
state. Using rotation invariance to put the polarization vector
in the direction of third axis we can write
Equations 共11兲 can be, equivalently, summarized as follows:
w共p,A兲 =
共1 + u0兲共p0 + m兲 − u · p + i共p à u兲 · ␴
. 共12兲
关2共1 + u0兲共p0 + m兲共m + up兲兴1/2
Let us now consider the relativistic spin − 21 particle. The
spin density matrix can be written in terms of Bloch vector
␮:
␳ = 21 共1 + ␮ · ␴兲, ␮ = Tr共␳␴兲.
␳␴␴⬘ =
冕
d 3p
a共p, ␴兲a共p, ␴⬘兲,
2p0
共14兲
while the Bloch vector is given by
␮=
冕
d 3p +
a 共p兲␴a共p兲,
2p0
where we denoted
a共p兲 =
冋 册
a共p,1兲
a共p,2兲
共15兲
共16兲
.
Let us now apply Lorentz transformation and calculate the
corresponding reduced density matrix. To this end it is sufficient to find the new Bloch vector
␮共␤兲 =
冕
␮ = e3 =
d 3p +
a 共p兲w+共p,A兲␴w共p,A兲a共p兲.
2p0
共17兲
␮共␤兲 =
冉
冊 冉
i
i
w 共p,A兲␴w共p,A兲 ⬅ exp − ⍀e · ␴ ␴ exp ⍀e · ␴
2
2
1 − ␮ 2共 ␤ 兲 =
⫻共e · ␴兲e
␮共␤兲 =
gives
␮共␤兲 =
冕
d 3p
cos ⍀a+共p兲␴a共p兲 + sin ⍀关a+共p兲␴a共p兲 ⫻ e兴
2p0
+ 共1 − cos ⍀兲关e · a+共p兲␴a共p兲兴e
共19兲
with e and ⍀ defined in Eq. (11).
Equation (19) provides the general expression for
Lorentz-transformed Bloch vector. It looks rather complicated but appears to be quite useful. However, before making
use of Eq. (19) we shall use Eq. (17) to show that any Lor-
冕
d 3p
兩a共p,1兲兩2e共p兲,
2p0
共22兲
冕
d 3p
兩a共p,1兲兩2关1 − ␮共␤兲 · e共p兲兴.
2p0
共23兲
冋冕
册
d3 p
cos ⍀兩a共p,1兲兩2 e3 .
2p0
共24兲
For ultrarelativistic observer, ␤ → ⬁:
冊
共18兲
共21兲
The necessary condition for 兩␮共␤兲兩 = 1 is that there exists a
subset
of
positive
measure
such
that
a+共p兲w+共p , A兲␴w共p , A兲a共p兲 has a constant direction if p belongs to this subset. Using Eq. (12) and a共p , 2兲 = 0 one easily
concludes that this leads to the relations of the form f共p兲
= 0 with smooth f; however, by standard arguments of differential geometry [10], the set of solutions to such equation
has vanishing measure.
Let us now analyze Eq. (19). Assume the initial state to
have pure spin density matrix. Therefore, we can arrange
things so that a共p , 2兲 ⬅ 0. Assume, further, that n = e3; then
the last term on the right hand side of Eq. (19) vanishes.
Moreover, ⍀共p , A兲 is axially symmetric. If we take 兩a共p , 1兲兩2
to be axially symmetric, the second term also vanishes and
cos ⍀ =
= cos ⍀ · ␴ + sin ⍀共␴ Ã e兲 + 共1 − cos ⍀兲
册
d 3p +
a 共p兲␴3a共p兲 e3
2p0
共20兲
where e共p兲 is the unit vector in the direction of
a+共p兲w+共p , A兲␴w共p , A兲a共p兲. Multiplying both sides of Eq.
(22) by ␮共␤兲 and using 兰共d3p / 2p0兲兩a共p , 1兲兩2 = 1 we get
Recalling properties (i) and (ii) of the Winger matrix
w共p , A兲 we easily conclude that ␮ transforms as standard
three vector under rotations while the identity
+
册
d 3p +
a 共p兲a共p兲 e3 ,
2p0
so that a共p , 2兲 ⬅ 0. Now, Eq. (17) can be rewritten as
共13兲
The case 兩␮兩 = 1 corresponds to pure state while ␮ = 0 describes maximally disordered one.
Consider the pure state of relativistic particle described by
the wave function a共p , ␴兲. The reduced density matrix reads
冋冕
冋冕
␮ = e3 =
2
共p0 − p3 + m兲2 − p⬜
2
共p0 − p3 + m兲2 + p⬜
, p⬜ = 共p1,p2,0兲.
共25兲
2
Ⰷ m2 one obtains cos ⍀ ⬇ 0. Taking a共p , 1兲
For p3 ⬇ 0, p⬜
axially symmetric and strongly peaked arround p3 = 0 and
2
= M 2 Ⰷ m2 we can arrange the integral (24) to attain arbip⬜
trary small values; the details can be supplied easily.
We conclude that the pure state may be recognized as
totally unpolarized by an observer moving sufficiently fast.
Although the initial wave function may not seem to be
very appealing, our reasoning shows that there are no a priori (i.e., not depending on the shape of initial wave packet)
bounds on disorder produced by Lorentz transformations.
Consider now the nonrelativistic particle resting in the
initial frame: 具p典 = 0 and 兩a共p , 1兲兩2 is supported in the region
兩p兩 Ⰶ m; again we assume that 兩a共p , 1兲兩2 is axially symmetric.
Expanding Eq. (25) in powers of p2 / m2 we find
034102-2
PHYSICAL REVIEW A 70, 034102 (2004)
BRIEF REPORTS
cos ⍀ ⯝ 1 −
2
p⬜
.
2m2
共26兲
Therefore
冉
冉
1
␮共␤ = ⬁兲 = 1 −
2m2
冕
冊
兩␮共␤ = ⬁兲兩 艋 1 −
d 3p 2
共p 兩a共p,1兲兩2兲 e3
2p0 ⬜
冊
frame everything is nonrelativistic, so all notions of nonrelativistic quantum mechanics, including that of position operator, are well-defined. Using Heisenberg uncertainty relations one can rewrite Eq. (27) as
冋
1
1
1
+
8m2 共⌬x1兲2 共⌬x2兲2
册
共28兲
共27兲
which gives the bound on polarization, as seen by ultrarelativistic observer, in terms of localization properties of the
state (cf. Ref. [4]).
2
The mean momentum vanishes and 具pជ ⬜
典 represents the
uncertainty of transverse momentum. However, in the rest
This work was supported by the Łódź University Grant
No. 690.
2
⬍p⬜
⬎
= 1−
e3 .
2
2m
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