Relativity of Entanglement

3 Pages·0.08 MB·English

by  W. Y. Hwang

Relativity of Entanglement
Won-Young Hwang∗, Jinhyoung Lee†, Doyeol (David) Ahn‡, and Sung Woo Hwang⋆
Institute of Quantum Information Processing and Systems, University of Seoul, 90 Jeonnong, Tongdaemoon, Seoul 130-743,
Korea
of entanglement does not increase under any LOCCs.
It has recently been suggested that various entanglement
The entanglement of quantum states has a rich struc-
measuresforbipartitemixedstatesdonotingeneralgivethe
tureandrequire,ingeneral,multipleparametersforcom-
same ordering even in the asymptotic cases [S. Virmani and
pletecharacterization. Itwasshownthatasingleparam-
M. B. Plenio, Phys. Lett. A 268, 31 (2000)]. That is, for
eter is sufficient to characterize pure entangled states of
two certain mixed states, the order of the degree of entan-
bipartite systems in the asymptotic case [14,15]. In this
glement depends on the measures. Therefore, incomparable
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pairs of mixed states which cannot be transformed to each special case, all entanglement measures are reduced to
0
otherwith unitefficiency byanycombinations of local quan- a unique measure [16,17]. In general, however, the en-
0
2 tumoperationsandclassicalcommunicationsexist. Wemake tanglement requires multiple parameters for characteri-
an analogy of the relativity of the order of the degree of en- zation,forexample,N−1parametersforpureentangled
n
tanglement to therelativity of temporal ordersin thespecial states of bipartite N-dimensional systems [15,18,19].
a
J theory of relativity. For better understanding and manipulation of entan-

8 gled states, quantum states must be classified as well as
PACS number(s);03.67.-a, 03.65.Bz
1 possible. The degree of entanglement of an entangled
pair can be compared by investigating the existence of
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Quantum entanglement led to a controversy over the LOCCs that transform one state to another. One state
v
0 Einstein-Podolsky-Rosenexperiment [1] and to the non- canbesaidtobe moreentangledthananotherifthe for-
1 locality of quantum mechanics [2]. On the other hand, mercanbetransformedtothelatterbyLOCCswithunit
0 entanglement is one of the key ingredients in quantum efficiency. However, some pairs of entangled states exist
1 information processing [3]. For example, a speedup suchthatonecannotbe transformedtothe otherbyany
1
in quantum computation [4] is obtained through paral- LOCCs. These pairs of states cannot be compared with
0
0 lel quantum operations on massively superposed states, each other; in other words, they are ‘incomparable’ (A
/ which are, in general, entangled. simple example is given in Ref. [20]).
h
Recently, it has been shown that different entangle- Consideration of orderings in degree of entanglement
p
- ment measures can give rise to different orderings for areimportantinanalyzingthestructureofentanglement.
nt mixed states of bipartite systems [5,6]. In this paper, Two certain entanglement measures EA and EB are de-
a we make ananalogybetween the non-uniqueness (or rel- finedtohavethe sameorderingifthey satisfythefollow-
u ativity) of the order of the degree of entanglement and ing condition for any two density operatorsρi and ρj;
q
the relativity of temporal order in the special theory of
: E (ρ )>E (ρ )⇔E (ρ )>E (ρ ). (1)
v relativity [7]. The temporal order of two certain events A i A j B i B j
i
X can be reversed, depending on the observers’ reference Recently, Virmani and Plenio showed that if two cer-
r frames when the two events are not causally conntected. tainentanglementmeasures,whichareidenticalforpure
a This is analogous to the following fact: The order of the states, give different degrees of entanglement for some
degree of entanglement for two certain mixed states can mixed states, the orderings of the two measures are dif-
bereversed,dependingontheentanglementmeasures. A ferent for those mixed states [5]. For examples, the exis-
trajectoryofquantumstatesinducedbyalocalquantum tence of a bound entangled state and some losses [21,22]
operationassistedby classicalcommunications(LOCCs) in purification processes cause the entanglement of cost
correspondsto a dynamic trajectory of a particle in spe- E to be strictly greater than entanglement of distilla-
C
cial relativity. tion E even in asymptotic cases. (In the finite case,
D
A few authors have proposed serveral entanglement it was explicitly shown by numerical analysis that the
measures, which quantify the degree of entanglement of entanglement of formation E [11] and the negativity of
f
quantum states, such as negativity of entanglement En entanglement En give different orderings [6].) Thus, the
[8–10],entanglementofcostEC,entanglementofdistilla- orderings in the degree of entanglement depend on the
tion ED [11], and quantum relative entropy of entangle- measures. This fact suggests that the various entangle-
mentEr [12]. Thesearereasonablemeasuresinthesense mentmeasuresdo nothaveto givethe same orderingfor
that they satisfy the three necessary conditions [11–13]: mixed states even in asymptotic cases.
(C1) E(ρ) = 0 if and only if the density operator ρ of a We note that, although the conclusion appears to be
bipartite system is separable, (C2) local unitary opera- odd, it does notgiverise to any barecontradictions. Let
tions leaveE(ρ) invariant,and(C3) the expected degree us consider two density operators ρ and ρ . The fact
i j
1
that the order depends on the entanglement measures twocertainmixedstates’orderinginthedegreeofentan-
obviouslymeans thatdegreeofentanglementofρ is less glement depends on the entanglement measures. How-
i
thanthatofρ inoneofthe twomeasuresandviceversa ever, such pairs of mixed states cannot be transformed
j
in the other. That is, we have either to each other by LOCCs, so they are incomparble. We
make an analogy between these facts and the relativity
EA(ρ1)>EA(ρ2) and EB(ρ1)<EB(ρ2), (2) of temporalorder in the special theory of relativity. The
non-uniqueness of the entanglement measure is also in
or
analogy with the non-existence of a preferred reference
EA(ρ1)<EA(ρ2) and EB(ρ1)>EB(ρ2). (3) frame. Our hope is that this analogy will inspire other
ideaswhichmayleadustoabetterunderstandingofthe
A quantum state with density operator ρ cannot be structure of entanglement measures. It is notable that
i
transformed to one with ρ by any LOCCs due to con- after this work the analogy was persued in the case of
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dition (C3) if the degree of entanglement of ρ is less finite pure-state entanglement [18].
i
thanρ inany ofthe reasonableentanglementmeasures.
j
(With less efficiency than one, forbidden paths may be
ACKNOWLEDGMENTS
allowed by LOCCs.) Thus, the pairs of states are in-
comparable [20], implying the necessity of multiple pa-
rameters for characterizing the entanglement of bipar- This work was supported by the Korean Ministry of
tite mixed states. Now, it is interesting to note that Science and Technology through the Creative Research
this fact is analogous to a fact of the special relativity. Initiatives Program under Contract No. 99-C-CT-01-C-
Thetemporalorderoftwocertaineventsdepends onob- 35.
servers’ reference frames [7]. Although it appears to be
odd,thisfactdoesnotgiverisetocontradictionsbecause
thetwoeventscannotbecausallyconnected(orcanonly
be space-likelyconnected)in this case. Letus consider a
map where each mixed state is coordinated by degree of
entanglementof two certainmeasures,E andE . (See
A B ∗ Present address: Department of Electrical and Com-
Fig. 1.) Wecaneasilyseethatapointinthemapcannot
puter Engineering, Northwestern University, Evanston,
bemovedbyanylocalunitaryoperationduetocondition
IL 60208-3118, USA:[email protected]
(C2). Thus,aclassofmixedstates,whichareequivalent † Previous address: Department of Physics, Sogang Uni-
within a local unitary operation, corresponds to a point versity,CPO Box 1142, Seoul 100-611, Korea
in the map. In general, the converse is not true. The ‡ Also with Department of Electrical
larger the number of reasonable entanglement measures Engineering, University of Seoul, Seoul 130-743, Korea;
we adopt, the more refined the coordination of density [email protected]
operators will become. If LOCCs are applied, a point ⋆ Permanentaddress: DepartmentofElectronicsEngineer-
ing, Korea University, 5-1 Anam, Sungbook-ku, Seoul
can flow through a trajectory that always points in the
136-701, Korea.
lower-left direction from a point in the map. This fact
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Fig. 1; Consider a map where each mixed state is co-
ordinated by the degrees of entanglement of two certain
measures, E and E . If LOCCs are applied, a point
A B
can flow through a trajectory that always points in the
lower-left direction from a point in the map. This fact
is in analogy with special relativity where each observer
goes through a path in space-time that always points to
somewhere within the light-cone from a point. A point
and a trajectory in the map, respectively, correspond to
aneventandanobserver’spathinspace-time. Thestates
correspondingtopandq areincomparablesincetheycan-
not be transformed, with unit efficiency, to each other by
any LOCCs.
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