Quantification of entanglement via uncertainties
Alexander A. Klyachko, Barış Öztop, and Alexander S. ShumovskyFaculty of Science, Bilkent University, Bilkent, Ankara, 06800 Turkey
共Received 14 June 2006; revised manuscript received 17 January 2007; published 12 March 2007兲 We show that entanglement of pure multiparty states can be quantified by means of quantum uncertainties of certain basic observables through the use of a measure that was initially proposed by Klyachko et al.关Appl. Phys. Lett. 88, 124102共2006兲兴 for bipartite systems.
DOI:10.1103/PhysRevA.75.032315 PACS number共s兲: 03.67.Mn, 03.65.Ud I. INTRODUCTION
Recent success in the realization of quantum key distribu-tion has been achieved through the use of quantum correla-tions between the parts in two-qubit systems, which are pe-culiar to entangled states共see 关1–3兴 and references therein兲. Further development of practical implementations of quantum-information technologies requires sources of robust entangled states and reliable methods of detection of the amount of entanglement carried by those states 共e.g., see 关4,5兴 and references therein兲.
There is no universal measure of entanglement suitable for all systems even in the case of pure states. For example, entanglement of two qubits is measured by means of the concurrence关6兴 for both pure and mixed states. In the case of pure states, the definition of concurrence has been ex-tended to bipartite systems with any dimension of the single-party Hilbert state dⱖ2 关7,8兴. At the same time, this defini-tion does not work for systems with the number of parts larger than two. In particular, concurrence is incapable of measuring the three-party entanglement in three-qubit sys-tems关9兴.
In our previous paper关10兴, we found a representation of concurrence valid for pure states of an arbitrary bipartite sys-tem which coincides with the Wootters concurrence关6兴 for the case of pure two-qubit states. A logical advantage of this representation is that it expresses the amount of entangle-ment in terms of variances共quantum uncertainties兲 of certain observables. In a sense, this reflects the physical nature of entanglement as a manifestation of quantum uncertainties at their extreme 关11–14兴 共see also the discussion in Refs. 关15–17兴兲.
The main objective of this paper is to prove the validity of the measure of Ref.关10兴 for pure states in general settings.
The paper is organized as follows. We start by giving a definition of the basic observables specifying a given physi-cal system. We further connect the notion of total variance with the measure of entanglement. Then we discuss applica-tion of this measure to pure states of two and three qubits. Further, we briefly consider how this measure works in the case of mixed states. Finally, in Appendix A, we put the proof of the validity of our measure in general settings.
II. QUANTUM DYNAMICAL SYSTEMS
An idealized von Neumann approach to quantum mechan-ics, based on the assumption that all Hermitian operators represent measurable quantities, was first put into question
by Wick, Wightman, and Wigner 关18兴 in 1952. Later, Her-mann关19兴 argued soundly that the basic principles of quan-tum mechanics require that measurable observables should form a Lie algebraL of 共skew兲 Hermitian operators acting in the Hilbert spaceH of the quantum system in question. We refer toL as the Lie algebra of observables and to the cor-responding Lie group G = exp共iL兲 as the dynamical symme-try group of the quantum system.
Restrictions on available observations are of fundamental importance for physics in general, and for quantum informa-tion specifically. The latter case usually deals with correlated states of a quantum system with macroscopically separated spatial components, where only local measurements are fea-sible. For example, the dynamical group of the bipartite sys-tem H=HA丢HB with full access to local degrees of free-dom amounts to SU共HA兲⫻SU共HB兲. Without such restrictions, the dynamical group G = SU共H兲 would act tran-sitively on pure states苸H, which makes them all equiva-lent. In this case there would be no place for entanglement and other subtle quantum phenomena based on intrinsic dif-ferences between quantum states.
III. TOTAL VARIANCE
Recall that the uncertainty of an observable X苸L in the state苸H is given by the variance
V共X,兲 = 具兩X2兩典 − 具兩X兩典2. 共1兲 Let us now choose an orthonormal basis X␣of the algebra of observablesL with respect to its Cartan-Killing form 共X,Y兲K 关20兴 and define the total variance by equation
V共兲 =
兺
␣ 共具兩X␣
2兩典 − 具兩X
␣兩典2兲. 共2兲
For example, for a two-qubit systemHA丢HB one can take the basis ofL=su共HA兲+su共HB兲, consisting of Pauli opera-torsiAandjBthat act on components A and B, respectively. For a general multipartite system, the sum 共2兲 is extended over orthonormal bases of traceless local operators for all parties of the system.
The total variance 共2兲 can be understood as the trace of the quadratic form
Q共X兲 = 具兩X2兩典 − 具兩X兩典2, X苸 L,
on the Lie algebraL, and therefore it is independent of the basis X␣. It measures the overall level of quantum fluctua-tions of the system in state.
The first sum in the total variance共2兲 contains the Casimir operator C =兺␣X␣2, which acts as a scalar CH in every irre-ducible representation G :H. As a result we get
V共兲 = CH−
兺
␣ 具兩X␣兩典
2. 共3兲
To clarify the second sum, consider the average of the basic observables X␣in state,
X=
兺
␣ 具兩X␣兩典X␣. 共4兲
It can be understood as the center of quantum fluctuations of the system in state . For example, in a spin system it is given by a suitably scaled spin projection onto the mean spin direction in state. The operator X is also independent of the basis X␣. This can be seen from the following property: 具兩X兩典 = 共X,X兲K, ∀ X 苸 L, 共5兲
which holds for basic observables X = X␣ by orthogonality 共X␣, X兲K=␦␣, and hence by linearity for all X苸L. Since
the Killing form is nondegenerate, Eq. 共5兲 uniquely deter-mines Xand provides for it a coordinate-free definition. We show in Appendix A that the operator Xis closely related to orthogonal projection of =兩典具兩 into the Lie algebra L. The operator X allows one to recast the total variance 共2兲 into the form
V共兲 = CH−具兩X兩典. 共6兲
In Appendix A, we explain how the total variance can be calculated and give an explicit formula for the multicompo-nent system H=丢AHA with full access to local degrees of freedom in terms of reduced statesA
V共兲 =
兺
A关dim HA− TrH A共A
2兲兴. 共7兲
IV. COMPLETELY ENTANGLED STATES We can infer from共3兲 the inequality
V共兲 ⱕ CH 共8兲
which turns into an equation if and only if
具兩X兩典 = 0, ∀ X 苸 L. 共9兲
For multiparty systemsH=丢AHA, the latter equation means that all one-party reduced states are completely disordered. In other words, there exists some local basis such that the reduced state is given by a diagonal matrixA, corresponding to a uniform probability distribution 共that is, A are scalar operators兲. This is a well-known characterization of maxi-mally entangled states. In general we refer to共9兲 as the en-tanglement equation and call the corresponding state com-pletely entangled.
The completely entangled states are characterized by maximality of the total variance. Therefore one may be tempted to consider entanglement as a manifestation of quantum fluctuations in a state where they come to their
extreme. The entanglement equation共9兲 just states that, in a completely entangled state, the quantum system is at the center of its quantum fluctuations, that is X= 0.
V. MEASURE OF ENTANGLEMENT
States opposite to entangled ones, to wit those with a minimal total level of quantum fluctuationsV共兲, for a long time were known as coherent states 关21兴 共see also Refs. 关11,22兴兲. For multicomponent systems like HA丢HB coher-ent states are just decomposable or uncoher-entangled states =A丢B.
Observe关10兴 that the square of the concurrence C共兲 for a two-component system coincides with the total variance V共兲 reduced to the interval 关0,1兴
C2共兲 =V共兲 − Vcoh Vent−Vcoh
, 共10兲
whereVentandVcoh are the total levels of quantum
fluctua-tions in completely entangled and coherent states, respec-tively. This clarifies the physical meaning of the concurrence as a measure of overall quantum fluctuations in the system and leads us to the natural measure of entanglement of pure states关10兴
共兲 =
冑
V共兲 − VcohVent−Vcoh
共11兲
valid for an arbitrary quantum system. It coincides with the concurrence for two-component systems, but we refrain from using this term in general, to avoid confusion with other multicomponent versions of this notion introduced in 关23兴. We explain how this measure can be calculated in Appendix A. For a multicomponent system H=丢AHA, it can be ex-pressed via local data, encoded in reduced statesA,
2共兲 =
兺
A 共1 − TrA2兲兺
A冉
1 − 1 dimHA冊
. 共12兲
For example, in a two-component system H=HA丢HB the reduced states A and B are isospectral. Hence TrA2 = TrB2 and for a system of square format d⫻d we arrive at the familiar formula for concurrence关7兴
C共兲 =
冑
dd − 1共1 − TrA
2兲 共13兲
共in 关7兴 the normalization factor is left adjustable兲. The isos-pectrality of single-party reduced states means that entangle-ment can be measured locally. For example, in the case of a bipartite spin-s system, measurement of only three observ-ables 共spin operators for either party兲 completely specifies the concurrence共see also the discussion in 关24兴兲.
An important application for the case of two qubits is provided by the polarization of photon twins共biphotons兲 that are created by type-II down-conversion关25兴. The spin opera-tors Sjcan be associated with the Stokes operators
Sx⬃ 共aH†aV+ aV † aH兲/
冑
2, Sy⬃ i共aH†aV− aV † aH兲/冑
2, Sz⬃ aH†aH− aV†aV, 共14兲 so that the measurement of concurrence共11兲 assumes mea-surement of three Stokes operators for either outgoing pho-ton beam. Here aH共aV兲 denotes the photon annihilation op-erator with horizontal 共vertical兲 polarization. The polarization of photons is known to be measured by means of either a standard six-state or a minimal four-state ellip-someter关26兴.Nevertheless, there is a certain problem with simultaneous measurement of polarization for one of two photons created at once and forming an entangled couple. Because of the commutation relation
关Sj,Sk兴 = i⑀jkmSm, j,k,m = x,y,z,
the three projections of spin共or three Stokes operators兲 can-not be measured independently. The minimal uncertainty re-lation by Schrödinger关27兴 states
V共;Sj兲V共;Sk兲 − 关Cov共Sj,Sk兲兴2ⱖ1
4円具兩关Sj,Sk兴兩典円
2,
共15兲 where V共; Sj兲 denotes the variance 共uncertainty兲 of observ-able Sjin the state and the covariance Cov共Sj, Sk兲 has the form
Cov共Sj,Sk兲 =1
2具兩SjSk+ SkSj兩典 − 具兩Sj兩典具兩Sk兩典. It is a straightforward matter to see that the uncertainty rela-tion is simply reduced to the following one:
0ⱕ 具兩X兩典 ⱕ 1/4, 共16兲
where Xis defined by Eq.共4兲. Thus, the uncertainty relation 共15兲 becomes an exact equality when =coh with 具兩X兩典=1/4. In other words, this is an unentangled
bipho-ton state in which each phobipho-ton has well-defined polarization. In the case of a completely entangled biphoton state, the quantity具兩X兩典 has zero value 关due to the condition 共9兲兴. In this case, the measurement performed on a single photon raises an additional question: how to distinguish between entanglement and classical unpolarized state.
Since Eq. 共16兲 is the only relation, connecting different components of the average spin vector in either party, the local quantity具兩X兩典 cannot be detected by either a single or even two measurements.
VI. MEASURE„… BEYOND TWO-PARTITE STATES Postponing consideration of the measure共兲 in general settings to Appendix A, we now note that, in the case of a multipartite system, it gives the total amount of entanglement carried by all types of interparty correlations.
For example, the Greenberger-Horne-Zeilinger 共GHZ兲 state of three qubits
兩G典 = x兩000典 +
冑
1 −兩x兩2兩111典, 兩x兩 苸 关0,1兴, 共17兲carries only three-party entanglement. This means that any two parties are not entangled. In fact, any reduced two-qubit state, say,
AB= TrC兩G典具G兩 = 兩x兩2兩00典具00兩 + 共1 − 兩x兩2兲兩11典具11兩,
clearly has zero concurrence. The amount of three-part en-tanglement in共17兲 is measured by the three-tangle 关9兴 or Cayley hyperdeterminant关28兴 共for the definition of the three-tangle, see Appendix B兲. It is easily seen that
共G兲 =2共G兲 = 4兩x兩2共1 − 兩x兩2兲.
Thus, the squared measure 共11兲, calculated for the three-qubit state共17兲, gives the same result as three-tangle.
Another interesting example is provided by the so-called W state of three qubits,
兩W典 =
冑
13共兩011典 + 兩010典 + 兩110典兲. 共18兲 This is a nonseparable state in three-qubit Hilbert space. Nevertheless, it does not manifest three-party entanglement because the corresponding three-tangle共W兲=0 关28兴. At the same time, the measure共11兲 gives
共W兲 =2
冑
23 ⬇ 0.94 共19兲
becauseV共W兲=8+2/3 and Vcoh= 6 in this case. The point is
that there is a two-qubit entanglement in the state 共18兲. To justify that the difference 2 + 2 / 3 is caused just by quantum pairwise correlations, let us calculate the total covariance
Cov共W兲 =
兺
i=x,y,zJ兺
⫽J⬘共具W兩iJiJ⬘兩W典 − 具W兩iJ兩W典具W兩iJ⬘兩W典兲. 共20兲 Here J , J
⬘
= A , B , C label the parties. It is a straightforward matter to see thatV共W兲−Vcoh= Cov共W兲. Similar results canbe obtained for the so-called biseparable states of three qu-bits
共兩001典 + 兩010典兲, 共兩001典 + 兩100典兲, 共兩010典 + 兩100典兲, 共21兲 which also manifest entanglement of two qubits and no en-tanglement of all three parts.
Examining entanglement of multiqubit systems in general 共the number of parts is greater than two兲, it is necessary first to determine classes of states with different types of en-tanglement共including the class of unentangled states兲. It is assumed that those classes are nonequivalent with respect to stochastic local operations assisted by classical communica-tion共SLOCC兲 关29兴. The point is that entanglement of a given type cannot be created or destroyed under action of SLOCC. In the case of three qubits, such a classification has been considered in Refs. 关28,30兴. In the case of four qubits, the
number of classes is much higher关31兴. A useful approach to classification is based on investigation of geometrical invari-ants for a given system共e.g., see Refs. 关11,32兴兲.
For example, the class of four-qubit entangled states can be specified by the generic GHZ-type state
x兩0000典 ±
冑
1 −兩x兩2兩1111典, 兩x兩 苸 关0,1兴, 共22兲 which becomes completely entangled at 兩x兩 =1/冑
2. In gen-eral, four-qubit completely entangled states can be defined by means of the condition共9兲 共see Appendix C兲. For the state 共22兲, the measure 共11兲 gives the amount of entanglement =冑
1 −共2兩x兩2− 1兲2, which becomes complete entanglement at 兩x兩 =1/冑
2 as expected.At the same time, there is another class of pairwise sepa-rable four-qubit states
1
冑
2共兩00典 + 兩11典兲丢 1冑
2共兩01典 + 兩10典兲, 共23兲 in which the first two pairs and the last two pairs separately manifest complete two-party entanglement, while there is no four-qubit entanglement关compare with the biseparable states of three qubits 共21兲兴. In this case, the measure 共11兲 again gives the total amount of entanglement carried by the parts of the system.VII. MIXED ENTANGLEMENT
The measure共11兲 cannot be directly applied to calculation of entanglement of mixed states because it is incapable of separation of classical and quantum contributions into the total variance 共2兲. Therefore, 共兲 always gives an estima-tion from above for the entanglement of mixed states. This can be easily checked for some characteristic states like the Werner state 关33兴 and the so-called maximally entangled mixed state of Ref.关34兴.
As far as we know, nowadays there is no universally rec-ognized protocol for separation of classical and quantum un-certainties in mixed states except for the case of two qubits 关6兴. A promising approach proposed in Refs. 关8,23兴 consists in the representation of concurrence of a mixed state as inf兺iC共i兲 of all properly normalized states such that =兺i兩i典具i兩.
VIII. SUMMARY
We have shown that the description of entanglement in a given system requires pre-definition of basic observables and that the entanglement of pure states can be adequately quan-tified in terms of the total variance of all basic observables. Unlike the conventional concurrence and three-tangle, which measure the amount of entanglement of different groups of correlated parties, our measure gives the total amount of multipartite entanglement carried by a given state. Other evi-dent virtues of the measure 共11兲 are its simple physical meaning, its applicability beyond bipartite systems, and its operational character caused by measurement of quantum uncertainties of well-defined physical observables.
At the same time, this measure cannot be directly applied to calculation of entanglement in mixed states. However, it may be used in the way that has been discussed in Refs. 关8,23兴 as follows:
共兲 = inf
兺
i共i兲.
ACKNOWLEDGMENTS
The authors thank Dr. S. J. van Enk, Dr. V. Korepin, and Dr. L. Viola for useful discussions and indication of their important works. One of the authors 共B.Ö.兲 would like to acknowledge the Scientific and Technical Research Council of Turkey共TÜBİTAK兲 for financial support.
APPENDIX A
Here we calculate the total variance V共兲 and the en-tanglement measure共兲.
Let Herm共H兲 be the space of all Hermitian operators act-ing in the Hilbert spaceH with trace metric TrH共XY兲. For the simple algebra L, restriction of the trace metric onto L is proportional to the Cartan-Killing form
TrH共XY兲 = DH共X,Y兲K, X,Y苸 L
with the coefficient DH known as the Dynkin index 关20兴. Consider now the orthogonal projection L of : =兩典具兩 苸Herm共H兲 into the subalgebra L傺Herm共H兲, so that TrH共X兲=TrH共LX兲, ∀ X苸L. The projection L is closely related to the mean operator共4兲
X=
兺
␣ TrH共X␣兲X␣ =兺
␣ TrH共LX␣兲X␣ = DH兺
␣ 共L ,X␣兲KX␣ = DHL. Therefore 具兩X兩典 = TrH共X兲 = TrH共LX兲 = DHTrH共L2兲 and the total variance共2兲 can be written in the formV共兲 = CH−具兩X兩典 = CH− DHTrH共L2兲. 共A1兲 For simple algebra the Casimir CHand Dynkin index DHare given by equations
CH=共, + 2␦兲, DH=dimH
dimL共, + 2␦兲, 共A2兲 where denotes the highest weight of the irreducible repre-sentation H and 2␦ is the sum of positive roots of L. For example, for full algebra of traceless Hermitian operators L=su共H兲 we have
CH= dimH − 1
dimH, DH= 1. 共A3兲 In general, the algebra L splits into simple components L=丣ALA and its irreducible representation H into tensor productH=丢AHA. In this case Eq.共A1兲 should be modified as follows: V共兲 =
兺
A 关CHA− DH ATrHA共A 2 LA 2 兲兴, 共A4兲whereA= dimH/dim HA.
In the quantum-information settingLA is the full algebra of traceless Hermitian operators XA:HA→HA. In this case everything can be done explicitly.
By definition of the reduced statesA we have TrH共XA兲 = TrH
A共AXA兲 =A
−1
TrH共AXA兲. Comparing this with the equation TrH共XA兲=TrH共L
AXA兲, ∀ XA苸LA, characterizing the projectionL
A苸LA we infer LA=A
−1A0,
whereA0=A−共1/dim HA兲I is the traceless part ofA. This allows us to calculate the trace
TrH A共LA 2 兲 =−2
冉
Tr HA共A 2兲 − 1 dimHA冊
.Plugging this into Eq.共A4兲 and using 共A3兲 we finally get V共兲 =
兺
A
关dim HA− TrH A共A
2兲兴. 共A5兲
As an example, consider the completely entangled statefor whichA=共1/dim HA兲I. This gives the maximum of the total variance, Vmax=Vent=
兺
A冉
dimHA− 1 dimHA冊
.The minimum of the total variance is attained for the coher-ent 共⫽separable兲 state , for which reduced states A are pure. Hence
Vmin=Vcoh=
兺
A
共dim HA− 1兲.
Combining these equations we can write down our measure of entanglement共11兲 explicitly for a multicomponent system H=丢AHAof arbitrary format 2共兲 =
兺
A 关1 − Tr共A2兲兴兺
A冉
1 − 1 dimHA冊
. 共A6兲 APPENDIX BFor an arbitrary normalized state of three qubits
兩典 =
兺
ᐉ,m,n=0 1
ᐉmn兩 ᐉ mn典
the three-tangle has the form关9,28兴
共兲 = 4兩0002 1112 +0012 1102 +0102 1012 +1002 0112 − 2共000001110111+000010101111 +000100011111+001010101110 +001100011110+010100011101兲 + 4共000011101110+001010100111兲兩. APPENDIX C
A general pure state of four qubits can be written in the form
兩典 =
兺
k,ᐉ,m,n=01
kᐉmn兩k, ᐉ ,m,n典 共C1兲
with the normalization condition兺k,1ᐉ,m,n=0兩kᐉmn兩2= 1. Thus,
there are 31 real parameters, defining any state. Condition共9兲 gives 12 equations for the coefficientskᐉmnin共C1兲:
具x共A兲典 = 共 0000 * 1000+0100* 1100+0010* 1010+0001* 1001 +0110* 1110+0101 * 1101+0011 * 1011+0111 * 1111兲 +共c.c.兲 = 0, 具x共B兲典 = 共 0000 * 0100+1000 * 1100+0010 * 0110+0001 * 0101 +1010* 1110+1001 * 1101+0011 * 0111+1011 * 1111兲 +共c.c.兲 = 0, 具x共C兲典 = 共0000 * 0010+1000 * 1010+0100 * 0110+0001 * 0011 +1100* 1110+1001* 1011+0101* 0111+1101* 1111兲 +共c.c.兲 = 0, 具x共D兲典 = 共 0000 * 0001+1000* 1001+0100* 0101+0010* 0011 +1100* 1101+1010 * 1011+0110 * 0111+1110 * 1111兲 +共c.c.兲 = 0, 具y共A兲典 = i共
1000 * 0000+1100* 0100+1010* 0010+1001* 0001 +1110* 0110+1101 * 0101+1011 * 0011+1111 * 0111兲 +共c.c.兲 = 0, 具y共B兲典 = i共 0100 * 0000+1100 * 1000+0110 * 0010+0101 * 0001 +1110* 1010+1101 * 1001+0111 * 0011+1111 * 1011兲 +共c.c.兲 = 0, 具y共C兲典 = i共 0010 * 0000+1010 * 1000+0110 * 0100+0011 * 0001 +1110* 1100+1011 * 1001+0111 * 0101+1111 * 1101兲 +共c.c.兲 = 0,
具y共D兲典 = i共 0001 * 0000+1001 * 1000+0101 * 0100+0011 * 0010 +1101* 1100+1011 * 1010+0111 * 0110+1111 * 1110兲 +共c.c.兲 = 0, 具z共A兲典 = 兩 0000兩2−兩1000兩2+兩0100兩2+兩0010兩2+兩0001兩2 −兩1100兩2−兩1010兩2−兩1001兩2+兩0110兩2+兩0101兩2 +兩0011兩2−兩 1011兩2−兩1101兩2−兩1110兩2+兩0111兩2 −兩1111兩2= 0, 具z共B兲典 = 兩 0000兩2+兩1000兩2−兩0100兩2+兩0010兩2+兩0001兩2 −兩1100兩2+兩1010兩2+兩1001兩2−兩0110兩2−兩0101兩2 +兩0011兩2+兩1011兩2−兩1101兩2−兩1110兩2−兩0111兩2 −兩1111兩2= 0, 具z共C兲典 = 兩 0000兩2+兩1000兩2+兩0100兩2−兩0010兩2+兩0001兩2 +兩1100兩2−兩1010兩2+兩1001兩2−兩0110兩2+兩0101兩2 −兩0011兩2−兩1011兩2+兩1101兩2−兩1110兩2−兩0111兩2 −兩1111兩2= 0, 具z共D兲典 = 兩 0000兩2+兩1000兩2+兩0100兩2+兩0010兩2−兩0001兩2 +兩1100兩2+兩1010兩2−兩1001兩2+兩0110兩2−兩0101兩2 −兩0011兩2−兩1011兩2−兩1101兩2+兩1110兩2−兩0111兩2 −兩1111兩2= 0,
where 具␣共i兲典=具ent兩␣共i兲兩ent典 and c.c. denotes the complex
conjugate. Thus, there are infinitely many completely en-tangled states and the state共22兲 at 兩x兩 =1/
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