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Entanglement of biphoton states: qutrits and ququarts

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New Journal of Ph ys ics
The open­access journal for physics

Entanglement of biphoton states: qutrits and ququarts
M V Fedorov1 , P A Volkov1 , J M Mikhailova1,2,4 , S S Straupe3 and S P Kulik3 1 A M Prokhorov General Physics Institute, Russian Academy of Science, Moscow, Russia 2 Max-Planck-Institut fÝr Quantenoptik, Garching, Germany 3 Faculty of Physics, M V Lomonosov Moscow State University, Moscow, Russia E-mail: fedorovmv@gmail.com, peter.volkov@gmail.com, j.mikhailova@gmail.com, straups@yandex.ru and sergei.kulik@gmail.com
New Journal of Physics 13 (2011) 083004 (32pp)

Received 5 April 2011 Published 8 August 2011 Online at http://www.njp.org/
doi:10.1088/1367-2630/13/8/083004

We investigate, in a general form, entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of, correspondingly, degenerate and non-degenerate spontaneous parametric down-conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and polarization entanglement are fully taken into account. In the case of qutrits, the most general three-parametric families of maximally entangled and nonentangled states are found, and anticorrelation of the degree of entanglement and polarization is shown to occur and to be characterized by a rather simple formula. Biphoton ququarts are shown to be two-qudits with the single-photon Hilbert space dimensionality d = 4, which differentiates them significantly from the often used two-qubit model (d = 2). New expressions for entanglement quantifiers of biphoton ququarts are derived and discussed. Rather simple procedures for a direct measurement of the degree of entanglement are described for both qutrits and ququarts.
Abstract.

4

Author to whom any correspondence should be addressed.

New Journal of Physics 13 (2011) 083004 1367-2630/11/083004+32$33.00

© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft


2
Contents

1. 2. 3. 4. 5. 6. 7.

Introduction Main ideas and outline State vectors and wave functions of biphoton qutrits Density matrices Degree of entanglement Schmidt modes of qutrits and subsystem entropy Ququarts 7.1. Definitions and wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Degree of entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Conclusions Acknowledgments Appendix A. Scheme for direct experimental measurement of the qutrit's degree of entanglement Appendix B. Measuring the ququart's parameters References
1. Introduction

2 3 6 8 10 13 16 16 19 23 23 24 29 31

The problem of providing meaningful entanglement quantifiers for systems of indistinguishable particles has a long history. Significant work was done and numerous papers were published on the subject [1­10]. Nevertheless, as is clear from analysis of these and other works, there is still no single unified and widely accepted approach to the problem. On the other hand, there is a highly developed field of quantum information, dealing with abstract distinguishable qubits. Entanglement of qubits is well understood (at least for the case when there are only two of them), and existing approaches to entanglement measures in two-qubit systems are generally accepted by the community. There is a number of experimental implementations of bipartite qubit systems as well, using various physical resources to encode quantum information. Quantum optical experiments are of great importance among them. Usually they make use of biphotons generated in the process of spontaneous parametric down-conversion, and all theoretical tools developed for qubits are straightforwardly applied to entanglement in various photonic degrees of freedom. We may be surprised by this fact: indeed, photons are indistinguishable bosons, and it is far from evident how one can construct two distinguishable qubits with two photons. Even less evident is the fact that entanglement of biphotons may be quantified with measures, derived for distinguishable qubits. Still we have something that is hard to argue about: the results of numerous experiments clearly show that under some experimental conditions biphotons behave like a system of qubits. This question is omitted in the majority of experimental works, and is usually addressed only in a general context. We believe that analysis of the situation considering simple and well-known objects, such as biphoton states with photons having only one or two discrete degrees of freedom, will be of interest for the community. The key insight to understanding the mentioned ambiguities is provided in Zanardi's work [5]. The simple idea is that entanglement can only be defined for a particular

New Journal of Physics 13 (2011) 083004 (http://www.njp.org/)


3 decomposition of a complex system into subsystems, and the amount of entanglement quantified with some measure may depend on the chosen decomposition. The Hilbert space of a quantum system is a unitized object with no a priori division into subsystems. It is up to us to choose such a division and to induce a corresponding tensor product structure, giving rise to possible entanglement. Entangled systems differ from separable ones in their behavior under local operations, but the mere definition of which operations we should consider as local relies on a particular choice of subsystems. Of course, sometimes such a choice is obvious, that is, for example, the case of two distinguishable particles. But this is not the case when particles are indistinguishable. In such cases, as we show, separation for subsystems can be done as separation in particles' variables. Although less obvious and somewhat more abstract, such separation provides conditions for describing intrinsic quantum correlations in systems of indistinguishable particles, which coexists with other types of entanglement and is accessible for experimental measurements.
2. Main ideas and outline

Specifically, we consider in this paper two kinds of states: biphoton qutrits and biphoton ququarts. In the first case, two one-photon modes are characterized only by polarization = H, V (horizontal or vertical). Biphoton purely polarization states (qutrits) can be formed in combinations of processes of degenerate spontaneous parametric down-conversion (SPDC), and their state vectors are given by arbitrary superpositions of three basis state state-vectors {|2H , 0 , |1H , 1V , |0, 2V }. Features of biphoton qutrits are discussed in sections 3­6 of this paper. In the second case, the underlying process is the non-degenerate SPDC, and photon modes are characterized by polarization and some other quantum number k , which can take only one of two values ka or kb . The quantum number k can mean either photon frequency, or angle, determining possible directions of photon propagation, or anything else. In this case, there are four one-photon modes (|1k a , and |1k b , ) and four two-photon modes (biphoton configurations or biphoton basis state-vectors) |1k a , 1 , 1k b , 2 , 1 , 2 = {H, V}. Superposition of four biphoton basis state-vectors forms a biphoton ququart, features of which are discussed in detail in section 7. It may be reasonable to emphasize that we do not consider here entanglement or correlations either between qutrits and between ququarts [11, 12] or in any multiphoton multimode states with amounts of photons >2 [13]. We assume here that the pump is not too strong and gives rise only to rare two-photon degenerate or non-degenerate SPDC pairs. We analyze only intrinsic entanglement of one-qutrit or one-ququart states. We assume that such investigation is important because qutrits and ququarts are key elements in many physical processes. Internal entanglement of these objects is one of their main fundamental characteristics, and knowing it is very important, especially because many widespread common opinions and evaluations in this field that need corrections are shown below. For any pure bipartite state its basis state-vector can be presented as a sum of basis statevectors, each of which is assumed to correspond to a given configuration of particles' occupation numbers in i th modes {n i }, | = {ni } C{ni } |{n i } . This gives rise to a density matrix =
{n i },{n j }

C

{n i }

C

{n j }

|{n i } {n j }|.

(2.1)

New Journal of Physics 13 (2011) 083004 (http://www.njp.org/)


4 By definition [5­8], the state determined by this state vector | and density matrix is entangled only if it does not have and cannot be reduced to the single-configuration form | = |{n i }0 , = |{n i }0 {n i }0 |, (2.2)

where the subscript `0' indicates some single given configuration. The entanglement defined by conditions (2.2) can be referred to as configurational entanglement. This definition is absolutely unambiguous in the case of bipartite states with distinguishable particles when all bipartite modes are non-degenerate, because two different distributions of two distinguishable particles in two different modes are physically different and distinguishable and represent two different configurations of the particles' distribution numbers. In the case of indistinguishable particles, such as photons, definition (2.2) is assumed to hold. But the question about the number of modes and configurations is not so clear. Let us discuss the situation that arises by using the simplest example of the state of two degenerate photons with different polarizations and the state vector aH aV |0 = |1H , 1V . The typical answer for the question about the number of biphoton modes and configurations in this state is `one mode' {H, V}, `one configuration' and, hence, no entanglement. But, first, in some works [2, 10] the state |1H , 1V is treated as entangled (similar to the two-electron state of the same kind [1]). And, second, it is rather difficult to draw conclusions about the numbers of modes by looking at only the state vector or density matrix in the symbolical operator form, = |1H , 1V 1H , 1V |. To reduce to the form of a normal matrix with clearly written matrix elements, let us first write down the orthogonality condition for biphoton purely polarization state-vectors given by 11 , 12 |1H , 1
V = 0|a1 a2 aH aV |0 =
1

,H 2 ,V

+

1

,V 2 ,H

.

(2.3)

Note that this expression follows directly from commutation rules for the photon creation and annihilation operators, and it would be wrong to leave only one term ( 1 ,H 2 ,V ) on its right-hand side (whereas the single-configuration description corresponds precisely to this approximation: only one term left in expressions like that of equation (2.3)). With the orthogonality relation (2.3) taken into account, we find that the matrix elements of the density matrix have the form 11 , 12 | |11 , 12 = (
1 ,H 2 ,V

+

1

,V 2 ,H

)(

1 ,H 2 ,V

+

1 ,V 2 ,H

).

(2.4)

Definitely, this matrix describes the entangled state. In terms of the number of modes and configurations in the state |1H , 1V , expressions (2.3) and (2.4) correspond to a two-mode and two-configuration state. To explain this result, we suggest an interpretation that is different from the traditional one. Instead of speaking about a single biphoton mode {H, V}, we say that the state we consider is characterized by two degenerate biphoton modes, {H, V} and {V, H}. `Degenerate' means in this case that the state vectors corresponding to these modes are identical, |1H , 1V |1V , 1H . Of course, the modes {H, V} and {V, H} are physically indistinguishable. But their existence is displayed clearly in the matrix elements (2.3) and (2.4). The suggested interpretation is valid not only for the state |1H , 1V but also for any other states of two photons in different single-photon modes, as well as for the general-form biphoton qutrits and ququarts, and for any biphoton states of higher dimensionality. In all cases, the key point is in the correct calculation of matrix elements as shown in equation (2.3). The same conclusions about entanglement of states like |1H , 1V can be obtained from consideration in terms of `coordinate'- dependent basis wave functions and their superpositions.
New Journal of Physics 13 (2011) 083004 (http://www.njp.org/)


5 For arbitrary bipartite states, the basis wave functions are defined as and their superposition is given by (x1 , x2 ) =
{n i } {n i }

(x1 , x2 ) = x1 , x2 |{n i } (2.5)

C

{n i }

x1 , x2 |{n i } =
{n i }

C

{n i }

{n i }

(x1 , x2 ).

The coordinate-dependent density matrix following from equation (2.1) has the form (x1 , x2 ; x1 , x2 ) =
{n i },{n i }

C C
{n i },{n i }

{n i }

C C

{n i }

x1 , x2 |{n i } {n i }|x1 , x
{n i }

2

=

{n i }

{n i }

(x1 , x2 )

{n i }

(x1 , x2 ).

(2.6)

For the simplest entangled biphoton state considered above, |1H , 1V , x1,2 = 1,2 , the sums (2.5) and (2.6) have only one term and the coordinate-dependent wave function is given by 1 (1 , 2 ) = (1 ,H 2 ,V + 1 ,V 2 ,H ). (2.7) 2 (1 , 2 ) coincides with the last expressions on the right-hand side of equation (2.3) divided by 2, and the coordinate density matrix (1 , 2 ; 1 , 2 ) = (1 , 2 )


(1 , 2 )

(2.8)

coincides with that of equation (2.4), divided by 2 for proper normalization. (Note that in the two cases of equations (2.3), (2.4) and (2.7) the symbols 1,2 play different roles: mode characterizers and `coordinates', or photon variables.) These coincidences illustrate equivalence of the considerations in terms of state vectors and coordinate wave functions, if only in the case state-vectors matrix elements are calculated correctly, as in equation (2.3). In terms of the coordinate-dependent wave functions, a bipartite state is entangled if its wave function cannot be factorized, i.e. cannot be presented in the form of a product of two single-particle functions, (x1 , x2 ) = (x1 ) â (x2 ). (2.9) A natural extension of this definition is given by the Schmidt decomposition, or Schmidt theorem [14, 15], according to which any entangled (unfactorable) bipartite wave function can be presented as a sum of factorized terms. These characteristics of entangled biphoton qutrits and ququarts are widely used in this work. In particular, in this way we find explicit conditions when the biphoton qutrits are entangled and, oppositely, factorized, in dependence on values of their parameters Ci . For comparison, it can be mentioned that an extreme but widely spread opinion is that all biphoton qutrits are factorable and disentangled, which is not confirmed by our consideration. We also find the relationship between entanglement and polarization of biphoton qutrits, and a series of other results is derived. Note that the orthogonality relation for state vectors (2.3), matrix elements of the density matrix (2.4), as well as the coordinate-dependent wave function (2.7) and the coordinate density matrix (2.8) are symmetric with respect to particle transpositions 1 2. In this sense, it is possible to speak about symmetry entanglement as entanglement existing only in systems of indistinguishable particles and arising exclusively owing to the symmetry of wave functions or matrix elements dictated by the Bose­Einstein or Fermi­Dirac statistics. The state |1H , 1V has
New Journal of Physics 13 (2011) 083004 (http://www.njp.org/)


6 only symmetry entanglement and nothing else. In more complicated states, such as the generalform biphoton qutrits and ququarts, the symmetry entanglement coexists with and is inseparable from the above-mentioned configurational entanglement, related to existence in such states of several, not necessarily degenerate, configurations (several terms in the sum of equation (2.1)). The Schmidt parameter K and the Schmidt decomposition take into account both symmetry and configurational entanglement. On the other hand, as shown above, separation for symmetry and configurational entanglement is rather conventional. In a slightly modified interpretation, the symmetry entanglement can be included in the generalized configurational entanglement and then, mathematically, the main requirement consists simply in a correct calculation of matrix elements of the type (2.3).
3. State vectors and wave functions of biphoton qutrits

In the form of state vectors, purely polarization biphoton states (qutrits) are given by a superposition | = C1 |2
H

+ C2 |1H , 1V + C3 |2V , 1 = aV2 |0 . 2

(3.1)

where the basis state-vectors are given by 1 |2H = aH2 |0 , 2
|1H , 1V = aH aV |0 ,

|2

V

(3.2)

|0 is the vacuum state-vector, and aH and aV are the creation operators of photons in modes with horizontal and vertical polarizations (with given equal frequencies and given identical propagation directions). C1,2,3 are arbitrary complex constants C1,2,3 = |C1,2,3 |ei 1,2,3 , obeying the normalization condition

|C1 |2 + |C2 |2 + |C3 |2 = 1.

(3.3)

Actually, as the total phase of the state vector (3.1) or wave function (see below) does not affect any measurable characteristics of qutrits, one of the phases 1,2,3 , or a linear combination of phases, can be taken equal to zero, and hence the general form of the qutrit's state vector (3.1) is characterized by four independent constants (e.g. |C1 |, |C3 |, 1 and 3 with 2 = 0). As qutrit (3.1) is a two-photon state, its polarization wave function depends on two variables. A general rule for obtaining multipartite wave functions from state vectors is known pretty well in quantum-field theory, and for bosons the corresponding formula has the form [16] (in slightly modified notation) (x1 , x2 , . . . , xn ) = x1 , x2 , . . . , xn |n 1 , n 2 , . . . , n k 1 = P (g1 (x1 )g1 (x2 ). . .g1 (xn 1 )g2 (x n 1 !n 2 !. . .n k !n ! P âg2 (x
n 1 +n
2

n 1 +1

)g2 (x

n 1 +2

). . . (3.4)

). . .gk (x

n 1 +n 2 +···+n

k -1

+1

). . .gk (xn )),

where x1 , x2 , . . . , xn are dynamical variables of identical boson particles, g j (xi ) are singleparticle wave functions (of j th modes and i th variables), P indicates all possible transpositions of variables xi in wave functions g j (xi ), n 1 , n 2 , . . . , n k are numbers of particles in modes, n is the total number of particles in all modes and k is the total number of modes; for empty modes, the corresponding single-particle wave functions have to be dropped.
New Journal of Physics 13 (2011) 083004 (http://www.njp.org/)


7 In the case of qutrits, we have two modes ( j = H or V) and two particles, n = k = 2. The polarization variables of two photons can be denoted as 1 and 2 . In terms of wave functions, the single-photon wave functions g j (xi ) are given by the Kronecker symbols. Thus, the qutrit basis wave functions corresponding to the basis state-vectors in equation (3.1) can be written as
HH

(1 , 2 ) = 1 , 2 |2

H

=

1 ,H 2 ,H

,
1 ,H 2 ,V

(3.5) +
2 ,H 1 ,V

HV

1 (1 , 2 ) = 1 , 2 |1H , 1V = [ 2 (1 , 2 ) = 1 , 2 |2
V

],

(3.6) (3.7)

VV

=

1 ,V 2 ,V

.

The same basis wave functions can be written equivalently in the form of two-row columns, which is more convenient for calculation of matrices 1 0 1 1 , (3.8) HH = 01 0 2 0 0 0 0 0 0 , (3.9) VV = 11 1 2 0 1 1 = 2 1 0 0 1 1 2 0 0 0 0 1 + 12 11 0 0 0 1 1 0 = , 1 2 1 0 0

HV


1

2

+

(3.10)

where the upper and lower rows in two-row columns correspond to the horizontal and vertical polarizations and indices 1 and 2 numerate indistinguishable photons. In a general form, the qutrit's wave function corresponding to the state vector (3.1) is given by =C
1 HH

+C

2

HV

+C

3

VV

,

(3.11)

where HH , HV and VV can be taken in the form of either (3.5)­(3.7) or (3.8)­(3.10). Alternatively, the same general qutrit's wave function (3.11) can be presented in the form of an expansion in a series of Bell states =C where C± = C1 ± C3 , 2 (3.13)
+ +

+C

2

+

+C

-

-

,

(3.12)

New Journal of Physics 13 (2011) 083004 (http://www.njp.org/)


8
+



2

(3.10) and
±

1 = 2 1 2

1 = 2 0 1 0 0 1 ± = 0 0 2 1 0
HH

±

VV

1 0


1

1 0

±
2

0 1


1

0 1

2

1 0 . 0 ±1



(3.14)

are the wave functions describing three Bell states. The fourth Bell state, 1 1 1 0 0 - = (3.15) - , 01 12 11 02 2 does not and cannot arise in expansion (3.12) because - (3.15) is antisymmetric with respect to the variable transposition 1 2, whereas all biphoton wave functions have to be symmetric. Nevertheless, in principle, the antisymmetric Bell state (3.15) can be included into expansion (3.12) but is obligatory with zero coefficient: = C+ + + C2 + + C- - + 0 â - . (3.16) This obligatory zero coefficient in front of - or the missing fourth antisymmetric Bell state in expansion (3.12) is related to restrictions imposed by the symmetry requirements for twoboson states. Thus, even the existence of biphoton qutrits as superpositions of only three basis wave functions occurs exclusively owing to symmetry restrictions eliminating the fourth (antisymmetric) basis Bell state.
4. Density matrices

±

and

+

The first step for finding function to the density matrix of qutrit (3.11) can 10 1 = |C1 |2 001 0 + 0 0
3 2

the degree of entanglement is related to a transition from the wave matrix of the same pure biphoton state = . The full density be presented in the following two forms: 00 00 |C2 |2 10 00 0 + + |C3 |2 011 012 001 012 02 2 1 0
1

0 1 0 0

1

0 0

+
2

01 00


1

00 10 00 10
1

+
2

00 10 00 10


1

01 00

2

+C1 C

1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0

01 00 01 00 00 10 00 01 00 01

+ C1 C 2

3


1

2

C1 C + 2
1


1

+
2

01 00 00 10 00 01 00 01

10 00 10 00 00 10 01 00

2

CC + 2 C3 C + 2
3

2


1

+
2


1

2

2


1

+
2


1

2

CC + 2

2


1

+
2


1

(4.1)
2

New Journal of Physics 13 (2011) 083004 (http://www.njp.org/)


9 and = |C1 |2 1 C1 C 2 1 C1 C 2
C1 C 3 2

2

1 C1 2 1 |C2 2 1 |C2 2 1 C3 2

C |2 |
2

2

C

2

1 C1 2 1 |C2 2 1 |C2 2 1 C3 2

C |2 |
2

2

C1 C

3



C

2

1 C3 C2 2 . 1 C3 C2 2 2 |C3 |

(4.2)

The next step is the reduction of the density matrix with respect to one of the photon variables, e.g. of photon 2. Mathematically, this means taking traces of all matrices with subscript 2 in equation (4.1), which gives |C2 |2 C1 C2 + C2 C3 2 |C1 | + 2 2 r = Tr2 = (4.3) . 2 C1 C2 + C2 C3 |C2 | 2 |C3 | + 2 2 It may be interesting to analyze a relation between the 4 â 4 density matrix (4.2) and the 3 â 3 coherence matrix introduced by Klyshko in 1997 [17]. The density matrix (4.2) is written in a natural two-photon basis, 1 1 0 0 0 1 0 0 , , , . (4.4) 0 0 1 0 0 0 0 1 The question is how it can be transformed to the basis of states HH , antisymmetric state - (3.15). Evidently, the transformation 0 1 1 0 0 0 0 0 0 1 0 0 1 1 1 1 , , , , , , 0 0 0 1 0 2 1 2 -1 0 1 0 0 0 1 0 is provided by the matrix 1 0 U = 0 0
VV

,

HV

plus the empty

0 0 0 1

(4.5)

0 0 0 1 1 0 2 2 1 1 - 0 2 2 0 0 1

.

(4.6)

New Journal of Physics 13 (2011) 083004 (http://www.njp.org/)


10 Now, transformed to the basis {
HH

,

VV

,

HV 2

,

-

}, the density matrix (4.2) takes the form
3

transf

|C1 |2 C1 C
3

0 C1 C

(4.7)

C1 C = UU = 0 C1 C

3

|C2 |2 0 C3 C2 . 0 0 0 2 C3 C2 0 |C3 |

A part of this matrix with non-zero lines and columns coincides with the 3 â 3 coherence matrix [17, 18] |C1 |2 C1 C2 C1 C3 coh = C1 C3 |C2 |2 C3 C2 . (4.8)
C1 C 3

C3 C

2

|C3 |

2

Although the coherence matrix coh (4.8) is widely used and analyzed in the literature, both coh and transf are hardly appropriate for reduction over one of the photon variables (e.g. 2) and for finding correctly the reduced density matrix r (4.3) because variables 1 and 2 are mixed not only in the matrix transf itself but also in the transformed basis of equation (4.5).
5. Degree of entanglement

As is known [14, 15], the trace of the squared reduced density matrix r2 determines purity of the reduced state coinciding with the inverse value of the Schmidt entanglement parameter K -1 . The result of its calculation for the reduced density matrix of equation (4.3) is given by K
-1

|C2 |2 = Tr( ) = |C1 | + 2
2 r 2

2

|C2 |2 + |C3 | + 2
2

2 + |C1 C2 + C2 C3 |2 .

(5.1)

With normalization condition (3.3) taken into account, equation (5.1) can be reduced to a much simpler form: 2 K= . (5.2) 2 2 - |2C1 C3 - C2 |2 It is also known [19] that in the case of bipartite states with the dimensionality of the oneparticle Hilbert space d = 2, there is a simple algebraic relation between Wootters' concurrence C [20] and the Schmidt entanglement parameter K , owing to which C= 2 1- 1 K
2 = |2C1 C3 - C2 |.

(5.3)

Finally, in terms of the constants C± (3.13), equations (5.2) and (5.3) take the form 2 2 2 2 , C = |C+ - C- - C2 |. K= 2 - C 2 - C 2 |2 2 - |C+ - 2

(5.4)

Note that expressions for the concurrence C (equation (5.3) and the last formula of equation (5.4)) can also be derived directly from the original Wootters' definition [20].

New Journal of Physics 13 (2011) 083004 (http://www.njp.org/)


11

Figure 1. The Schmidt entanglement parameter K

real

(5.3) and the von Neumann subsystem entropy Sr real (3.12) with real constants C1,2,3 , C± versus C+ (3.13).

(5.2), concurrence Creal (6.7) of the qutrit (3.11),

Indeed, for a pure bipartite state with dimensionality of the one-particle Hilbert space d = 2, concurrence is defined as C = | | ~ |, (5.5) where ~ is the function or state vector arising from after the `spin-flip' operation, ~ = y y , (5.6) 1 2 and y is the Pauli matrix, y = 0 -i . For qutrits, is given by equation (3.11) or (3.12). i0 The rules of the `spin-flip' transformation for one-photon wave functions are 1 -i 0 and 0 1 0 i 1 . From here we easily find the spin-flip transforms of the qutrit basis wave functions 1 0 (3.8)­(3.10) ~
HH

=-

VV

,

~

VV

=-

HH

,

~

HV

=

HV

,

~+ =-

+

,

~- =

-

(5.7)

and of the general-form qutrit wave function (3.11) ~ = -C
1 VV

+C

2

HV

-C

3

HH

= -C

+

+

+C

-

-

+C

2

HV

.

(5.8)

Substitution of these expressions into equation (5.5) gives
C = | - 2C1 C3 + C2 | = |C
2

2 -

-C

+

2

+ C2 |,

2

(5.9)

in complete agreement with equations (5.3) and (5.4). In a special case of real constants C1,2,3 and C± , owing to normalization (3.3), the Schmidt entanglement parameter and concurrence (5.4) appear to be determined by the only real parameter C+ , 2 2 K real = , Creal = |2C+ - 1|. (5.10) 2 4 1 + 4C+ - 4C+ The functions K real (C+ ) and Creal (C+ ) are shown in figure 1 together with the subsystem entropy found in the following section.
New Journal of Physics 13 (2011) 083004 (http://www.njp.org/)


12 This picture demonstrates 2 C+ = ±1/ 2 and, hence, C- + functions with real coefficients 1 NE real = ( 2 = cos2 clearly that qutrits are non-entangled ( K real = 1 and Creal = 0) if 2 C2 = 1 . Consequently, the family of non-entangled qutrit wave 2 is given by
+

+ sin
HH

HV

+ cos

-

)

sin 2 + (5.11) HV + sin VV 2 2 with arbitrary . If the constants C1,2,3 , C± are complex, a general condition of no-entanglement is the same: C = 0. As seen from the general expression for C (5.3), the concurrence depends on phases 1,2,3 of the constants C1,2,3 only via the combination 1 + 3 - 22 . Hence, if 2 = 1 (1 + 3 ), in the case of complex constants C1,2,3 , equation (5.3) is reduced to C = 2 |2|C1 ||C3 | - |C2 |2 |, i.e. this case appears to be equivalent to the case of real constants C1,2,3 . From here we find that the general three-parametric family of wave functions of non-entangled qutrits is given by
NE general

2

sin ( , 1 , 3 ) = e( 2

i/2)(1 +3 )

HV

+ cos2

i e 2

1

HH

+ sin2

i e 2

3

VV

(5.12)

with arbitrary , 1 and 3 . With these wave functions found explicitly, we can reconstruct the corresponding family of state vectors of non-entangled qutrits, |
NE general

1 = sin e( 2

i/2)(1 +3 ) aH aV

+ cos2

i3 i1 2 e aH + sin2 ea 2 2

2 V

|0 .

(5.13)

Qutrits are maximally entangled when C = 1 and K = 2, and for wave functions with real constants C1,2,3 , C± this occurs in two cases: C+ = ±1 and C+ = 0. In the first of these cases, = ± + . In the second case the maximally entangled wave function has the form of an arbitrary superposition of HV and - , = sin HV + cos - . As before, this result can be 1 generalized for the case of wave functions with complex coefficients such that 2 = 2 (1 + 3 ). As a result, we obtain the following three-parametric family of wave functions of maximally entangled qutrits cos i1 (i/2)(1 +3 ) i3 (5.14) max ( , 1 , 3 ) = sin e HV + (e HH - e VV ) 2 and the corresponding family of state vectors |
max