Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qi.phys.msu.su/papers/2006-pra-73-063810.pdf Äàòà èçìåíåíèÿ: Wed Jun 21 09:57:06 2006 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:42:48 2012 Êîäèðîâêà:

PHYSICAL REVIEW A 73, 063810 2006

Polarization states of four-dimensional systems based on biphotons
Yu. I. Bogdanov
Institute of Physics and Technology, Russian Academy of Sciences, 117218 Moscow, Russia

E. V. Moreva*
Moscow Engineering Physics Institute (State University), 115409 Moscow, Russia

G. A. Maslennikov
Department of Physics, National University of Singapore, Singapore, 117542 Singapore

R. F. Galeev, S. S. Straupe, and S. P. Kulik
Faculty of Physics, Moscow State University, 119992 Moscow, Russia Received 17 February 2006; published 9 June 2006 We discuss the concept of polarization states of four-dimensional quantum systems based on the frequency nondegenerate biphoton field. Several quantum tomography protocols were developed and implemented for the measurement of an arbitrary state of ququart. A simple method that does not rely on the interferometric technique is used to generate and measure the sequence of states that can be used for quantum communication purposes. DOI: 10.1103/PhysRevA.73.063810 PACS number s : 42.50.Dv, 03.67. a

I. INTRODUCTION

Recently multi-dimensional quantum systems quantum dits, or qudits attracted much attention in context of quantum information and communication. It is partially caused by fundamental aspects of quantum theory since the usage of qudits allows one to violate Bell-type inequalities longer than with two-dimensional systems qubits 1 . Much interest in qudits also comes from their possible usage in the applied quantum key distribution QKD . Multilevel systems are proved to be more robust against noise in the transmission channel, although measurement and preparation procedures of such systems seem to be much more technically complicated than in the case of qubits. Different aspects of the security of qudit-based QKD protocols have been analyzed 2 including those where the reduced set of bases 3 is used. Lately a proof-of-principal realization of a QKD protocol with qudits 4 and with entangled qutrits D =3, where D is the dimensionality of the Hilbert space 5 have been demonstrated. For the last several years elegant experiments were performed in which different kinds of optical qudits were introduced. Most of them are based on the states of light emitted via spontaneous parametric down-conversion SPDC . There are qudits obtained using spatial degrees of freedom: With either single photons 4 or twin photons 6 ; orbital angular momentum of photons 7 ; time-bin technique 8 ; multi-arm interferometers 9 ; postselection of polarization qutrits from four-photon states 10 ; and polarization states of single-mode biphotons 11 . Although several ways of qudit state preparation have recently been introduced, it seems that until now, there has been only one experiment, demonstrating the complete control over a multilevel state during the preparation stage

within one setup. In this experiment an arbitrary polarization qutrit state pure or mixed was generated, exploiting the properties of the single-mode biphoton field 11,12 . However, this method does not us allow to create entangled qutrits. Another disadvantage of using polarization qutrits rises from the fundamental impossibility to switch between arbitrary qutrit states using SU 2 optical elements, like retardation plates, rotators, etc. Meanwhile it is these transformations that would be extremely useful for quantum communication purposes. In this paper we present the results of the experimental preparation and measurement carried out with polarization based ququarts or quantum systems with dimensionality D = 4. The paper is organized as follows. In Sec. II we discuss the main properties of ququarts that are generated using the polarization state of the two-photon biphoton field. Such operational notions as the coherency matrix and the polarization degree are introduced. The criterion of separability for qubits forming ququarts is discussed as well. Section III describes different experimental techniques of the biphotonbased ququarts preparation and measurement. In Sec. IV we consider a specific set of ququarts which can be used for the practical quantum key distribution.
II. POLARIZATION QUQUARTS AND THEIR PROPERTIES A. Polarization states of the biphoton field

The biphoton field generated via the spontaneous parametric down-conversion process is represented by a coherent mixture of two-photon Fock states and the vacuum state 13 : = vac + 1 2 F
kiks ki,ks

1ki,1

ks

,

1

*Electronic address: ekaterina.moreva@gmail.com
1050-2947/2006/73 6 /063810 13

where 1ki ,1ks describes the state with one idler photon in the mode ki and one signal photon in the mode ks. Since
063810-1 ©2006 The American Physical Society


BOGDANOV et al.

PHYSICAL REVIEW A 73, 063810 2006

FIG. 1. Photograph of lithium iodate crystal LiIO3 frequencyangular spectrum taken with the helium cadmium laser pump operated at 325 nm. The angle between the optical axis of the crystal and the laser pump is 59.22°. Dashed lines select the part of the spectrum frequency degenerate, collinear regime of SPDC that contributes to biphoton-based qutrits. Vertical lines at the bottom belong to the spectrum of neon gas which serves as a frequency reference.

FIG. 2. Nondegenerate regime of SPDC. The angle between the crystal optical axis and the laser pump is 58.97°. Dashed lines select the part of the spectrum contributed to ququarts.

squared modulus of the coefficient F

ki,ks

gives the probability

to register two photons in modes ki and ks, this quantity is called the biphoton amplitude 14 . The pure polarization state of the down-converted light field, which is often referred as the two-photon field or biphotons , can be written as follows: = c1 2H,0
V

+ c2 1H,1

V

+ c3 1V,1

H

+ c4 0H,2V .

2

In order to distinguish between the middle terms in 2 one must induce the distinguishability between downconverted photons either in frequency or momentum or detection time. In experiments described in this paper we used the collinear, frequency nondegenerate regime of SPDC so twin photons that form a biphoton had different frequencies and propagated along the same pump direction. The twodimensional SPDC spectrum that describes this situation is shown in Fig. 2. In order to obtain this spectrum it is sufficient to tilt slightly the crystal by a small angle from the orientation that is used for the frequency-degenerate regime. Two dashed rectangles indicate the angular-frequency ranges of the spectrum contributing to ququarts. The state 2 can be thus rewritten in the form: = c 1 H 1, H 2 + c 2 H 1, V 2 + c 3 V 1, H 2 + c 4 V 1, V 2 . 4 a vac , V1 2 b vac , where a , Here H1 2 are the creation operators of down-converted photons b with central wavelength 1 2 in the horizontal and vertical polarization mode. The alternative way of representing a ququart would be to introduce the distinguishability between spatial modes, while keeping the same central wavelengths of photons: ks ki, s i. The detailed description of polarization properties of such a state of two qubits has been done by James and co-authors 15 . The case of ququart with two frequency modes attracts much attention because for some types of the tasks it is convenient to operate with states in the single spatial mode.
12 12 12 12

Here ci = ci ei i, i4=1 ci 2 = 1 are complex probability amplitudes. The first n and second m symbols place in kets correspond to the number of photons with the definite either horizontal H or vertical V polarization, the total photon number m + n = 2. If down-converted photons have equal frequency and momentum, then a ququart state 2 converts to a qutrit state, i.e., middle terms in 2 become indistinguishable: ks ki, s i and s + i = p, where p is the laser pump frequency. In this case any arbitrary pure polarization state of the biphoton field that can be expressed in terms of three complex amplitudes c1 , c2, and c3 has the following form: c = c1 2, 0 + c2 1, 1 + c3 0, 2 , 3

3 2 The vector c with ci = ci exp i i , i=1 ci = 1. = c1 , c2 , c3 represents a three-state system or a qutrit. In Fig. 1 we show a photo of the two-dimensional SPDC spectrum taken in wavelength-angle coordinates. By the convention the left-and-upper side in respect with doubled pump wavelength belongs to the "signal" range whereas the right-anddown side belongs to the "idler" one. The dashed lines limit the central part of the spectrum, which corresponds to the frequency-degenerate and collinear regime when photons forming the biphoton have approximately the same wavelengths and propagate along the pump direction. During the experiment, in order to select this part of the spectrum that corresponds to qutrit states, a set of pinholes and/or narrowband filters is typically used.

B. Is ququart a separable or entangled state of two qubits?

In general, the state 4 cannot be written as a direct product of polarization states of two photons:
1 2

= p1 H1 + q1 V

1



p2 H2 + q2 V

2

.

5

That is why it is useful to write becomes a separable state. The subsystem second photon can state over all states of the first

down the condition when 4 reduced density matrix of the be found by tracing the joint photon:

063810-2


POLARIZATION STATES OF FOUR-DIMENSIONAL¼

PHYSICAL REVIEW A 73, 063810 2006

2

= Sp

1

=

c c

1

2

+c +c

3

2

c 1c * + c 3c 2 c
2 2

* 2c 1

* 4c 3

+c

4

* 4 2

.

6

It is straightforward to show that the eigenvalues of 6 become 1,2 =0,1 and also the entropy of each subsystem drops to zero if c 1c *c 4c * + c 3c *c 2c * - c 2c *c 3c * - c 4c *c 1c * = c 1c 4 - c 2c 1 4 3 2 1 4 3 2
3 2

=0 . 7

For the two-frequency and single-spatial mode field, the Stokes parameters will contain time and frequency dependent terms exp i j - k t that describe "beats" of frequency modes and do not describe the biphoton light polarization. However, these terms vanish if one considers the finite detection time that allows one to classically average these "beatings." So in the case of the two-frequency biphoton field, the Stokes operators are given by the sum of corresponding operators in two frequency modes. S 0 = a a 1 + a a 2 + b b 1 + b b 1 2 1 2
1 1 2 1 2

Thus, if c1c4 = c2c3 holds, the state 4 can be factorized by definite states of each qubit 1, 2 and vice versa. In other words, 7 plays the role of criterion for the biphotonbased polarization ququart to be a separable state of a couple of polarization qubits belonging to two different modes. Subindexes in 4 label either frequency or spatial modes. As an example, we show different ququart product states that can be generated from a single nonlinear crystal via SPDC: 1 ty pe I: H 1H 2 = 0 0 0 0 ty pe II: V 1H 2 = 0 1 0 , H 1V 2 = , V 1V 2 = 0 0 0 1 8 0 1 0 0 , ,

S

1 0 2

+S

2 0

=2;

S 1 = a a 1 + a a 2 - b b 1 - b b 2 1 2 1 S +S =2 c
2 ö 1 1 2

-c

4

2

;

S 2 = a b 1 + a b 2 + b a 1 + b a 2 1 2 1 S
1 2

+S

2 2

=2 Re,,c c2 + c3 + c4 cö + cö ...; 3 2
2 ö 1

S 3 = a b 1 + a b 2 - b a 1 - b a 1 2 1 2 S
1 3

+S

2 3

=2 Im,,c c2 + c3 + c4 cö + cö ... . 2 3

10

The same definition applies to the second wavelength 2 and we take into account that these operators do not commute for different frequency modes.
D. Fourth-order of the field: Coherency matrix

and maximally entangled Bell states it requires multiple crystals in one setup : 0 1 ¯ 2 ± 1 ¯ 2 0 1 ¯ 2 ,
±

Polarization properties of biphoton-based ququarts are determined completely by the coherency matrix. It is a matrix consisting of ten fourth-order moments of the electromagnetic field. An ordered set of such moments can be obtained using the direct product of 2 2-coherency matrices for both polarization qubits forming the biphoton: A EFG E* B I K F* I* C L G* K* L* D

±

=

= ±

0 0 1 ¯ 2

K . 9

4

K

2 1

K

22

=

,

11

where K2 1,2 are 2 2-coherency matrices for single photons with different frequencies marked by indices j =1,2: K a a j
2j j b jj

At the end of this section we would like to mention that in the general case, a pure polarization ququart state cannot be mapped as a pair of points on the PoincarÈ sphere as it occurs for biphoton-based qutrits 16,17 . It is so because the state 4 cannot be factorized in general. However, if criterion 7 is satisfied, the subset of separable qubit states admits its visual representation on the PoincarÈ sphere.
C. Second-order of the field: Stokes parameters

a b j b b j

j j

a

.

12

The diagonal elements of 11 are formed by moments, which characterize the intensity correlation in two polarization modes H and V: A C a a a 1a 2 = c 1 2, 12 b a b 1a 2 = c 3 2, 12 B D a b a 1b 2 = c 12
2 4 2

, . 13

b b b 1b 2 = c 12

2

Polarization properties of a pure state 4 can be described by Stokes parameters which are defined as mean values of Stokes operators averaged over the state 4 . Although the description of the light polarization can be introduced only for the quasimonochromatic plane waves, it is possible to use P-quasispin formalism 18 to describe the polarization of arbitrary quantum beams with n modes, frequency or spatial.
063810-3

Nondiagonal moments are complex: E G K a a a 1b 2 = c *c 2, 12 1 a a b 1b 2 = c *c 4, 12 1 a b b 1b 2 = c *c 4, 12 2 F I L a a b 1a 2 = c *c 3 , 12 1 a b b 1a 2 = c *c 3 , 12 2 b a b 1b 2 = c *c 4 . 12 3 14


BOGDANOV et al.

PHYSICAL REVIEW A 73, 063810 2006

The averaging in 13 and 14 is taken over the state 4 . The polarization reduced density matrix of ququart state coincides with coherency matrix K4.
E. Polarization degree and transformations of ququarts

the photons with different frequencies. For example, to perform the following transformation
4

= V 1, V

2

÷

2

= H 1, V

2

,

19

The polarization degree is given by
3 3

one needs to use the wave plate at 1 and case when ququart is transformation matrix 15

retardation plate that serves as a halfas a wave plate at 2. In the general not a product state of two qubits the ^ G:
out

S P=
k=1

k

2

S =
k=1

1 k

+S
2 0

22 k

^ =G

in

20

S

0

S

1 0

+S

.

This definition of polarization degree is just generalization of the commonly used classical one. It differs from the definition suggested in 19 , where it serves as a witness of the state purity. In the case of polarization qutrit states 11,12 , the polarization degree is P3 = c
2 1

can be found simply in Heizenberg representation. We use the equivalent representation of 4 using creation and annihilation operators:
in

= c1aa + c2ab + c3ba + c4bb vac . 12 12 12 12

21
j

SU 2 transformation between input a j, b j and output a j , b operators is given by: a j = t ja j + r jb j , b j =- r*a j + t*b j , j j where t j = cos r j = i sin
j j

-c

2 3

+2 c1*c2 + c2*c

2 3

16

with c1 = c1 , 2c2 = c2 = c3 , c3 = c4. The polarization degree of a qutrit state is invariant under unitary polarization transformations, which is reasonably straightforward 20 . Indeed it is impossible to prepare all demanded pure states that can be used, for example, in QKD protocols unless one uses interferometric schemes with several nonlinear crystals 11 , or introduces losses. In particular, there is no way to transform the basic qutrit state 4 = V , V with P 1 into orthogonal state 2 = H , V with P 0 using SU 2 transformations retardation plates, etc. 21 . However, in the case of polarization ququarts, this quantity is no longer invariant. Indeed, the polarization degree 15 can be rewritten as
2 j=1 3

22

+ i sin
j

j

cos 2 j , no j - ne j h/ j . 23

sin 2 j,

=

,,Tr K

2

2j

- 2 det K2 j... +2
2

Sk1 S
k=1

2 k

P4 =

.

Tr K
j=1

2j

Here t j and r j are the amplitude transmission and reflection coefficients of the wave plate, j is its optical thickness, h is the geometrical thickness, j is the orientation angle between the optical axis of the wave plate and vertical direction. Lower indices j =1,2 label the frequency modes of photons. Substituting 22 into 21 one can immediately find that the unitary transformation on state 4 is given by 4 4 matrix which is obtained by a direct product of two 2 2 matrices describing the transformation performed on each photon 20 : t 1t ^ G - - r = t
1 * 1 2

17 In 17 we used trivial links between coherency matrices and Stokes parameters 23 . Both the expression in the round brackets and the denominator are invariant under unitary transformations as well as the denominator. The second term under the square root in the numerator can be expanded in terms of fourth-order moments:
3

t 1r t - -

t 1r * 2 r *t 2 1 ** 1r 2 r t
1 * 1

2 t* 12 r *r 2 1 r *t * 12

r 1t -r -t t
2 * 2

2

r 1r r 1t t *r 1 t .

* 1r 2

2 * 2

t *t 1

2 ** 1r 2

2 ** 1t 2

-r



r t

-r

2 * 2

24

Sk1 S
k=1

2 k

= TrK4 -2 Re I + B + C .

18

Since unitary transformations keep the number of photons, then TrK4 is invariant. At the same time it is easy to prove that the combination of moments Re I + B + C is not invariant, so the whole expression 17 changes under SU 2 transformations. As a consequence, the polarization degree changes by applying local unitary transformations in each frequency mode. This can be done in an experiment by using dichroic polarization transformers, which act separately on

It is important to note that the optical thickness depends on the wavelength, so parameters t and r which determine the result of transformation differ for photons forming the biphoton-based ququart. Thus for the ququart transformation without taking a nonessential phase term into account considered above in 19 , matrix G becomes 0010 ^ g 0001 1000 0100 . 25

063810-4


POLARIZATION STATES OF FOUR-DIMENSIONAL¼

PHYSICAL REVIEW A 73, 063810 2006

FIG. 3. Preparation of states I. A nonlinear crystal generates one of the basic ququart state s via type-I SPDC. Then the state is transformed by the wave plate QP with the parameters depending on its orientation angle .

FIG. 4. Preparation of states II. Two nonlinear crystals generate a pair of the basic ququart states via type-I SPDC. The relative phase between states 14 is controlled by two quartz plates QP1, QP2. The basic state amplitudes are controlled with the help of the half-wave plate.

The same dichroic plate performs unitary transformations between polarization Bell states: g ± = ± , and g ± = ± , where ± and ± are represented by ququarts with c1 =± c4 =1/ 2 and c2 =± c3 =1/ 2 correspondingly. Similar transformations with frequency nondegenerate biphotons propagating in the single spatial mode have been realized in 22 .
III. EXPERIMENTAL IMPLEMENTATION

In order to introduce a phase shift 14 between horizontally and vertically polarized type-I biphotons, two 1 mm quartz plates QP1,2 that can be rotated along the optical axis were used. By tilting the plates the following family of ququarts were generated: c
II 1

=

0 0 c4 e
-i 14

,

27

In this section we describe several methods of ququart state preparation and measurement.
A. Preparation

In general to prepare an arbitrary ququart state 4 it is necessary to use four nonlinear crystals arranged in such a way that each crystal emits coherently one basic state in the same direction. But in special cases, a reduced set of crystals is sufficient to generate the specific ququart states that can be used in practice. In the experiments described below we used the following methods to generate a subset of ququart states.
1. Separable states

where real amplitudes c1 , c4 were controlled by the half wave plate inserted into the linearly polarized pump beam. -like Remarkable interference phenomena with II states were observed. For example, when the phase of 1-photon is altered, then the modulation with the same wavelength is observed in the coincidence rate, while varying the phase of second photon 2-wavelength modulation is observed 24 . Transformations of frequency nondegenerate Bell states experimentally have been studied in 22 . Another effect related to biphoton-based ququarts that is widely discussed in quantum optics is "hidden polarization."
B. Measurement

If a nonlinear crystal generates one of the basic states, 8 then applying the transformation 24 one can get a subset of ququarts which are the product states of polarization qubit pairs Fig. 3 . We used the basic state V1V2 , so the states generated by this method have the following structure: r 1r
I 2 * r 1t 2 t *r 2 1 t *t * 12

=

.

26

Since coefficients r j and t j depend on the orientation angle j, there is a simple way to transform the state by rotating the retardant plate. As we will show below, a use of a single crystal allows one to prepare the whole subset of ququarts which can be used for the quantum key distribution.
2. Entangled states

To prepare the ququarts with c1c4 c2c3 it is sufficient to use two crystals as it was done when frequency nondegenerated Bell states have been generated 24 Fig. 4 .

Basically it is necessary to perform D2 independent measurements for the complete reconstruction of the arbitrary qudit density matrix. So for D = 4 the total number of measurements is equal to sixteen. Since the only way to measure a biphoton is to register a coincidence of photocounts, we choose the Brown-Twiss scheme supplied with polarization elements to perform projective measurements. A coincidence click that occurs when signals from two detectors fall into the coincidence window is considered a registered event. Since the coincidence click appears at the output with some probability, a statistical treatment of outcomes becomes very important. Another point that we would like to note is that according to Bohr 's complementarity principle, there is no way to measure all moments 13 and 14 at the same time, dealing with a single ququart only. So to perform a complete set of measurements one needs to generate a lot of representatives of a quantum ensemble. In order to measure the ququarts we developed two protocols.

063810-5


BOGDANOV et al.

PHYSICAL REVIEW A 73, 063810 2006

a b

j j

=

00 01 1 2 0

D

j/2

j

= /2, D

j/4

j

= /4,

0 1 2

a b

j j

.

28

We label different output ports of the beam-splitter which correspond to different wavelengths 1,2 with indices j =1,2. Four 2 2 matrices in the right-hand side of 28 describe the action of the nonpolarizing beam-splitter, /4-, / 2-plates and vertical polarization prism on the state vector of each photon;
FIG. 5. Measurement block for Protocol 1. Brown-Twiss scheme for measuring the intensity correlation between two polarization modes. After the spatial separation at the nonpolarizing 50/50 beam-splitter NPBS , signal 2 and idler 1 photons propagate through the quarter- QWP1,2 and half- HWP1,2 -wave plates, polarizing prisms PP1,2 placed in both channels. The interference filter centered at 1 IF placed in the upper arm selects one frequency mode of a ququart. Finally, photons are registered by detectors D1,2 . The coincidence rate at the output of the coincidence circuit CC is proportional to the fourth moment in the field R.

D

j/2, j/4

=

t

j * j

r t

-r

j * j

,

where r j and t j are the coefficients introduced in Eq. 23 , so for a j / 4-plate j = /4 t and for a
j/4

=

1 2

1+ i cos 2
j

j

,

r

j/4

=

i 2

sin 2

j

29a

j

/ 2-plate t
j/2

= /2
j

= i cos 2

,

r

j/2

= i sin 2

j

.

29b

1. Protocol 1

The idea of the first method is to split the ququart state in into two frequency modes and to perform transformations independently on each photon from the pair Fig. 5 . Usage of a dichroic beam-splitter allows one to separate frequency modes in space and reduce this protocol to that one developed in 15 for two spatially separated polarization qubits. We called this method "frequency selective" because frequency mode separation is basically applied to the ququart before subjecting it to polarization projective measurements. The polarization transformations can be done by using polarization filters placed in front of detectors. Each filter consists of a sequence of quarter- and half-wave plates and a polarization prism which transmits vertical polarization. In our experiments we kept the protocol developed in 15 unchanged, although there are many other configurations of polarization filters leading to the matrix K4 reconstruction. Instead of a dichroic beam-splitter we used a 50/50 % polarization insensitive beam-splitter and a narrow-band filter centered at 1 that was put in front of the first detector. In such a way we postselected the desirable events in registered coincidences. Approximately in a quarter of trials, the photons with central wavelength 1 forming a biphoton are going to the first detector, while the conjugated ones with central wavelength 2 are going to the second detector. In the remaining cases, the events are not registered because either they do not contribute to coincidences or corresponding photons do not pass through the interference filter. In Heisenberg representation the polarization transformation for each output port of the dichroic beam-splitter is given by:

Thus, there are four real parameters two for each channel that determine polarization transformations. Namely, these parameters are orientation angles for two pairs of wave plates: 1, 1, 2, 2. Also we would like to note that there is another parameter that affects strongly the result of polarization transformations. It is a wavelength s of down-converted photon s . For example, varying the frequency spectrum of SPDC by tilting a crystal which generates biphotons, one can significantly change the transformed state 20 using fixed parameters of the wave plates such as geometrical thickness h and its orientation . This property makes biphoton-based ququarts more controllable than biphoton-based qutrits and allows one to choose convenient regimes for dealing with them. As it was mentioned above, the registered signal from Brown-Twiss scheme is the coincidence rate of the pulses coming from two detectors D1 and D2. The fourth-order moment of the field that corresponds to a registered coincidence event has the following structure: R
1,2

b 1 b 2 b 1 b 2 = R

1

,

1

, 2,

2

.

30

In the most general case this moment contains a linear combination of ten moments 13 and 14 that form matrix K4. So the main purpose of the state reconstruction procedure is to extract these moments from the registered coincidence rate for different orientations of the polarization filters in the Brown-Twiss scheme. Let us consider some special examples which give the corresponding lines in the complete protocol introduced below Table I . The measurement of the first four moments 13 is a trivial procedure. For instance, the third line in Table I corresponds to filtering out the basic state V1V2 . The other three upper lines correspond to the

063810-6


POLARIZATION STATES OF FOUR-DIMENSIONAL¼

PHYSICAL REVIEW A 73, 063810 2006

TABLE I. Protocol 1. Each line contains the orientation of the half- s,i and quarter- s,i wave plates in the measurement block. Last two columns show the corresponding moment R and the process amplitude M =1, . . . ,16 . Sign "-" in the orientation of the plates for the reflection channel is introduced to compensate -phase shift occurring after the reflection from the beam-splitter. Set-up parameters
s s i i

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

0 0 0 0 0 0 45° 45° 45° 45° 0 0 0 0 0 0

45° 45° 0 0 22.5° 22.5° 22.5° 22.5° 22.5° 22.5° 22.5° 45° 0 0 45° 22.5°

0 0 0 0 0 0 0 0 0 -45° -45° - - - - - 45° 45° 90° 90° 90°

-45° 0 0 -45° -45° 0 0 -45° -22.5° -22.5° -22.5° - - - - - 22.5° 22.5° 22.5° 22.5° 22.5°

1 / 16 1 / 16 1 / 16

1 / 16

Moments to be measured R =M 2 A /4 B /4 D /4 C /4 1 / 8 A + C +2 Im F 1 / 8 B + D +2 Im K 1 / 8 B + D -2 Re K 1 / 8 A + C -2 Re F A + B + C + D -1/8 Im E +Re -Im G +Im I +Re K +Im L A + B + C + D -1/8 Re F -Re +Re G +Re I +Re K -Re L A + B + C + D +1/8 Im F +Re +Im G +Im I +Re L +Im K 1 / 8 A + B +2 Re E 1 / 8 C + D +2 Re L 1 / 8 C + D +2 Im L 1 / 8 A + B +2 Im E A + B + C + D +1/8 Im F +Im -Re G +Re I +Im L +Im K

F E E

Amplitude of the process M c1 /2 c2 /2 c4 /2 c3 /2 1/2 2 c1 - ic3 1/2 2 c2 - ic4 1/2 2 c2 - c4 1/2 2 c1 - c3 1/4 c1 - c3 + i c2 - c4 1/4 c1 + c2 - c3 - c
4

1/4 c1 + c2 - i c3 + c 1/2 1/2 1/2 1/2 1/4 c1 - 2 2 2 2 c4 c1 + c2 c3 + c4 c3 - ic4 c1 - ic2 - i c2 + c

4

E

3

measurements of other basic states H1H2 , H1V2 , and V1H2 . Remaining lines of protocol show the orientations of the wave plates that were used to measure the complex moments 14 . For example, in order to measure the real part of moment E, let us set the wave plates in the Brown-Twiss scheme in the following way. The first channel: /4: =0°, D = 1 1+ i 0 ; 2 0 1- i 0i i0 31a

R=

b b b 1b 12

2

= 1/8 A + C +2 Re E .

1

/4

A complete set of projective measurements that is called a tomography protocol is presented in Table I. Each row corresponds to the setting of the wave plates that were used to measure the moment highlighted in the sixth column. The last column corresponds to the amplitude of the process see below . Amplitude M of the quantum process describing the passage of down-converted photons through the measurement setup for each configuration of wave plates is determined by the following equation: M
1

/2: The second channel: /4:

1

=45 ° ,

D

/2

=

.

31b

, k, 2,

l

=

1 a 1a 2c 1 + a 1b 2c 2 + b 1a 2c 3 + b 1b 2c 4 , 2 33

where D
/4

2

=-45 ° ,

=

1

1

-i 1 -i

2 -i 1 i

;

32a

a1 =- r a2 =- r

/2

1

t t

* /4 * /4 * /4 * /4 2 1 2 1

-t -t +t +t

/2

1

r r t t

/4

1

, , , .
in

/2

2

/2

2

/4

2

/2:

2

= - 22.5 ° ,

D

/2

=

2 -i -i

.

32b

b1 =- r b2 =- r

/2

1

r r

/2

1

/4

1

Substituting these matrices into Eq. 28 and taking into account the commutation rules for the creation and annihilation operators, it is easy to get the final moment measured in the experiment:

/2

2

/2

2

/4

2

34

was perThe complete reconstruction of input state formed according to the algorithm considered in our previous work 12 .

063810-7


BOGDANOV et al.

PHYSICAL REVIEW A 73, 063810 2006

front of each single photon detector in the Brown-Twiss setup. So the number of coincidences is R
kl

V 1V

2

out kl

2

.

37

FIG. 6. Measurement block for Protocol 2. After the input state is subjected to the transformations by wave plates QP1, QP2 , the projective measurements selecting vertical polarizations of ququart components are performed.

In order to reconstruct the state it is sufficient to perform at least four measurements corresponding to different orientations of the second plate l and repeat this procedure at least four times varying the orientation of the first plate k. In order to increase the accuracy, we actually performed an excess number of measurements--36 orientations of the second wave plate for each four orientations of the first one. Thus, in this protocol was the total number of measurements equal to 4 36 = 148. Therefore, Protocol 2 includes 148 lines instead of 16 lines used in Protocol 1.
C. Experimental setup

Concluding this section we would like to stress that if a usual frequency insensitive broadband beam-splitter is used and no interference filters are inserted either in front of or behind this beam-splitter, then a "nonselective" method of ququart measurement is possible. In this case one does not need to use the interference filter to select the wavelength in each channel of Brown-Twiss scheme. In other words, each detector is allowed to register photons with different frequencies and the fourth moment of the field that is measured in the experiment is given by R
1,2

b1 + b

2 Detector2

b1 + b

2 Detector1

35

instead of 30 . In a nonselective method the expression connecting observable value R1,2 with components of coherence matrix 13 and 14 is too cumbersome and it is not reasonable to use it for the state reconstruction. However, in special cases we applied a nonselective method to the ququart reconstruction procedure and developed the special protocol, which is introduced in the next section.
2. Protocol 2

In the second protocol the whole ququart state 2 is subjected to the linear transformations that are done by a set of the retardation plates 25 . Then projective measurements are applied to state in after it being transformed by the wave plates. Using the retardation wave plates with a fixed optical thickness, one can reconstruct the state varying the orientation of the plates Fig. 6 . Mathematically one needs to derive a set of equations linking real and imaginary parts of moments 13 and 14 with a coincidence rate. It turns out that the complete set of moments 13 and 14 can be extracted from the projective measurements if two different wave plates are used.
out kl

^ =G

1

,

k

^ G

2

,

l

in

,

36

where k and l are the orientation angles of the first and second plates. The parameters of the plates, i.e., optical thicknesses for different wavelengths 1,2 and their orientaj tions, are supposed to be known. The projective measurements were carried out by means of a pair of polarization prisms, transmitting the vertical polarization and settled in

For the generation of biphoton-based ququarts we used lithium-iodate 15 mm crystal pumped with 5 mW cwhelium-cadmium laser operating at 325 nm. The orientation of the crystal was chosen at 58.98° with respect to the pump wave vector. The wavelengths of the collinear downconverted photons were 1 = 702 nm and 2 = 605 nm. For some particular cases we selected biphoton-based ququarts at wavelengths 1 = 667 nm and 2 = 635 nm. Therefore, either V702V605 or V667V635 states were used as initial states. Then, these states were subjected to transformations done by dichroic wave plates to prepare a subset of ququarts with c1c4 = c2c3 26 . That subset of states was used to be reconstructed. In particular, we used the 0.988 mm or 0.315 mm length quartz plate and changed its orientation , which served as a parameter. Since the thickness of the plate, quartz dispersion and orientation were supposed to be known, we were able to calculate the result of the state transformation with high accuracy. When Protocol 2 was applied we used the quartz plates with thicknesses 0.821 mm QP1 and 0.715 mm QP2 . Both protocols were applied to reconstruct . In the case when sub-set c1c4 the initial state I c2c3 was generated we used two consecutive 1.8 mm thick type-I BBO beta-barium borate crystals with optical axes oriented perpendicularly with respect to each other and at 36.33° with respect to the pump wave vector. An interference filter centered either at 702 nm or at 635 nm and with a full width at half maximum FWHM bandwidth of 3 nm was placed in the transmitted arm to make the spectral selection of one photon of the pair, while the photon with conjugated frequency was sent to the reflected arm. As it was already mentioned above, this scheme is equivalent to that one used in 15 . The only difference between our scheme and the one developed in 15 is the following. The scheme described above operates with frequency nondegenerate collinear regime of SPDC instead of noncollinear, frequency degenerate regime used in 15 . Without the loss of generality, this scheme also allows one to reconstruct any arbitrary polarization ququart state by registering the coincidence rate for different projections that are done by the polarization filters located in each arm. Each filter consists of a zero-order quarter- and half-wave plate and a fixed analyzer. Two SiAPD's linked to a coincidence scheme with 1.5 ns time window were used as single photon detectors.

063810-8


POLARIZATION STATES OF FOUR-DIMENSIONAL¼

PHYSICAL REVIEW A 73, 063810 2006

FIG. 7. Dependence of the coincidence rate on tilting angle . D. Results and discussion TABLE II. Results of the state reconstruction measured with Protocol 1. The left column indicates the orientation of the quartz plate QP, determining the state to be measured. The table contains theoretical and experimental state vectors, as well as their fidelities, placed in the right column.

We applied both protocols to measure the set of states I that were obtained by transforming the initial state V667 nmV635 nm with quartz plate QP, as it is shown in Fig. 3. We tested the states which were generated for two orientation angles of plate QP = 0°, 90°. According to Protocol 2, four sets of measurements were performed for each input state, so totally we performed 148 measurements of the coincidence rate as a function of orientation angles l of QP1 and k of QP2. The dependence of the coincidence rate on the orientation l for k = 90° Fig. 7 a , 105° Fig. 7 b , 120° Fig. 7 c , and 135° Fig. 7 d is shown in Fig. 7. These curves correspond to the measurement of an input state created from V667 nmV635 nm by QP oriented at = 20° to the vertical direction. The solid curves give the theoretical behavior of the normalized fourth moment, the dots with bars show the experimental data. Tables II­IV show the results of the statistical reconstruction of ququarts I performed with the root estimation method. The tables show theoretical and experimental four-dimensional state-vectors ctheory, cexp as well as fidelity F defined by F= c
theory

Protocol 1 deg 0 Theory 0 0 0 1 -0.0015 - 0.0229i 0.0038 - 0.0050i 0.2013 + 0.1735i 0.9638 -0.0021 - 0.0386i 0.0154 - 0.0162i 0.3430 + 0.3625i 0.8654 -0.0015 - 0.0445i 0.0337 - 0.0225i 0.3530 + 0.5716i 0.7383 -0.0005 - 0.0440i 0.0511 - 0.0116i 0.1601 + 0.7466i 0.6421 Experiment -0.0295 - 0.0306i 0.0543 - 0.0202i -0.0154 - 0.0093i 0.9972 0.0584 - 0.1926i 0.0073 + 0.0170i 0.1633 + 0.1233i 0.9577 0.0015 - 0.0326i 0.0019 - 0.0660i 0.3967 + 0.2085i 0.8909 -0.0310 - 0.0542i 0.0166 - 0.0416i 0.4526 + 0.5386i 0.7065 0.0112 - 0.0823i 0.1174 - 0.0328i 0.1087 + 0.8363i 0.5167 F 0.995

10

0.963

20

0.976

c

ex p

2

.

38

30

0.991

In Table II we present the data measured with Protocol 1 ququarts with 1 = 702 nm and 2 = 605 nm, h QP = 0.315 mm . Table III corresponds to measurements performed with Protocol 2 ququarts with 1 = 667 nm and 2 = 635nm, h QP = 0.988 mm . Finally, we demonstrate the validity of preparation, transformation, and measurement procedures performed over biphoton-based ququarts by the reconstruction of Bell states ± 1 = 702 nm and

40

0.970

063810-9


BOGDANOV et al. TABLE III. Results of the state reconstruction measured with Protocol 2. The left column indicates the orientation of the quartz plate QP, determining the state to be measured. The table contains theoretical, experimental state vectors and their fidelities, placed in the right column. Protocol 2 deg 0 Theory 0 0 0 1 0.8097 -0.4568 - 0.3527i -0.0103 - 0.0859i -0.0316 + 0.0529i Experiment -0.0555 - 0.0204i -0.0059 + 0.005i -0.0425 + 0.0052i 0.9973 0.8067 -0.4847 - 0.3304i 0.023 - 0.0554i -0.0174 + 0.0413i F 0.996

PHYSICAL REVIEW A 73, 063810 2006 IV. POLARIZATION QUQUARTS IN QKD PROTOCOL

The complete QKD protocol with four-dimensional polarization states exploits five mutually unbiased bases with four states in each 27 . In terms of biphoton states, the first three bases consist of product polarization states of two photons and the last two bases consist of two-photon entangled states: I. II . III . IV . H 1H 2 ; D 1D 2 ; R 1R 2 ; H 1V 2 ; D 1D 2 ; R 1L 2 ; V 1H 2 ; D 1D 2 ; L 1R 2 ; V 1V 2 , D 1D 2 , L 1L 2 ,

R 1H 2 + L 1V 2 ; L 1H 2 + R 1V 2 ; H 1R 2 + V 1L 2 ; H 1L 2 + V 1R 2 ;

R 1H 2 - L 1V 2 ; L 1H 2 - R 1V 2 , H 1R 2 - V 1L 2 ; H 1L 2 - V 1R 2 . 39

20

0.998

V.

2 = 605 nm , measured with Protocol 1 and shown in Table IV. As it is seen from the tables, the accuracy of the state reconstruction is a little bit lower when Protocol 1 is applied. It results from the fact that Protocol 1 exploits a minimal number of measurements 16 needed for the four-state system reconstruction. At the same time, since an excess number of measurements 148 was involved when applying the second protocol, the highest accuracy was achieved as in the case of biphoton-based qutrits 12 . The respectively low fidelity for Bell states reconstruction Table IV is explained by experimental imperfections at the preparation stage. Preliminary estimation of the prepared state quality can be extracted from visibility that is measured during interference experiments with Bell states. For example, typical visibility observed with space-time interference was 0.9 26 . Another factor limiting the accuracy of the state reconstruction is the finite number of registered events. In the case of I -set of states we collected about several thousands of coincidences during the accumulation time whereas in the case of II -set only a few hundred coincidences were totally registered.

¯ Here H 1, V 0 , D 1/ 2 1 + 0 , D 1/ 2 1 - 0 , R 1/ 2 1 + i 0 , L 1/ 2 1 - i 0 indicate horizontal, vertical, +45° linear, -45° linear, right- and left-circular polarization modes correspondingly. Lower indices numerate the frequency modes of two photons. It has been proved 3 that it is sufficient to use only the first two or three bases for the efficient QKD. When using an incomplete set of bases one sacrifices the security but enhances the keygeneration rate. Since the fulfillment of Bell measurements for the last two bases requires a big experimental effort on both preparation and measurement stages of a protocol, we will restrict ourselves to the first three bases. As we will show experimentally, all states from the first three bases can be prepared with the help of a single nonlinear crystal and local unitary transformations. This is fundamental compared with biphoton-based qutrits, where SU 2 transformations between states from mutually unbiased bases are prohibited. We also present a measurement scheme that allows one to distinguish the states belonging to one basis deterministically thus allowing its implementation in QKD protocol with polarization ququarts.
Experimental procedure

TABLE IV. Bell states reconstruction. The table contains theoretical, experimental state vectors, and fidelities. The measurements performed with Protocol 1. Protocol 1 Experiment 0.7326 0.0818 - 0.0963i 0.0003 - 0.0281i 0.6131 + 0.2657i 0.6597 0.2518 + 0.4692i 0.0897 - 0.0319i -0.6155 + 0.3261i

The experimental setup for the generation of ququart states that belong to the first three bases 39 is shown in Fig. 8. To measure a prepared state either Protocol 1 or Protocol 2

Theory 0.707 0 0 0.707 0.707 0 0 -0.707

F 0.941

0.934

FIG. 8. Setup for preparation and measurement of ququarts which can be used in QKD. Wave plates DP1, DP2 oriented at 45° with respect to the vertical axis serve as the dichroic retardation plate with variable optical thickness which is controlled by tilting angle . Two zero-order plates ZP1, ZP2 allow one to choose the basis. 063810-10


POLARIZATION STATES OF FOUR-DIMENSIONAL¼

PHYSICAL REVIEW A 73, 063810 2006 TABLE V. Fidelities for the states belonging to the first three bases. I F II F III F H 1H
2

H 1V

2

V 1H

2

V 1V

2

0.98 D 1D
2

0.94 D 1D
2

0.98 D 1D
2

0.98 D 1D
2

0.97 R 1R
2

0.96 R 1L
2

0.95 L 1R
2

0.99 L 1L
2

0.95

0.95

0.96

0.97

dence of the coincidence rate on the plate optical thicknesses i is given by the formula: R
FIG. 9. Dependence of single counts upper and coincidences rates lower on tilting angle .
coin

aba1b2 = sin2 12

1

cos2

2

,

40

whereas the single counts distribution in the upper channel is given by I
702 nm

can be used. Usually we applied Protocol 1 for checking the ququart state. As an example let us consider the preparation of state H1V2 from initial state V1V2 . This transformation can be achieved by a dichroic wave plate that introduces a phase shift of 2 for a vertically polarized photon at 605 nm, a phase shift of for the conjugated photon. The wave plate is oriented at 45° to the vertical direction. Using quartz plates as retardation material it is easy to calculate that the thickness of the wave plate that does this transformation should be equal to 3.406 mm 28 . Since these wave plates were not readily available and the result of transformation is extremely sensitive to small variations of thickness, we used the following method to achieve the desired thickness. Two quartz plates with geometric thicknesses of 3.716 mm DP1 and 0.315 mm DP2 with the orthogonally oriented optical axis were placed consecutively in the biphoton beam. The consecutive action of these two wave plates correspond to the action of the quartz wave plate with an effective thickness of 3.401 mm. If then one can tilt these wave plates towards each other by a finite angle , then the optical thickness of the effective wave plate formed by DP1 and DP2 will change, and, at a certain value of , the desired transformation will be achieved. This corresponds to the maximal coincidence rate when the measurement part Protocol 1 is tuned to select state H1V2 . Monitoring the coincidences, one can obtain the value of for which the maximum occurs. Then, fixing the tilting angle at this value, one can perform a complete quantum state tomography protocol in order to verify if the state really coincides with the ideal. In order to change the basis from I to II III , zero-order half- quarter wave plates ZP1 ZP2 oriented at 22.5 ° 45 ° were used. This procedure is repeated for the generation of any of the states. Figure 9 shows the coincidences and single count rates versus the change of tilting angle which determines the optical thickness of the effective wave plate. If the measurement setup is tuned to select state H1V2 then the depen-

aa1 = sin2 1

1

,

41

where the optical thickness depends on the tilting angle as follows: h
j

n n
ej 2

j

=

ej

2

n - n
oj 2

oj

2

- sin2

- sin2

.

42

The solid lines in Fig. 9 show the theoretical curves. We performed tomography measurements for both the maxima as well as for the minimum. The minimum in coincidences occurs when intensity in any channel drops to zero, so it is not a necessary condition to distinguish the orthogonal state from that one selected by given settings of polarization filters. Nevertheless according to calculations and our measurements, the minimum in the coincidences in Fig. 9 refers exactly to state V1H2 . Starting with V702 nmV605 nm , we also prepared and measured the whole set of states from 39 belonging to the first three bases. The result of the state reconstruction is presented in Table V.

FIG. 10. Measurement part at Bob's station.

063810-11


BOGDANOV et al.

PHYSICAL REVIEW A 73, 063810 2006 TABLE VI. Coincidence rate and density matrix components.

D 4D 0

2

11

D 4D

1

22

D 3D 6

2

33

D 3D 0

1

44

0.0 0

220

0.973 1

0.027 0

0.0 0

At the same time the method discussed in this section allows one to distinguish unambiguously all states forming chosen bases. The measurement setup that is fulfilling this task and that has been already tested in our experiments is shown in Fig. 10. It consists of the dichroic mirror, separating the photons with different wavelengths, and a pair of polarization beamsplitters, separating photons with orthogonal polarizations. The four-input double-coincidence scheme linked with the outputs of single-photon detectors registers the biphotonbased ququarts. For example, for the first basis the scheme works as follows, provided that Bob's guess of the basis is correct: i If state H1H2 comes, then detectors D4, D2 will fire; ii If state H1V2 comes, then detectors D4, D1 will fire; iii If state V1H2 comes, then detectors D3, D2 will fire. iv If state V1V2 comes, then detectors D3, D1 will fire. The same holds for any of the remaining correctly guessed bases, since the quarter- and half-wave plates transform the polarization into HV basis in which the measurement is performed. The registered coincidence count from a certain pair of detectors contributes to the corresponding diagonal component of the density matrix. So if the basis is guessed correctly, then the registered coincidence count deterministically identifies the input state. We illustrate this statement by Table IV which shows the total number of registered events per 30 s for input state R1L2 measured in the circular basis and calculated components of the experimental theoretical density matrix. The main obstacle for the practical implementation of the free-space QKD protocol based on ququarts is that one needs to perform fast polarization transformation at the selected wavelengths. There are different ways of overcoming this problem and we will discuss them elsewhere. In this section we mention briefly the possible ways. Since it is not practical to tune the tilting wave plates every time one wants to encode a ququart value, we suggest either using a polarization modulator that operates on two wavelengths or splitting the photons with dichroic mirrors and perform these transformations on halves of a biphoton independently in a MachZehnder-like configuration. It is important to note that interferometric accuracy in Mach-Zehnder interferometer is not

needed, since it is used only for spatial separation of photons. The practical solution would be to couple the downconverted photons in a single mode fiber to ensure a perfect spatial mode overlap and then to split them with the wavelength division multiplexer WDM . Then, the switching between the states can be done with the polarization modulators that introduce a or 2 phase shifts for the selected wavelength. The choice of basis on Alice's side is performed by zero-order quarter- and half-wave plates, which can be realized within the Pockel Liquid Crystal cell driven by a randomly selected voltage. Free space communication is proposed since it is not practical to distribute a polarization state within an optical fiber. On Bob's side, the random choice of basis random number generator RNG is performed in the same way as on Alice's side. Then photons are spatially separated with the help of WDM or a dichroic mirror and each of the photons is sent to the Brown-Twiss scheme with a polarizing beam-splitter that projects an arrived photon on a H or V state as it is shown in Fig. 10. Moreover, registering coincidences allows one to circumvent the problem of the detection noise that is common for single photon based protocols. If the coincidence window is quite small, it is possible to assure a lower level of accidental coincidences for the usual dark count rate of single photon detectors Table VI . In conclusion, we have suggested and tested a method of preparation and measurement of subset of four-dimensional polarization quantum states. Since for this class of states the polarization degree is not invariant under SU 2 transformations, it is possible to switch between orthogonal states using linear transformations namely geometrical rotations and phase shifts . Note added. Recently, we learned about closely related work performed by R. B. R. Adamson and co-authors 29 .

ACKNOWLEDGMENTS

Stimulating discussions with G. BjÆrk and V. P. Karassiov are gratefully acknowledged. This work was supported in part by the Russian Foundation of Basic Research Project Nos. 05-02-16391a, 06-02-16769, 06-02-16393a and the Leading Russian Scientific Schools Project No. 4586.2006.2 . S.P.K. acknowledges support from the Russian Ministry of Science and Education Project Nos. 2006-RI-19.0/001/593 and 2-435/2005 .

063810-12


POLARIZATION STATES OF FOUR-DIMENSIONAL¼ 1 D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, Phys. Rev. Lett. 85, 4418 2000 ; D. Kaszlikowski, D. K. L. Oi, M. Christandl, K. Chang, A. Ekert, L. C. Kwek, and C. H. Oh, Phys. Rev. A 67, 012310 2003 ; T. Durt, N. J. Cerf, N. Gisin, and M. Zukowski, ibid. 67, 012311 2003 ; D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, Phys. Rev. Lett. 88, 040404 2002 . 2 H. Bechmann-Pasquinucci and A. Peres, Phys. Rev. Lett. 85, 3313 2000 ; D. Bruú and C. Macchiavello, ibid. 88, 127901 2002 ; 88, 127901 2002 ; D. Horoshko and S. Kilin, Opt. Spectrosc. 94, 691 2003 . 3 H. Bechmann-Pasquinucci and W. Tittel, Phys. Rev. A 61, 062308 2000 ; M. Bourennane, A. Karlsson, and G. BjÆrk, ibid. 64, 012306 2001 ; N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, Phys. Rev. Lett. 88, 127902 2002 ; F. Caruso, H. Bechmann-Pasquinucci, and C. Macchiavello, Phys. Rev. A 72, 032340 2005 . 4 S. Walborn, D. Lemelle, M. Almeida, and P. Ribeiro, Phys. Rev. Lett. 96, 090501 2006 . 5 S. GrÆblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, New J. Phys. 8, 75 2006 . 6 M. O'Sullivan-Hale, I. Ali Khan, R. Boyd, and J. Howell, Phys. Rev. Lett. 94, 220501 2005 ; L. Neves, G. Lima, J. G. Aguirre Gomez, C. H. Monken, C. Saavedra, and S. Padua, ibid. 94, 100501 2005 . 7 A. Vaziri, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 89, 240401 2002 ; A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, ibid. 91, 227902 2003 ; N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, ibid. 93, 053601 2004 . 8 H. de Riedmatten, I. Marcikic, V. Scarani, W. Tittel, H. Zbinden, and N. Gisin, Phys. Rev. A 69, 050304 R 2004 . 9 R. T. Thew, A. AcÌn, H. Zbinden, and N. Gisin, Quantum Inf. Comput. 4 2 , 93 2004 ; R. T. Thew, A. AcÌn, H. Zbinden, and N. Gisin, Phys. Rev. Lett. 93, 010503 2004 . 10 J. C. Howell, A. Lamas-Linares, and D. Bouwmeester, Phys. Rev. Lett. 88, 030401 2002 . 11 Yu. Bogdanov, M. Chekhova, S. P. Kulik, G. Maslennikov, M. K. Tey, and C. Ch. Oh, Phys. Rev. Lett. 93, 230503 2004 . 12 Yu. Bogdanov, M. Chekhova, L. Krivitsky, S. P. Kulik, L. C. Kwek, M. K. Tey, C. Ch. Oh, and A. A. Zhukov, Phys. Rev. A 70, 042303 R 2004 . 13 D. N. Klyshko, Photons and Nonlinear Optics Gordon and Breach, New York, 1988 .

PHYSICAL REVIEW A 73, 063810 2006 14 A. V. Belinsky and D. N. Klyshko, Laser Phys. 4, 663 1994 . 15 D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A 64, 052312 2001 . 16 A. V. Burlakov and M. V. Chekhova, JETP Lett. 75, 432 2002 . 17 M. V. Chekhova, L. A. Krivitsky, S. P. Kulik, and G. A. Maslennikov, Phys. Rev. A 70, 053801 2004 . 18 V. P. Karassiov, J. Phys. A 26, 4345 1993 . 19 G. BjÆrk, J. SÆderholm, A. Trifonov, P. Usachev, L. L. Sanchez-Soto, and A. V. Klimov, Proc. SPIE 4750, 1 2002 ; A. Sehat, J. Soderholm, G. BjÆrk, P. Espinoza, A. B. Klimov, and L-L. Sanchez-Soto, Phys. Rev. A 71, 033818 2005 . 20 D. N. Klyshko, JETP 84, 1065 1997 . 21 Of course, it is possible to chose specific orthogonal qutrit states that might be transformed into each other by geometrical rotations and phase shifts. These states are: HV , RL , and ¯ DD . According to our definition these states possess P =0 whereas its unit polarization degree is introduced in 19 . The concept of biphoton-based qutrits orthogonality and their transformation properties are discussed in detail in 17 . 22 A. V. Burlakov, S. P. Kulik, Yu. Rytikov, and M. V. Chekhova, JETP 95, 639 2002 . 23 L. Mandel and E. Wolf, Optical Coherence and Quantum Optics Cambridge University Press, Cambridge, MA, 1995 . 24 Y. H. Kim, S. P. Kulik, and Y. Shih, Phys. Rev. A 63, 060301 R 2001 . 25 Yu. I. Bogdanov, R. F. Galeev, S. P. Kulik, G. A. Maslennikov, and E. V. Moreva, JETP Lett. 82, 164 2005 . 26 When studying "space-time" interference for polarization-type entangled states 27 the dependence of coincidence rate on phase delay 14 is analyzed by setting the polarization filters at ±45°. 27 Bases are mutually unbiased if the inner products between all possible pairs of vectors belonging to different bases have the same magnitude equal to 1 / D, where D is the dimension of the Hilbert space. 28 The thickness of the retardation plate that performs the transformation is determined by the number of order mi that is chosen for the specified wavelength according to m1 1/ n1 = 2m2 +1 2/2 n2. Here ni is the value of the birefringence for the specific wavelength. We chose order m1 = 51 for = 702 nm and m2 = 47 for = 650.2 nm. 29 R. B. A. Adamson, L. K. Shalm, M. W. Mitchell, and A. M. Steinberg, e-print quant-ph/0601134.

063810-13