Entanglement of qutrits and ququarts
We investigate in a general form entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of collinear and, correspondingly, degenerate and non-degenerate Spontaneous Parametric Down-Conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and and polarization entanglement are fully taken into account. In the case of qutrits the most general 3-parametric families of maximally entangled and non-entangled states are found, and anti-correlation 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 , which differs them significantly from the often used two-qubit model (). 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.
pacs:03.67.Bg, 03.67.Mn, 42.65.Lm
Optical qutrits and ququarts are promising objects of the modern quantum information and quantum cryptography (1); (2); (3); (4); (5); (6); (7). Formally, qutrits and ququarts are defined as superpositions of, correspondingly, three and four basis states. In practice, most often, the basis states for qutrits and ququarts are formed by biphoton states arising in the processes of Spontaneous Parametric Down-Conversion (SPDC). For qutrits it is sufficient to use the collinear degenerate SPDC processes, i.e., such processes in which wave vectors of two photons in a SPDC pair are strictly parallel to each other and frequencies are also given and equal to each other. For constructing ququarts, one has to use either the non-collinear frequency-degenerate or collinear but frequency-non-degenerate SPDC processes, i.e., processes in which either directions of wave vectors or frequencies of two photons in SPDC pairs differ from each other. Below we will refer both of these possibilities as ‘’non-degenerate. In theory, from the very beginning (8), biphoton qutrits were considered as arbitrary superpositions of three Fock states, corresponding to three possibilities of distributing two indistinguishable photons of an SPDC pair in two polarization modes, horizontal () and vertical () ones. Each of these basis states is a direct product of two one-qubit single-photon states, and biphoton qutrits are two-qubit states. There were many works devoted to biphoton qutrits (9); (10); (11); (12); (13); (14); (15). Most of them study polarization properties of qutrits and much less entanglement. Giving a comprehensive picture of entanglement of qutrits is one of the goals of this paper. In particular, we will find and describe general families of maximally entangled and non-entangled qutrits, their entanglement quantifiers such as the Schmidt entanglement parameter, concurrence, and the subsystem entropy, relations between entanglement and polarization of qutrits, etc.
An important point to be discussed is the role of the symmetry of biphoton wave functions with respect to permutation of particles’ variables. Existence of entanglement related to symmetry was realized by many authors rather long ago, (16); (17); (18); (19); (20) and so on. But a general attitude to the entanglement related to symmetry is rather sceptical, up to the opinion that this type of entanglement is unimportant and can be forgotten. In such approach all basis states of qutrits and ququarts would be non-entangled and the only reason for entanglement would be related to various choice of coefficients in superpositions of basis states. This is a configurational entanglement. But in reality, owing to symmetry, basis states of qutrits and ququarts are entangled (at least one basis states in the case of qutrits and all basis states of ququarts). This is a fundamental, unchangeable entanglement. In superpositions of basis states the symmetry-related and configuration entanglement exist together and each of them is so strongly built into a general picture of entanglement of qutrits and ququarts, and into their entanglement quantifiers, that it’s impossible to separate these two types of entanglement and to take off the symmetry-related one without hurting significantly the general picture and results. Thus, all kinds of entanglement have to be taken into account together and none of them can be ‘’forgotten.
In the case of ququarts, the symmetry-related entanglement arises from both polarization and frequency (or angular) degrees of freedom. This makes the traditional two-qubit model of biphoton ququarts invalid. As we show, biphoton ququarts are two-qudit states with the dimensionality of the one-photon Hilbert space and dimensionality of the two-photon Hilbert space . This makes ququarts significantly different from qutrits (where and ), and a series of new results on entanglement of ququarts is derived and discussed in Section 7. This new understanding of the physics of ququarts raises a question about changes in applications of ququarts analyzed earlier in the frame of the two-qubit model when the latter is substituted by the qudit picture. We hope to return to such analysis elsewhere.
In addition to the above-mentioned entanglement quantifiers, we use widely the Schmidt decomposition of biphoton wave functions (21); (22). As known (23), for pure biphoton states with the dimensionality of the one-photon Hilbert space the maximal amount of terms in the Schmidt decomposition equals , and such states are unseparable. Only if all but one coefficients in the Schmidt decomposition are equal zero, and the remaining exceptional coefficient equals unit, the Schmidt decomposition is reduced to a single product of Schmidt modes, and such state is separable. We consider this criterion as the ultimate indication of whether states are separable or not.
2 State vectors and wave functions of biphoton qutrits
In the form of state vectors, purely polarization biphoton states (qutrits), are given by a superposition
where the basis state vectors are given by
is the vacuum state vector, and are the creation operators of photons in the modes with horizontal and vertical polarizations (with given equal frequencies and given identical propagation directions). are arbitrary complex constants , obeying the normalization condition
Actually, as the total phase of the state vector (2.1) or wave function (see below) does not affect any measurable characteristics of qutrits, one of the phases , or a linear combination of phases, can be taken equal zero, and, hence, the general form of the qutrit state vector (2.1) is characterized by four independent constants (e.g., , , , and with ).
As qutrit (2.1) is a two-photon state, its polarization wave function depends on two variables. A general rule of obtaining multipartite wave functions from state vectors is known pretty well in the quantum-field theory, and for bosons the corresponding formula has the form (24) (in slightly modified notations)
where are dynamical variables of identical boson particles, are single-particle wave functions (of -th modes and -th variables), indicates all possible transpositions of variables in wave functions , are numbers of particles in modes, is the total amount of particles in all modes, is the total amount of modes; for empty modes the corresponding single-particle wave functions have to be dropped.
In the case of qutrits we have two modes () and two particles, . The polarization variables of two photons can be denoted as and . In terms of wave functions, the single-photon wave functions are given by the Kronecker symbols. Thus, the qutrit basis wave functions corresponding to the basis state vectors in Eq. (2.1) can be written as
The same basis wave functions can be written equivalently in the form of 2-row columns, which is more convenient for calculation of matrices
where the upper and lower rows in two-row columns correspond to the horizontal and vertical polarizations and the indices 1 and 2 numerate indistinguishable photons.
In a general form the qutrit wave function corresponding to the state vector (2.1) is given by
Alternatively, the same general qutrit wave function (2.11) can be presented in the form of an expansion in a series of Bell states
and are the wave functions describing three Bell states. The fourth Bell state,
does not and cannot arise in the expansion (2.12) because (2.15) is antisymmetric with respect to the variable transposition , whereas all biphoton wave functions have to be symmetric. Nevertheless, in principle, the antisymmetric Bell state (2.15) can be included into the expansion (2.12) but obligatory with the zero coefficient:
This obligatory zero coefficient in front of or missing forth antisymmetric Bell state in the expansion (2.12) is related to restrictions imposed by the symmetry requirements for two-bozon states. Thus, even existence of biphoton qutrits as superpositions of only three basis wave functions occurs exclusively owing to the symmetry restrictions eliminating the forth (antisymmetric) basis Bell state.
3 Density matrices
The first step for finding the degree of entanglement is related to a transition from the wave function to the density matrix of the same pure biphoton state . The full density matrix of the qutrit (2.11) can be presented in following two forms
The next step is the reduction of the density matrix with respect to one of the photon variables, e.g., of the photon 2. Mathematically this means taking traces of all matrices with the subscript 2 in Eq. (3.1), which gives
It may be interesting to analyze a relation between the density matrix (3.2) and the coherence matrix introduced by Klyshko in 1997 (25). The density matrix (3.2) is written in a natural two-photon basis
The question is how it can be transformed to the basis of states , , plus the empty antisymmetric states (2.15)? Evidently, the transformation
is provided by the matrix
Now, transformed to the basis , the density matrix (3.2) takes the form
Though the coherence matrix (3.8) is widely used and analyzed in literature, both and are hardly appropriate for reduction over one of the photon variables (e.g., 2) and for finding correctly the reduced density matrix (3.3) because the variables 1 and 2 are mixed not only in the matrix itself but also in the transformed basis of Eq. (3.5).
4 Degree of entanglement
As known (21); (22), the trace of the squared reduced density matrix determines purity of the reduced state coinciding with the inverse value of the Schmidt entanglement parameter . The result of its calculation for the reduced density matrix of Eq. (3.3) is given by
It’s known also (23), that in the case of bipartite states with the dimensionality of the one-particle Hilbert space , there is a simple algebraic relation between the Wootters’ concurrence (26) and the Schmidt entanglement parameter , owing to which
Note that the expressions for the concurrence [Eq. (4.3) and the last formula of Eq. (4.4)] can be derived also directly from the original Wootters’ definition (26). Indeed, for a pure bipartite state with the dimensionality of the one-particle Hilbert space , the concurrence is defined in Ref. (26) as
where is the function or state vector arising from after the ‘’spin-flip operation
and is the Pauli matrix, . For qutrits, is given by Eqs. (2.11) or (2.12). The rules of the ‘’spin-flip transformation for one-photon wave functions are and . From here we easily find the spin-flip transforms of the qutrit basis wave functions (2.8)-(2.10)
and of the general-form qutrit wave function (2.11)
Substitution of these expressions into Eq. (4.5) gives
The functions and are shown in Fig. 1 together with the subsystem entropy found in the following section.
This picture demonstrates clearly that qutrits are non-entangled ( and ) if and, hence, . Consequently, the family of non-entangled qutrit wave functions with real coefficients is given by
with arbitrary . If the constants , are complex, a general condition of no-entanglement is the same: . As seen from the general expression for (4.3), the concurrence depends on phases of the constants only via the combination . Hence, if , in the case of complex constants , Eq. (4.3) is reduced to , i.e. this case appears to be equivalent to the case of real constants . From here we find that the general three-parametric family of wave functions of non-entangled qutrits is given by
with arbitrary , , and . With these wave functions found explicitly we can reconstruct the corresponding family of state vectors of non-entangled qutrits
Qutrits are maximally entangled when and , and for the wave functions with real constants , this occurs in two cases: and . In the first of these cases . In the second case the maximally entangled wave function has a form of an arbitrary superposition of and , . As previously, this result can be generalized for the case of wave functions with complex coefficients such that . As a result we get the following three-parametric family of wave functions of maximally entangled qutrits
and the corresponding family of state vectors
In the case Eqs. (4.14) and (4.15) show that the state () belongs to the family of maximally entangled states too (contrary to a met opinion that the state is factorable). A way of seeing explicitly whether qutrits are factorable or not consists in finding their Schmidt decompositions, which can contain either two products of Schmidt modes or only one product. This analysis is carried out in the following section.
But before switching to the Schmidt-mode analysis let us discuss briefly the problem of qutrit polarization. If we define the biphoton polarization vector, as suggested by Wang (28), , where is the vector of Pauli matrices, from Eq. (3.3) we easily find
A direct comparison with the results obtained in 1999 by Burlakov and Klyshko (8) for polarization characteristics of qutrits shows that the polarization vector (4.16) coincides exactly with one half of the vector of Stokes parameters of Ref. (8), and the absolute value of coincides with the degree of polarization (introduced also in (8))
Note that the Stokes parameter and the degree of polarization were found in Ref. (8) in a way, absolutely different from that used above for derivation of the polarization vector . For this reason the coincidences (4.17) are rather non-trivial.
The derived expression for the polarization vector (4.16) can be compared with the general expression for the qutrit concurrence (4.3) to show that they obey the relation obtained by Wang in 2000 (28). Note, however, that for qutrits we have used the concurrence of Eq. (4.3), , rather than obtained in Refs. (28) for the state to be commented below in section 7. In terms of the degree of polarization the relation can be rewritten as
Thus, the degrees of polarization and entanglement anti-correlate with each other: the maximally entangled qutrits are non-polarized, and maximally polarized states are non-entangled. Though, maybe, intuitively more or less expected, as far as we know, anticorrelation of polarization and entanglement has never been presented in a rigorous mathematical form of Eq. (4.18).
5 Schmidt modes of qutrits and the subsystem entropy
Entanglement means that the biphoton wave function cannot be factorized whereas no-entanglement means that it is factorable. A transition from non-factorable to factorable wave functions can be reasonably explained in terms of Schmidt modes. Schmidt modes are eigenfunctions of the reduced density matrix, i.e., solutions of the equation . As in the case of qutrits (3.3) is the matrix, it has two eigenvalues , its eigenfunctions are 2-row columns , and the eigenvalue-eigenfunction equation has the form
The modes can be normalized and they are orthogonal to each other . In terms of the Schmidt entanglement parameter equals to .
In accordance with the Schmidt theorem, the biphoton wave function can be presented as a sum of products of Schmidt modes (Schmidt decomposition). In the case (two-qubit states or qutrits) the Schmidt decomposition contains only two terms
where arguments of the Schmidt modes indicate variables of photons ‘’1 and ‘’2. This decomposition shows that in a general case the wave function is nonseparable. Exceptions occur when one of the eigenvalues of the reduced density matrix, or , becomes equal zero.
Eigenvalues of the matrix (3.3) can be found rather easily and they can be reduced to a very simple form being expressed via the concurrence
In the case , when entanglement is maximal, , i.e., two products in the Schmidt decomposition (5.2) are presented with equal weights and, clearly, the wave function is nonseparable.
In the case (no entanglement) Eq. (5.3) gives and . In this case one of the products of Schmidt modes () in the Schmidt decomposition disappears because the coefficient in front of it vanishes. This is the reason of separability of the wave function in the case of no-entanglement. As for the remaining product in the Schmidt decomposition, the corresponding eigenfunction of the reduced density matrix at can be found easily and has a reasonably simple form (4.12)
With this expression we find immediately that in the case the Schmidt decomposition (5.2) gives the following representation for the factored biphoton wave function
Identity of Eqs. (4.12) and (5.5) can be checked directly, be means of substitution in Eq. (4.12) instead of , , their column representations (2.8)-(2.10), calculation of direct products of columns in all terms of Eq. (4.12) and in Eq. (5.5), and presenting in both cases in the form of the following 4-row column
In the case of qutrits with real coefficients the concurrence itself is a function of (4.10). With substituted instead of into Eq. (5.3) and then substituted into Eq. (5.7), we get a function , which is plotted in Fig. 1. Though different from both and , the subsystem entropy has the same main features as two other entanglement quantifiers: is minimal and equals zero at and is maximal and equals 1 at and . Therefore, though the Schmidt entanglement parameter , concurrence and the subsystem entropy characterize the degree of entanglement in different metrics, their behavior is very similar, which confirms all conclusions made above about conditions of separability and nonseparability of qutrits and their wave functions.
As mentioned above, one of the basis wave functions of qutrits, (2.6), is maximally entangled (, ) and, hence, unseparable. Eigenvalues of the reduced density matrix of this state are degenerate, , and the Schmidt decomposition has the form
with the Schmidt modes given by
Thus, entanglement of the state (2.6) is proved here in several ways, by direct calculations of the entanglement quantifiers , and , and by showing that its Schmidt decomposition contains two products of the Schmidt modes (5.8). In literature the opinion about entanglement of the state is shared by some authors (20); (17), though sometimes the state is claimed to be separable (30). Actually, this difference of opinions reflects a rather popular point of view that all basis states of qutrits and ququarts (see below) are separable and non-entangled. In a more general form, this statements can be reformulated as saying that all states of the type generate separable wave functions in both cases of coinciding and non-coinciding mode indices. Our analysis shows that for biphoton states the latter is not true. In reality, for arbitrary photon variables (polarization, angular, frequency, or their combinations), if and are the wave functions of the and modes and , the state vector generates the wave function of the type (2.6),
6 Finding the degree of entanglement of qutrits from direct polarization measurements
In this section we will show that there is a rather simple method of measuring experimentally the degree of entanglement of qutrits (supposedly not known in advance). The key idea consists in splitting the original biphoton beam for two identical parts by a non-selective beam splitter (BS) and performing a series of coincidence photon-counting measurements in the arising channels I an II (see Fig. 2).
Measurements have to be done with polarizers installed in the horizontal () and vertical () directions, as well as along the axes and turned for the angle with respect to, correspondingly, - and -axes (Fig. 2)). As shown below this set of measurements is sufficient for determining the degree of entanglement as well as for a full reconstruction of all parameters of arbitrary biphoton qutrits.
6.1 Beam splitter
In terms of biphoton wave functions, BS adds an additional degree of freedom to each photon - the propagation angles and , which can take only two value, and . In a nonselective BS each photon has equal probabilities of transmitting or being reflected. Thus, if the biphoton wave function in front of BS is , after BS it takes the form