Intrinsic Measure of Coherence in High-Dimensional Systems
Proper quantification of coherence is crucially important. The widely-adopted resource theoretical framework to quantify coherence (Baumgratz et al., PRL, 113, 140401 (2014)) is manifestly basis dependent and leads to unphysical implications including existence of mixed states more coherent than the pure states. In contrast, the degree of polarization, developed in the classical coherence theory, is an intrinsic basis-independent measure of coherence for two-dimensional systems and does not lead to any such unphysical implications. Recently, a Frobenius norm based measure of coherence has been conjectured by generalizing the geometric interpretation of the degree of polarization to higher dimensions. In this paper, we establish the suitability of this quantity as an intrinsic measure of coherence in higher dimensions by generalizing two other interpretations, namely, the visibility and Stokes parameters. We further provide an experimental protocol to measure this quantity for a four-dimensional two-photon state through sixteen coincidence measurements. Using this measure, we show that in parametric down-conversion, the coherence of the pump field predetermines the coherence of the down-converted polarization-entangled two-photon state while predetermining only an upper bound on its entanglement.
pacs:Valid PACS appear here
Light comprises of oscillations of the electromagnetic field which propagate through space and time. The central objective of coherence theory is to quantify the correlations between these oscillations at different space-time points Mandel and Wolf (1995); Born and Wolf (1959). One of the useful features of this theory is that it has been formulated in terms of quantities that can be directly measured in experiments. The simplest manifestation of coherence is interference of light. As the measure of coherence, correlation functions were defined in such a way that they are directly related to the visibility of certain interferometric experiments. These correlation functions have been successfully used in two dimensions (2D) to understand coherence between a pair of space-time points and in polarization domain. Advent of quantum mechanics in early twentieth century called for fundamental rectification of all known concepts including coherence. Accordingly, theory of quantum coherence was developed to incorporate quantum mechanics Glauber (1963a, b); Sudarshan (1963). In this theory, the quantum versions of correlation functions were redefined preserving their visibility interpretation. Although these measures of coherence can directly be measured through experiments and have clear physical meaning, no way to generalize them to arbitrary dimensions is yet known. It is noteworthy that within the framework of quantum mechanics, due to the wave particle duality, coherence is a property not only of light but of any physical system.
In recent years, attempts have been made to quantify coherence for an arbitrary finite dimensional system Aberg (2006); Baumgratz et al. (2014); Winter and Yang (2016); Marvian et al. (2016) from resource theoretical viewpoint. In the paper by Baumgratz et al Baumgratz et al. (2014) [BCP] the authors identified a set criteria which, they demanded, every reasonable measure of coherence should satisfy. Following this paper, several measures of quantum coherence have been proposed Baumgratz et al. (2014); Winter and Yang (2016); Yuan et al. (2015); Girolami (2014); Streltsov et al. (2015); Napoli et al. (2016); Chiribella and Yang (2017). It has been argued that this formalism has potential applications in the field of thermodynamics Lostaglio et al. (2015a); Misra et al. (2016); Lostaglio et al. (2015b), quantum algorithm Hilery (2016); Shi et al. (2017); Anand and Pati (2016), quantum metrology Giorda and Allegra (2016); Marvian and Spekkens (2016) and quantum phase transition B. et al. (2015); Karpat et al. (2014); Malvezzi et al. (2016), but some fundamental issues with it have come out. The definition of incoherent state given by BCP is not immune to objection. In a particular basis, incoherent states have been defined as the states with zero non diagonal elements. This definition emerges from the idea that non diagonal terms represent coherence between the basis states. So the non diagonal terms being zero means that there is no coherence between each pair of orthogonal basis components. But the physical sense in which it is justified to call the state incoherent is not clear. This issue needs to be properly addressed, because, from the theory of entanglement, on which the structure of resource theory of coherence is based, it is known that counter intuitive results may emerge due to the increase in dimensionality. For example, in case of multipartite entanglement, the state may not be fully separable even if there is no entanglement between all pairs Wurflinger et al. (2012). So, the claim that pairwise incoherence implies complete incoherence requires further justification. The BCP formalism has one more major problem. As all pure states are unitarily connected it is reasonable to consider them as equally coherent. Mixed state in quantum mechanics is understood as the incoherent mixture of pure states. It implies that mixed state is less coherent than pure state. According to BCP formalism, the norm of coherence, defined as the sum of the modulus of non diagonal terms of a density matrix, is a valid measure. This measure does not consider all pure states to be equally coherent. Also, there can be mixed states more coherent than the pure states. This implication is not congruent with the basic understanding of pure and mixed states. Moreover, issues with BCP formalism have been pointed out from resourece theoretical viewpoint as well Chitambar and Gour (2016); Marvian and Spekkens (2016). The axiomatic approach to quantify coherence is directly analogous to the resource theory of entanglement Vedral et al. (1997); Horodecki et al. (2009). In a generic resource theory, free states (states with no resource) and free operations (operations which can not generate any resource) are defined. For a resource theory, all mathematically valid free operations do not satisfy the property of physical consistency as described in Refs. Chitambar and Gour (2016); Marvian and Spekkens (2016). A free operation is said to be physically consistent if all possible free operations can be realized by putting some physically implementable experimental restrictions. In the case of entanglement there is a natural choice of free operation called local operation and classical communication (LOCC), which also turns out to be physically consistent in the sense explained in Refs. Chitambar and Gour (2016); Marvian and Spekkens (2016) becauese all LOCC operations can be realized by prohibiting the use of quantum channels. BCP theory has defined a class of operations called the incoherent operation as the free operation of the resource theory of coherence but until now it has not been shown that the given definition satisfies the criteria of physical consistency.
Considering the issues related to the formalism given by Baumgratz et al, we have adopted an alternative approach. A quantity called degree of polarization () was introduced in classical optics to quantify the intrinsic degree of polarization of light Wolf (1959). is a basis independent quantity. It can be interpreted in terms of visibility and can be evaluated by measuring the Stokes parameters Stokes (1851). For an arbitrary state we call the intrinsic measure of coherence in 2D. In the next section we will explain why does not have the drawbacks of BCP formalism. So, if one can suitably generalize to higher dimensions, the generalized quantity, too, will not have the issues of BCP formalism. A basis independent measure of coherence has some important benefits. By virtue of being basis independent such quantity provides a way to intrinsically characterize states. Moreover, an intrinsic measure is very helpful to investigate propagation of coherence in a physical process Jha and Boyd (2010); Kulkarni et al. (2017); Monken et al. (1998); Miguel Angel Olvera (2015); Hioe and Eberly (1981).
Recently, a Frobenius norm based measure of intrinsic coherence () has been proposed for finite dimensional states based on a geometric argument by Yao et al Yao et al. (2016). They proposed this measure in an attempt to generalize but they did not show if the measure satisfies similar properties as . In this paper, we establish the suitability of this measure by showing that generalization of two additional physically meaningful derivations of lead to the same measure. Furthermore, we present an explicit experimental scheme to measure the intrinsic measure for a four dimensional (4D) system consisting of two photon, through sixteen coincidence measurements. Using this measure, we show that in spontaneous parametric down conversion (SPDC), the coherence of the pump field predetermines the intrinsic coherence of the down converted two photon states. For conceptual clarity, first we will provide a brief introduction to degree of polarization.
Ii Degree of Polarization
The degree of polarization, as the name suggests, quantifies the intrinsic amount of polarization correlations of light. For an arbitrary state of light the degree of polarization is defined as
is a basis independent quantity and is bounded between and . Note that attains its maximum value 1 for any pure state. For a mixed state is less than 1. So, unlike the measures of BCP formalism, for a mixed state is less than that of the pure states. is zero only for the identity matrix. That is, the identity matrix is the completely incoherent state. As the identity matrix is invariant under unitary transformation, its non diagonal elements are zero in all bases. This property justifies to consider it the completely incoherent state. After showing that does not have the issues of BCP formalism, we note the following properties of .
is equal to the radius of state in Bloch’s sphere. It is the distance between and identity matrix when distance is quantified using Frobenius norm Golub and Van Loan (1996)
where is the Frobenius norm.
can be derived from the Stokes parameters, introduced by E C G Stokes to describe the polarization of light Stokes (1851). The concept of Stokes parameter, like degree of polarization, can be extended into a general 2D density matrix. Any 2D state can be expanded as a linear combination of identity and Pauli matrices. The coefficients of this linear expansion are called Stokes parameters.
Here are Stokes parameters, , , , . can be expressed in terms of Stokes parameters in a nice compact form.
Stokes parameter representation of is very useful from experimental perspective because value of Stokes parameters can be extracted by four properly chosen intensity measurements. From these measurements, using (4) one can also find .
can be interpreted in terms of visibility which gives it a physical meaning. Visibility () of an interference pattern was defined by Michelson as
where are maximum and minimum of the intensity. The interpretation of in terms of visibility can be illustrated using a simple experiment with a partially polarized light Wolf (1959). Let the light pass through a wave plate, which introduces a phase difference between two polarizations and a polarizer whose axis is at an angle . Monitoring the light which passes through polarizer one can find the maximum and minimum of intensity upon variation of and . From the maximum and minimum intensity, visibility calculated according to (5) gives the value of .
One can show that any state can be uniquely decomposed into maximally and minimally coherent state with certain weightage. That is
where is a pure state, is 2D identity matrix and is the weight factor. (6) has a unique solution for
This decomposition means that is the weightage of maximally coherent state.
Another quantity, related to , called polarization correlation function () is used in quantum optics. It is basis dependent and quantifies the correlation between two orthonormal components. In a chosen basis is defined as
where are two bases. This quantity is always bounded below by , that is . The equality is satisfied in the basis where the diagonal terms of are equal. is 1 for all pure states and for a mixed state it is lower than the pure states. So, in spite of being basis dependent, does not have one issue of BCP formalism.
Iii Intrinsic Measure of Coherence
To quantify the amount of coherence of an arbitrary dimensional state one needs to generalize to higher dimensions. Yao et al Yao et al. (2016) have generalized property 2, following up the work by Luis Luis (2007), to propose in dimensional systems as
The pre factor in equation (9) is chosen in such a way that . We have discussed four other properties of . In what follows, we will show that generalization of three of these properties [4, 5, 8] lead to . This will establish as the suitable generalization of .
First we will generalize property 4, the derivation of using Stokes parameters. In the derivation, a 2D density matrix is expanded as a linear combination of identity and Pauli matrices. Observing that Pauli matrices are generators of special unitary group of degree 2, denoted as SU(2), Sètala et al have extended this derivation for three-dimensional systems Setala et al. (2002a, b). They expanded a 3D state, , as a linear combination of identity and Gellmann matrices, which are the generators of SU(3).
Here s are 3D spectral Stokes parameters and s are Gellmann matrices. The 3D analog of has been defined as
Following up their work, we have expanded N dimensional state as the linear combination of identity and generators of SU(N). The generators of SU(N) are known as generalized Gellmann matrices Bertlmann and Krammer (2008). Among the Gellmann matrices, are symmetric, are antisymmetric and are diagonal. In a chosen basis the generators can be written in a compact form.
Here stand for symmetric, antisymmetric and diagonal respectively. The indices satisfy and . Using these generators we expand a dimensional state as
Here is the identity matrix. We call the coefficients generalized Stokes parameters. From (15) we find the generalized Stokes parameters in terms of the density matrix elements.
Using these results one can show that
Here is the normalization factor to restrict between 0 and 1. This analysis is direct generalization of the interpretation of in terms of Stokes parameter. Thereby it firmly establishes the validity of (9) as the intrinsic measure of coherence for a finite dimensional system.
Now we will generalize property 5, the visibility interpretation of . The simplest definition of visibility is given in (5). Visibility quantifies the contrast of an interference pattern. It is 1 when and it is 0 when , which physically makes sense. We have discussed the correspondance between and visibility in the previous section. In the case of , explicit calculation shows that , where are the eigenvalues of the input state such that and . Substituting the value of and in (5) we can write
can be interpreted as visibility because it is equal to . In order to generalize this interpretation one thing should be noted that the definition of visibility given in (5) has limited applicability. It is unambiguous when there is one maximum and minimum. But for a more general case definition of visibility needs to be suitably modified. The knowledge of provides us the hint on how the definition of visibility should be modified. From (21) we see that, in this case, instead of intensities it is more helpful to work with eigenvalues of the state. So, for a dimensional system we propose a generalized definition of visibility in terms of eigenvalues
Where are the eigenvalues of the corresponding dimensional state. The motivation behind the definition given in (22) is that can be written in the same form.
It is expected from physical consideration that visibility should be 1 when all except one eigenvalues are 0 and it should be 0 when all eigenvalues are equal. The given definition of visibility is a sensible one because it satisfies these properties.
Finally we will generalize property 8 by constructing a N dimensional basis dependent quantity similar to . We call this generalized correlation function . The form of this function is the following
The proof of the claim that satisfies the properties of is given in appendix. According to (23), can be described as the weighted average of the all possible pairwise coherence between the orthogonal basis states. Note that is 1 for all pure states in any basis and for a mixed state is less than that of pure states.
Iv Measuring Coherence of a Four Dimensional System
One can find the degree of polarization () of a photon through four intensity measurements Mandel and Wolf (1995). As we have claimed (9) to be the generalization of , it should also be measurable using similar technique. In this section, we will provide an explicit procedure to measure coherence for a four-dimensional system. Expression of can be obtained from (9)
The four-dimensional system we consider consists of two photons in polarization basis. Value of for the system can be found through sixteen coincidence measurements.
Required experimental set up for the measurements is shown in Fig. 1. A general two-photon two-qubit polarization-entangled state has been generated by a source. The most general form of this two photon state is
where denotes horizontal and vertical polarization state of the photon. Each of the photons passes through a phase retarder and a polarizer before being detected. The phase retarders induce a phase difference between two polarizations and the axis of the polarizers are set at an angle for the photon A and B respectively. The value of coincidence counts () is given by the first diagonal elements of the operator .
where and the unitary operators for have the following form in basis
Measuring the coincidence counts for sixteen different sets of these parameters one can find the values of sixteen two photon Stokes parameters. A convenient set of measurements is given in Table 1. The two photon Stokes parameters can be evaluated from these measurements using following algebraic equations
Now can be evaluated using the relation
The measurements tabulated in Table 1 can be performed using widely used optical devices like polarizers and quarter wave plate. A possible platform to practically implement this measurement is spontaneous parametric down conversion.
V Transfer of Coherence in Spontaneous Parametric Down Conversion
Spontaneous parametric down conversion (SPDC) is a non linear optical process where single photon of an optical pump field is down converted into two entangled photons. In this process, polarization coherence of the input field can be characterized by degree of polarization (). But to characterize the polarization of the output two photon state in a similar way, four dimensional analog of is required. Our proposed measure of coherence as in (24) can serve this purpose. Transfer of correlation for a generic SPDC process has previously been studied by Kulkarni et al Kulkarni et al. (2016). But, in that paper, due to the lack of any intrinsic measure of coherence in 4D, the authors considered the entanglement of the output state quantified by concurrence. A SPDC process can be characterized by the eigenvalues of input and output state. The eigenvalues of the 2D state () of the pump with degree of polarization are and . Suppose the eigenvalues of the output two photon states (), in descending order, are and . If trace is preserved and no coherence is gained from external degrees of freedom, it has been shown that the eigenvalues of output state is majorized by the eigenvalues of the input state Kulkarni et al. (2016), that is . To explain the meaning of majorization wConsider two vectors and such that and for , that is the elements are arranged in decreasing order. A is defined to be majorized by B iff for and the equality is satisfied for Bhatia (2013). In the particular case of SPDC, the required majorization is satisfied if
In terms of eigenvalues is
Using (46) and (47) we have found the upper bound on for SPDC. The bound is obtained by repeated use of the theorem that if and then . The bound is saturated when the process is unitary, that is no coherence is lost. For such processes coherence of the two photon state () is predetermined by coherence of the pump field () according to the following equation
If the process involves non unitary evolution then naturally the coherence will be lower than (48). In Kulkarni et al. (2016) it has been shown that concurrence () of the output state satisfies but the equality may not be satisfied even when the process is unitary. So, imposes a condition more stringent than that imposed by entanglement on the evolution of coherence in SPDC. Note that state with can produce states with non zero . This is because the produced two photon state has a larger Hilbert space accessible to it.
In fig. 2 , maximum value concurrence and entanglement of formation of two photon state has been plotted with respect to degree of polarization of pump field. We see that entanglement of formation is always lower than but there is a crossover between concurrence and .
We have established the suitability of the quantity as the measure of intrinsic coherence of an arbitrary finite dimensional system by showing that the generalizations of two physically meaningful interpretations of , namely visibility and Stokes parameters, lead to this quantity. We have provided an experimental protocol to measure this quantity for a four dimensional two-photon state through sixteen coincidence measurements. Using this measure, we have shown that in parametric down-conversion, the coherence of the pump field predetermines the coherence of the down-converted polarization-entangled two-photon state while predetermining only an upper bound on its entanglement. Entanglement is a special type of non local correlation which quantifies the correlation between different parties. We think further analysis of can shed light on the understanding of entanglement. Furthermore, as provides a systematic way to investigate the evolution of coherence it may be used in the study of decoherence.
We acknowledge helpful discussion with Shaurya Aarav and financial support through an initiation grant no. IITK/PHY/20130008 from Indian Institute of Technology (IIT) Kanpur, India and through the research grant no. EMR/2015/001931 from the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India.
viii.1 Properties of Generalized Correlation Function
Generalized correlation function,, should satisfy the same properties as . We identify four properties which should satisfy. A. It reduces to in two-dimension, B. for all pure states, C. and D. the equality in condition C is satisfied when all the diagonal elements are equal. Now we will show that defined in (23) satisfy these criterion.
It is straightforward to check that satisfies A.
For a pure state implies that for all pure states.
To prove property C, is bounded above by , note that
Now the inequality can be rewritten as . Substituting this we find that , which completes the proof.
The equality in property C is satisfied when . It will happen when all the diagonal elements of the state are equal (D)
- preprint: APS/123-QED
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