Quantum state tomography with informationally complete POVMs generated in time domain

The article establishes a framework for dynamic generation of informationally complete POVMs for quantum state tomography. Assuming that the evolution of quantum systems is given by random unitary dynamics, one can switch to the Heisenberg representation and define the measurements in time domain. Consequently, starting with an incomplete set of positive operators, one can obtain sufficient information for quantum state reconstruction by multiple measurements. The framework has been tested on qubits and qutrits. For some types of dynamical maps, it suffices to initially have one measurement operators. The results demonstrate that quantum state tomography is feasible even with limited measurement potential.


I. INTRODUCTION
Quantum communication and quantum computation require well-defined resources in order to encode quantum information [1][2][3]. For this reason, the ability to characterize quantum systems based on measurements is crucial in these fields [4,5]. Quantum state tomography (QST) provides a variety of techniques which allow to reconstruct the accurate representation of a physical system.
Quantum state tomography originated in 1933 when Wolfgang Pauli posted the question whether a particle's wave function can be determined from the position and momentum probability densities [6]. However the problem of recovering the state of a physical system dates back to 1852 when George G. Stokes introduced his famous 4 parameters which allow to uniquely determine the polarization state of a light beam [7]. For years there have been many proposals of tomographic techniques which differ in accuracy, types of measurement and other properties. The notion of quantum state tomography refers to reconstruction of any mathematical representation of a system, e.g. quantum wave functions, density matrices, state vectors, Wigner functions. In this article we shall focus on the tomographic reconstruction of an unknown density matrix [8,9].
According to fundamental postulates of quantum mechanics, any d−level physical system is associated with Hilbert space H such that dim H = d. Pure states of the system are represented by complex vectors from the Hilbert space, whereas mixed states are expressed by density operators ρ which belong to the space of all linear operators acting on H (denoted by L(H)). By selecting a basis in the space, the density operators can be represented by a matrix which has to be Hermitian, positive semi-definite and trace one [10]. The set of all physical density matrices is called the state set and shall be denoted by S(H) (see more in [11]). * aczerwin@umk.pl The classic approach to quantum state tomography involves polarization measurements along with estimation methods such as maximum likelihood estimation [12][13][14][15][16]. Another possibility to implement a feasible framework for state reconstruction is based on measurements defined in the mutually unbiased bases (MUBs) [17,18]. There are also tomographic techniques which solve the density matrix reconstruction problem by means of expectation values of Hermitian operators [19,20]. Finally, contemporary quantum state tomography methods usually utilize the concept of generalized quantum measurements, which is the focus of this article.
Regardless of the particular state reconstruction method one considers a total number of measurements as a resource. For this reason, economic frameworks, which aim at reducing the amount of required data, are gaining in popularity [21][22][23][24]. One of such approaches, i.e. the so-called stroboscopic tomography which inspired this article, utilizes information about quantum dynamics encoded in the generator of evolution e.g. [25]. There have also been proposals involving continuous measurement defined in time-domain [26][27][28][29]. Special attention should be paid to the methods, both theoretical and experimental, which demonstrate the possibility to obtain the complete information about an unknown quantum state from a single measurement setup [30,31].
In this article we contribute to the search for optimal methods of quantum tomography as we introduce a framework for dynamic generation of informationally complete set of measurement operators. In Sec. II we revise the definitions and concepts connected with the generalized quantum measurements. Then in Sec. III we present the framework and in Sec. IV we demonstrate its performance in state reconstruction of qubits subject to random unitary dynamics. Sec. V is devoted to qutrits tomography. The results prove that one can reduce the amount of resources needed for quantum tomography provided the system dynamics is known. The framework has a potential to enhance the efficiency of experimental tomographic techniques.

II. QUANTUM MEASUREMENT IN STATE TOMOGRAPHY
Properly defined measurements play a crucial role in quantum state tomography since they provide information about the unknown system. According to postulates of quantum mechanics, measurements are described by a collection {M k } of measurement operators [32]. The index k refers to the results of measurement that may occur in the experiment.
In our framework we postulate that the source can repeatedly perform the same procedure of preparing quantum systems in an unknown quantum state ρ(0). Thus, we have access to a relatively large number of identical quantum systems which are at our disposal. We can assume that each copy from the ensemble is measured only once. For this reason, the post-measurement state of the system is of little interest, whereas all attention is paid to the probabilities of the respective measurement outcomes. Therefore, we follow the Positive Operator-Valued Measure (POVM) formalism [33].
A general measurement is defined by a set of positive semi-definite operators {M k }, acting on a finitedimensional Hilbert space H, such that where 1 denotes the identity operator. The set of operators {M k } is referred to as a POVM. As already mentioned, in this approach we are not interested in the structure of post-measurement state. The set of operators {M k } is sufficient to determine the probabilities of the possible measurement outcomes which are computed according to the Born's rule [34]: Note that the probabilities have to sum up to one, i.e. k p(k) = k Tr{M k ρ} = 1, which is equivalent to the condition Eq. 1 since Trρ = 1.
The probabilities Eq. 2, which are accessible from an experiment, provide knowledge about an unknown state ρ. The goal of quantum state tomography is to estimate the state by using the results of measurement. If the measurement operators provide sufficient information for the state reconstruction, they are called a quorum [10]. In the case of POVM, if the measurement provides complete knowledge about the system state, it is said to be an informationally complete POVM (IC-POVM) [35][36][37]. For a given system there might be various different sets of operators which lead to full state characterization.
We shall follow an operational definition of IC-POVM [38]. As a consequence, an IC-POVM has to comprise at least d 2 operators. IC-POVMs which have exactly d 2 elements are called minimal. Note that a spanning set can be considered a generalized basis for a vector space: any vector from the space can be expressed as a linear combination of the set elements, but in general the set does not need to be composed of linearly independent (or normalized) vectors.
Usually, special attention is paid to a particular case of POVM which is called a symmetric, informationally complete, positive operator-valued measure (SIC-POVM) [39]. Originally, SIC-POVMs are constructed from rank-one projectors, but their general properties have also been excessively studied [40].
Definition 2 (Symmetric IC-POVM). Let us assume there is a set of d 2 normalized vectors |φ k ∈ H such that Then the set of rank-one projectors {Π i } defined as constitutes a symmetric, informationally complete, positive operator-valued measure (SIC-POVM).
The concept of SIC-POVMs can be mathematically expressed in a very elegant manner by utilizing the frame theory [39]. Nonetheless, for the sake of current analysis, the key property of SIC-POVMs relates to their maximum efficiency in quantum state estimation.

III. FRAMEWORK FOR DYNAMIC GENERATION OF IC-POVMS
Information stored in an unknown density matrix can be retrieved from a properly defined set of measurements. If one can implement measurements corresponding with an IC-POVM, then one obtains complete knowledge needed for density matrix reconstruction. Let us first assume that we do not have access to an IC-POVM or we want to perform quantum tomography by a fewer number of distinct measurement operators. Thus, at the beginning there is an incomplete set of positive semidefinite operators: Starting from an incomplete POVM, we seek a way to dynamically generate an IC-POVM. We consider a scenario that there is an ensemble of identically prepared quantum systems which evolve and different operators can be measured at distinct time instants (each copy of the system is measured only once). In order to achieve this goal, we have to assume that we possess knowledge about the system dynamics. Changes in quantum systems are usually described by completely positive and trace-preserving (CPTP) maps, i.e. quantum channels. For the convenience of this analysis we shall adopt the Kraus representation [41,42].
The condition in Eq. 5 ensures that the CP map Λ preserves trace. It is worth mentioning that the Kraus representation is non-unique.
In order to keep track of changes in the quantum system over time, one needs to introduce time-dependent CPTP maps Λ t . Maps which are legitimate from the physical point of view are called dynamical maps.
The last condition in the definition 4 is natural, because if the family of maps {Λ t , t ∈ R 1 + } is to describe evolution of density operators, it has to satisfy the initial Bearing in mind the definition of the dynamical map, we shall assume that the evolution of our quantum system is given by a special kind a map, called random unitary dynamics [43,44].
By applying the random unitary dynamics into the measurement result we can write a time-dependent formula for the probability Eq. 2 associated with k−th operator from the set M: (8) Note that the probability formula was rewritten due to algebraic properties of the matrix trace. Physically speaking, this is equivalent to the shift from the Schrödinger picture to the Heisenberg representation since the unitary operators {U α } are now acting on M k .
If the number of unitary operators U α equals κ, we can select a discrete number of time instants {t 1 , t 2 , . . . , t κ } and obtain a system of equations expressing probabilities: Let us formulate a theorem concerning the solvability of the system Eq. 9.
Proof. One can observe that the system of equations Eq. 9 can be converted into a matrix equation with the matrix [π α (t j )] (where α = 1, . . . , κ and j = 1, . . . , κ) multiplying the vector of the unknown quantities. The condition det[π α (t j )] = 0 is sufficient for the matrix to be invertible, which leads to the ability to solve the system and obtain the figures In general the matrix [π α (t j )] does not have to be square. Nonetheless, in the context of quantum tomography it does not seem sensible to consider a case with number of measurements higher than κ.
Due to repeated measurements of the same operator M k (over distinct copies of the system) we compute the In other words, repetition of one kind of measurement defined by the operator M k leads a set of probabilities which are in accordance with the Born's rule and POVM formalism.
The measurement procedure is performed for each operator from the initial set {M 1 , M 2 , . . . , M r }.
In order to be able to reconstruct the initial density matrix ρ(0) from the probabilities . . , r and α = 1, . . . , κ has to span L(H), i.e. the space to which ρ(0) belongs, and additionally: In other words, the set {U † α M k U α } where k = 1, . . . , r and α = 1, . . . , κ has to constitute an informationally complete POVM. Note that some elements of the set {U † α M k U α } may be redundant. Thus, in specific circumstances one can either select a sufficient number of operators which shall constitute an IC-POVM or decide to implement an over-complete set of measurement operators. The latter approach can be especially facilitative in a realistic scenario when the measured probabilities are burdened with errors [15,45].
The efficiency of the framework described in Sec. III depends on the analytical properties of the functions π α (t). Thus, lest us consider two specific examples which demonstrate the performance of the framework.

B. Example 1: qubit dephasing
Let us analyze a specific kind of qubit random unitary dynamics which is called dephasing. In this case we have: which allows us to express the dynamical map as where γ > 0 denotes a decoherence parameter. For a quantum system with evolution given by Eq. 14 we can formulate and prove a theorem.
Proof. First, one can quickly verify that M 1 , M 2 > 0, which means that initially we have two positive operators at our disposal that do not constitute an IC-POVM. Thus, single measurement of probability corresponding with each operator does not provide sufficient data for density matrix reconstruction. Therefore, we see the need for the dynamic approach.
If the probability associated with the operator M 1 is measured at two time instants, we get a system of equations according to Eq. 9: (16) One can compute: which holds true based on the assumption that t 1 = t 2 . This means that from the system Eq. 16 we can compute the figures Tr{M 1 ρ(0)} and Tr{(σ 3 M 1 σ 3 )ρ(0)} which according to the Born's rule Eq. 2 are equivalent to measurement probabilities of the operators M 1 and σ 3 M 1 σ 3 for the unknown state ρ(0).
In the same vein one can write a matrix equation for double measurement of probaility associated with the operator M 2 , which can be solved under the same condition Eq. 17 and leads to figures: Tr{M 2 ρ(0)} and Tr{(σ 3 M 2 σ 3 )ρ(0)}. Now one can observe that It is easy to verify that M 1 +M 1 + M 2 +M 2 = 1 2 , which means that it remains the check whether the operators satisfy the spanning condition form Def. 1. One can consider the following equation in order to investigate whether the operators are linearly independent: which can be equivalently expressed by means of a matrix equation: This matrix has a unique solution in the form a = b = c = d = 0, which implies that the operators As it was mentioned in Sec. II a given quantum state may be reconstructed from different IC-POVMs. Thus, the theorem 2 is not restricted to the proposed operators M 1 and M 2 . One may expect that for a different pair of positive operators, we could be able to conduct a similar reasoning.

C. Example 2: general RUD for qubits
Let us now consider a more general model of qubit dynamics such that the functions π α (t) from Eq. 12 have the forms π 0 (t) = 1 + e −γ1t + e −γ2t + e −γ3t 4 where γ 1 , γ 2 , γ 3 are positive decoherence rates such that γ 1 = γ 2 = γ 3 . We propose to formulate and prove the following theorem. Proof. First, one can notice that M 0 > 0, which means that this operators fits to the idea of generalized quantum measurement. Naturally, probability of one measurement Tr{M 0 ρ(0)} would not be sufficient to determine the unknown density matrix ρ(0). This justifies the need for the dynamic approach.
First, one can notice that since the functions π α (t) are linearly independent (which can be demonstrated by calculating the Wronskian), the matrix [Γ ij ] is invertible as long as t 1 = t 2 = t 3 = t 4 . This means that experimentally accessible probabilities p(t 1 ), p(t 2 ), p(t 3 ), p(t 4 ) allow us to compute the set of figures: Tr{M 0 ρ(0)}, Tr{(σ 1 M 0 σ 1 )ρ(0)}, Tr{(σ 2 M 0 σ 2 )ρ(0)}, Tr{(σ 3 M 0 σ 3 )ρ(0)}. Now one finds: from which it is easy to verify that M 0 +M 1 +M 2 +M 3 = 1 2 . It remains to prove that the set of operators spans L(H). Let us investigate an equation: a M 0 + bM 1 + cM 2 + dM 3 = 0 0 0 0 , which can be put into the matrix form: (25) One can verify that the determinant of the left-hand side matrix is non-zero and, for this reason, the equation has one unique solution a = b = c = d = 0, which means that the operators M 0 ,M 1 ,M 2 ,M 3 are linearly independent and consequently they span L(H). Since they satisfy all necessary conditions, the operators constitute an IC-POVM and can be used as a source of information for quantum state tomography, which finishes the proof.

D. Discussion and analysis
In the case of quantum tomography of qubits one traditionally needs to define 4 distinct operators in order to have an IC-POVM [32]. In practical applications these 4 operators are often defined by means of polarization measurements [12,13,46]. In the current article we have showed that, for qubits subject to dephasing, one can start with two positive operators and then dynamically generate an IC-POVM by double measurement of probabilities corresponding with these operators. This results suggest that we can reduce the amount of resource needed for quantum state tomography provided we can utilize the knowledge about system evolution.
The result concerning the second example proves that the effectiveness of the framework depends on the type of dynamics. For the qubit evolution given by the random unitary dynamics Eq. 12 with the probabilities {π α (t)} defined in Eq. 21 one positive operator is sufficient to generate an IC-POVM. Such kind of dynamics can be considered optimal in the context of state recovery. Practically, it means that one needs to prepare one experimental setup and then repeat the same kind of measurement at 4 distinct time instants. Such a procedure provides complete knowledge about the unknown system. This approach appears to be more convenient than preparing a number of distinct experimental setups. This result is in line with the current discoveries which indicate that quantum state tomography based on one type of measurement is feasible [30,31].
It should also be noted that the positive operators proposed in the theorems are not the only ones that suit to the framework. IC-POVM can be realized by different sets of operators. Thus, one may expect that in the case of the theorem 3 the optimal state reconstruction would be possible if the initial operator M 0 was different than Eq. 22. A natural question may concern the possibility to implement an operator from the SIC-POVM in the dynamic framework. For dim H = 2 the SIC-POVM is defined by 4 vectors (|0 , |1 denote the standard basis): and the operators are given by Π i = 1/2 |φ i φ i |.
It can be proved that none of the operators Π 1 , Π 2 can be substituted for M 0 from Eq. 22 since neither of the sets {Π i , σ 1 Π i σ 1 , σ 2 Π i σ 2 , σ 3 Π i σ 3 } for i = 1, 2 contains 4 linearly independent vectors. However, it turns out that either Π 3 or Π 4 can be implemented to generate an IC-POVM for a qubit subject to random unitary dynamics since we have: for j = 3, 4. This observation means that when the measurement potential is limited one can perform only one kind of measurement, associated with either Π 3 or Π 4 , at four distinct time instants in order to obtain complete information about the quantum state with dynamics defined by Eq. 21.

A. Preliminaries
Let us consider a qutrit subject to random unitary dynamics: where π kl (t) is defined: π 00 (t) := 1 + and {γ α } denotes 8 positive decoherence rates such that γ j = γ i for i = j. In addition, U kl stands for 3−dimensional Weyl matrices: where ω = e 2πi/3 and ω 2 = e −2πi/3 = ω † . A common practice in quantum tomography of qutrits involves applying a SIC-POVM which for dim H = 3 is characterized by means of the vectors [47]: where {|0 , |1 , |2 } denotes the standard basis in H. These vectors allow one to define the measurement operators: Π j i := 1/3 |ψ j i ψ j i | which satisfy all necessary conditions for a SIC-POVM. Thus, according to the standard approach if one wants to reconstruct the quantum state of a qutrit one needs 9 probabilities representing the outcomes of SIC-POVM.

B. Results: qutrits subject to random unitary dynamics
We can formulate and prove a theorem.
Theorem 4. Assuming that qutrits evolution is given by the random unitary dynamics Eq. 29 with the functions π kl (t) defined as Eq. 29, one can extract complete knowledge about the state ρ(0) from one operator Π j i := 1/3 |ψ j i ψ j i | provided its probability is measured at 9 distinct time instants.
Proof. Let us assume that we can prepare a measurement setup which gives us the probability associated with one elements from the SIC-POVM set: Π j i := 1/3 |ψ j i ψ j i |. The ensemble of quantum systems evolves and we perform 9 measurements at distinct time instants: {t 1 , . . . , t 9 }. The results of measurements, denoted by {p(t 1 ), . . . , p(t 9 )}, can mathematically be expressed by means of a matrix equation (cf. Eq. 23): . . .
First, one should notice that the matrix equation has a unique solution. Since the functions {π α (t)} are defined as linearly independent, which can be demonstrated by calculating the Wronskian, we have det[Γ βα ] = 0 as long as t 1 = t 2 = . . . = t 9 . Thus, from the matrix equation we can compute the figures Tr{U † kl Π j i U kl ρ(0)}, where (i, j) is fixed and k, l = 0, 1, 2. The values computable from the equation Eq. 32 can be understood as probabilities according to the Born's rule since U † kl Π j i U kl > 0 One can notice a connection between the Weyl matrices and the SIC-POVM operators: where ⊕ denotes addition modulo 3. This property means that U † 00 Π j i U 00 = Π j i and for any starting Π j i the other operations let us generate the remaing 8 operators from the SIC-POVM set. In other words, the set {U † kl Π j i U kl } where k, l = 0, 1, 2 is equivalent to the SIC-POVM for any starting operator Π j i , which finishes the proof.

C. Discussion and analysis
The standard approach to quantum state tomography of qutrits requires the ability to realize 9 distinct measurements associated with each operator from the SIC-POVM [48]. If one decides to reconstruct the quantum state of two entangled qutrits, then the number of measurement operators rises up to 81 [49]. In this section we have proved that the number of distinct operators can be reduced if one implements the dynamic approach to quantum measurement. The evolution model defined by Eq. 29 can be considered optimal since one measurement operator at our disposal is sufficient to generate the SIC-POVM. This can be achieved by the interdependence between the Weyl matrices and the SIC-POVM. Thus, we can start with only one element of the SIC-POVM and dynamically generate the remaining operators by multiple measurements. Although relations between the Weyl operators and SIC-POVMs have been studied [50], in this article we have proved that these algebraic properties can be applied to facilitate quantum state tomography.

VI. CONCLUSIONS
We have proposed a dynamic approach to quantum measurement which can enhance the efficiency of quantum state tomography. The framework can reduce the amount of resource since one can generate an IC-POVM (or SIC-POVM) starting from an incomplete set of measurement operators. Therefore, it offers a possibility to perform quantum state tomography with limited measurement potential. Alternatively, by generating additional measurements the framework can facilitate experimental state estimation, where there is a tendency to implement over-complete sets of measurement operators.
The results correspond well to current trends in quantum tomography which focus on reducing the number of distinct measurement setups needed for state reconstruction. A key difference between the present model and other proposals which rely on quantum dynamics is the fact that here we deal with quantum tomography within the POVM formalism, whereas other works utilize expectation values of Hermitian operators e.g. [19,27].
Furthermore, the present framework is not restricted to specific types of density matrices (e.g. low rank), unlike other proposals which aim at minimizing the number of measurement operators, cf. [21,22].
Further research is needed to answer remaining questions concerning the model. First of all, more types of quantum dynamics should be tested in terms of their efficiency in generating IC-POVMs. Additionally, the framework shall be applied to multilevel quantum systems (e.g. entangled qubits and qutrits). Finally, the ultimate goal is to experimentally verify the efficiency of the framework.