Quantum Tomography of Pure States with Projective Measurements Distorted by Experimental Noise

ACTA PHYSICA POLONICA A No. 2 Vol. 139 (2021)

A. Czerwinski∗

Institute of Physics, Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University, Grudziądzka 5, 87-100 Toruń, Poland

Received: 23.12.2020 & Accepted: 21.01.2021

Doi: 10.12693/APhysPolA.139.164 ∗e-mail: [email protected]

The article undertakes the problem of pure state estimation from projective measurements based on photon counting. Two generic frames for qubit tomography are considered — one composed of the elements of the SIC-POVM and the other defined by the vectors from the mutually unbiased bases. Both frames are combined with the method of least squares in order to reconstruct a sample of input qubits with imperfect measurements. The accuracy of each frame is quantified by the average fidelity and purity. The efficiency of the frames is compared and discussed. The method can be generalized to higher-dimensional states and transferred to other fields where the problem of complex vectors reconstruction appears. topics: quantum state tomography, mutually unbiased bases, complex vector reconstruction, phase retrieval

1. Introduction Moreover, |ψi is normalized such that hψ|ψi = 1, where h·|·i denotes the inner product in the Hilbert Quantum state tomography (QST) originated space. Then, the goal of QST is to reconstruct the in 1852, when G.G. Stokes derived the polariza- accurate representation of |ψi on the basis of data tion state of a light beam based on intensity mea- accessible from an experiment. Naturally, multi- surements [1]. The problem of state identifica- plying the state vector by a scalar of unit modulus tion remains relevant since well-characterized quan- does not change the measurement results. Thus, tum resources are required for quantum informa- the state vector can be determined up to a global tion processing [2] and quantum key distribution phase factor. (QKD) [3]. Apparently, there are many approaches In the case of pure states tomography, we are an- to QST which differ from one another in the kind alyzing the problem of recovering a complex vec- of measurement(s) and the number of their repeti- tor from intensity measurements — the very same tions [4, 5]. One popular method involves polariza- kind of a problem is considered in many other ar- tion measurements and can be applied to recovering eas of science, from pure mathematics to speech the quantum state of photons [6]. Other frameworks recognition or signal processing [14]. Thus, there is connected with photonic state tomography rely on vast literature concerning phase retrieval, see [15]. registering the Hong–Ou–Mandel interference [7, 8]. In particular, in recent years, a lot of attention A fundamental problem of QST has always con- has been paid to the connection between the com- cerned determining a set of measurement operators plex vector reconstruction and the theory of frames, sufficient to identify an unknown state [9]. Some e.g., [16, 17]. theoretical proposals aim at reducing the number of Let us recall a definition. By an M-element com- d operators to the necessary minimum [10, 11], while plex frame in C , denoted Ξ = {|ξ1i,..., |ξM i} d experimental frameworks tend to apply overcom- (where |ξii ∈ C ), one should understand a set plete sets of operators in order to overcome noise of complex vectors that span Cd. In articles not and errors [12, 13]. connected to quantum tomography, authors usually To clarify the problem investigated in the ar- consider in general the problem of reconstructing d ticle, let us postulate that the achievable infor- an unknown complex vector |xi ∈ C from its in- mation about a d-level pure quantum system is tensity measurements, i.e., it is discussed whether encoded in a complex vector — called the state the knowledge about the non-linear map 2 vector and denoted by |ψi which belongs to the JΞ : |xi → |hξi|xi| (1) Hilbert space H =∼ Cd, such that dim H = d < ∞. i=1,...,M

164 Quantum Tomography of Pure States. . . is sufficient to determine the complex vector |xi. Theorem I.1 states clearly the necessary and suf- In physics, this type of measurement is referred to ficient condition that needs to be satisfied so that as a projective measurement. the frame Ξ defines injective measurements and, To formulate a sufficient condition for complex therefore, it is possible to reconstruct a complex vector reconstruction, let us first revise general def- vector on the basis of the intensity measurements. initions involving the question when phase retrieval For a given frame, one can relatively easily verify is possible. In [18], the authors propose to as- whether the condition stated in Theorem I.1 is ful- sume that phase retrieval is possible when any two filled or not. However, so far there has been no vectors |ψi and |ψ0i with identical intensity mea- concrete proposal concerning a feasible procedure surements differ only by a scalar of norm one, i.e., to obtain such a sufficient frame. 0 |ψi = e i φ |ψ0i. In other words, the same postu- In this article, we investigate qubit state recon- late can be stated that it is possible to reconstruct struction by two frames which differ in the number a complex vector |ψi if and only if the non-linear of elements. One is defined by the elements from the symmetric, informationally complete, positive map JΞ is injective and Ξ is a frame. Thus, hence- forth, in the situations when the phase retrieval is operator-valued measure (SIC-POVM) [21] and the possible, we shall say that the frame Ξ generates other consists of the vectors from the MUBs [22]. (or defines) injective measurements. The accuracy of each frame is quantified by two figures of merit — the average fidelity and pu- In [19], Bandeiraa et al. postulated a conjecture rity, which are computed and presented on graphs. according to which if one wants to reconstruct a vec- d In Sect. 2, we present the framework of QST for tor |xi ∈ C , then a frame that contains less than pure states and assumptions concerning experimen- 4d − 4 vectors cannot generate injective intensity tal noise. Then, in Sect. 3, the results are intro- measurements, i.e., according to the authors fewer duced and discussed. The findings demonstrate how than 4d − 4 modulus of inner product of |xi with efficient the frames are at overcoming the experi- other vectors is not sufficient to obtain the struc- mental noise. ture of |xi. Furthermore, in the same paper the authors postulated the second part of the conjecture 2. State reconstruction framework that a generic frame with 4d − 4 vectors (or more) generates injective measurements on Cd. The sec- In this work, we assume that the initial state of ond part of the conjecture has been proved in [18], a qubit can be presented as a vector where the authors explained the notion of a generic θ cos 2 frame and demonstrated that for a generic frame Ξ |ψini = , (3) e i φ sin θ which contains at least 4d − 4 elements the corre- 2 sponding map J is injective. where 0 ≤ φ < 2π and 0 ≤ θ ≤ π. An un- Ξ known quantum state of the form (3) can be re- Another recent paper [20] proves a result that constructed from projective measurements, which contradicts the first part of the conjecture from [19]. in the case of photons are based on photon count- Vinzant proposed a frame in 4 which consists of C ing. Let us denote a frame by Ξ = {|ξ1i,... }, where 11 vectors and proved that it defines injective mea- |ξ i ∈ 2. Next, we take into account experimental 4 i C surements on C . Therefore, the current knowl- noise connected with each measurement. In partic- edge about the phase retrieval problem does not ular, we impose the Poisson noise which is a typical give an answer to the question what is the minimal kind of uncertainty arising in photon counting [23]. number of elements of the frame Ξ so that the map Thus, assuming that the total number of photons JΞ is injective. It remains unknown, in general, equals N , we shall introduce a formula for the mea- how many intensity measurements are required to sured photon count associated with the k-th projec- reconstruct an unknown complex vector. However, tive measurement in [19], it was proposed a relatively efficient way to nM = N |hξ |ψ i|2, (4) verify whether a given frame Ξ generates injective k k k in measurements. Their approach is presented below where Nk stands for a number generated randomly as a theorem. from the Poisson distribution characterized by the expected value N . Theorem I.1 (Bandeiraa et al. 2014). For a given frame Ξ , we can numerically generate d A frame Ξ = {|ξ1i,..., |ξM i} (where |ξii ∈ C ) experimental data for any specific input state (3) defines injective measurements, i.e., one can recon- (we shall consider a sample consisting of 400 input struct an unknown vector |xi ∈ Cd from intensity states). However, in a quantum state reconstruc- 2 measurements |hξi|xi| for i = 1,...,M, if and only tion problem, we postulate that there is no a priori if the linear space knowledge about an unknown state. For this rea- son, an output state, which results from the tomo- LΞ := (2) graphic algorithm, is assumed to be represented by n d×d o Q ∈ C : hξ1 Q ξ1i = ··· = hξM Q ξM i = 0 a density matrix [6, 24]: T †T does not contain any non-zero Hermitian matrix of ρout = (5) the rank ≤ 2. Tr (T †T )

165 A. Czerwinski with and purity defined as [28] ! t 0 γ := Tr ρ2 . (12) T = 1 (6) out t3 + it4 t2 The quantity F quantifies the overlap between which is equivalent to the Cholesky factorization. the actual state |ψini and the result of estimation In other words, the problem of state reconstruction ρout [29], whereas γ measures how close the recon- of qubits means that we strive to estimate the val- structed density matrix is to the pure state. In the case of both measurement scenarios, we perform ues of the four parameters t1, t2, t3, t4 which fully characterize the density matrix. Consequently, the QST with each frame for a sample of 400 qubits expected photon count for the k-th frame vector defined as (3), with φ and θ covering the full range. takes the form Then, the performance of the frames is discussed based on the average fidelity F and purity γ E av av nk = N Tr ξkihξk ρout . (7) computed over the sample. See also [27]. In order to estimate the parameters t1, t2, t3, t4, we shall apply the method of least squares (LS) [25]. 3. Results and discussion This method has been used to study the performance of QST frameworks with simulated measure- The goal of this section is to compare the perfor- ment results [26, 27]. According to the LS method, mance of two frames in QST of pure states. One can the minimum value of the following function needs utilize SIC-POVMs in order to reconstruct an un- to be determined: known quantum state [30]. For dim H = 2, we as- X E M 2 sume that {|0i, |1i} denotes the standard basis in H. fLS(t1, t2, t3, t4) = nk − nk = k Then, the SIC-POVM consists of four projectors de- X 22 fined by the vectors: N Tr |ξkihξk|ρout − Nk hξk|ψini , 1 r2 k (8) |ξSICi = |0i, |ξSICi = √ |0i + |1i, 1 2 3 3 which allows one to perform QST for any input state r |ψini and a given frame Ξ . SIC 1 2 i 2π |ξ i = √ |0i + e 3 |1i, Next, a scenario with extended measurement er- 3 3 3 rors will be considered. Apart from the Poisson r SIC 1 2 i 4π noise, dark counts shall be taken into account. |ξ i = √ |0i + e 3 |1i. (13) In practice, it means that the detector receives not 4 3 3 only the desirable signal but also some number of Mathematically speaking, the vectors (13) consti- photons which come from the background. Math- tute a frame which defines injective measurements ematically speaking, the background noise shall be according to Theorem I.1. The frame comprising modeled by adding to the input state a component the SIC-POVM shall be denoted by Ξ SIC. proportional to the maximally mixed state (pertur- The other frame, denoted by Ξ MUB, consists of bation term), i.e., six vectors which correspond to the elements of the MUB for, i.e.: ρin = (1 − ) ψinihψin + 112, (9) 2 ! ! MUB 1 MUB 0 where 112 denotes the 2 × 2 identity matrix and |ξ1 i = |ξ2 i = , shall be referred to as the noise parameter (nat- 0 1 urally: 0 ≤ ≤ 1). This gives that a modified ! ! formula for the measured photon counts MUB 1 1 MUB 1 1 |ξ3 i = √ , |ξ4 i = √ , M 0 2 1 2 −1 nk = NkTr |ξkihξk|ρin = ! ! 2 1 1 1 1 (1 − ) Nk hξk|ψini + Nk , (10) |ξMUBi = √ , |ξMUBi = √ . 2 5 2 i 6 2 −i which can be substituted into the function (8) in order to estimate the state in the other scenario. (14) MUB We shall compare the quality of qubit estimation The frame Ξ also generates injective mea- with two frames: one defined by the symmetric, surements. From the physical point of view, in- informationally complete, positive operator-valued tensity measurements associated with the frame MUB measure (SIC-POVM) and the other by the mu- Ξ can be realized on photons through polar- tually unbiased bases (MUBs). Firstly, the sce- ization measurements since the vectors are com- nario with the Poisson noise alone will be consid- monly used to represent: vertical/horizontal, di- ered. In the next step, the perturbation term will agonal/antidiagonal, right/left circular polarization be added to the input states. To evaluate the effi- states, respectively, e.g., [31]. ciency of the frames, we utilize the notion of quan- In order to investigate the efficiency of each frame tum fidelity, given by in pure state reconstruction, numerical simulations were conducted, assuming a different number of q√ √ 2 F := Tr ρout|ψinihψin| ρout , (11) photons involved in measurements. A sample of 400 input states of the form (3) was considered and each

166 Quantum Tomography of Pure States. . .

TABLE I TABLE II

Average ﬁdelity Fav(N ) and purity γav(N ) in pure Average ﬁdelity Fav() in pure state estimation with state estimation with two distinct frames. The two distinct frames. Each value was computed as method of least squares was applied with the func- the mean for a sample of 400 input qubits of the tion (8). Each value was computed as the mean for form (9). For the measured photon counts (10) one a sample of 400 input qubits of the form (3). applies N = 10.

Frame Frame N Ξ MUB Ξ SIC 0.1 0.2 0.3 0.4 0.5 MUB Fav(N ) γav(N ) Fav(N ) γav(N ) Ξ 0.8960 0.8632 0.8231 0.7867 0.7476 1 0.7714 0.9142 0.7313 0.9036 Ξ SIC 0.8919 0.8584 0.8298 0.7900 0.7457 5 0.9036 0.9251 0.8788 0.9104 10 0.9334 0.9354 0.9080 0.9128 25 0.9564 0.9406 0.9461 0.9368 50 0.9721 0.9597 0.9655 0.9550 100 0.9793 0.9679 0.9761 0.9652 1000 0.9940 0.9890 0.9925 0.9864 10000 0.9981 0.9964 0.9979 0.9959 state was reconstructed with the measured photon counts distorted by the Poisson noise (4). The results are gathered in Table I. It was expected that the impact of the Poisson noise should be greater if we utilize fewer photons per measurement. Thus, we can observe that the accuracy of both frames in QST increases along with the number of photons. For N = 10 000, both frames lead exactly to the unknown state. How- ever, if the number of photons decreases, one can observe a substantial discrepancy between the state obtained from the algorithm ρout and the original state |ψini. It is worth noting that, for smaller num- MUB bers of photons, Ξ results in better quality of Fig. 1. Plots present the average fidelity Fav() (a) SIC pure state estimation than Ξ . This feature is in and the purity γav() (b). Each point was obtained agreement with the common practice in QST to em- by the method of least squares for a sample of 400 ploy overcomplete sets of measurement operators in input states including a noise parameter (9). For- order to combat experimental noise. When we in- mula (10) for the measured photon counts was ap- crease the number of photons, the figures of merit plied, assuming that N = 1000. for both frames converge. A further insight into the efficiency of the frames Finally, let us investigate the difference between can be provided by investigating input states in- the frames in the case of few photons and the pres- fluenced by the error parameter , as in (9). Let ence of dark counts (9). If we reduce the number us assume that the number of photons is fixed: of photons to N = 10, then from Table II we ob- N = 1 000. Then, for each frame, we can consider serve that Ξ MUB has no significant advantage over the average fidelity and purity as functions of , de- Ξ SIC for non-zero values of the error parameter. Al- noted by Fav() and γav(), respectively. The plots though there are tiny differences between the fig- of the functions are presented in Fig. 1. ures in Fig. 1, they should be considered negligible. For the number of photons N = 1000, the results These results prove that for noisy measurements, demonstrate that if we consider the input states distorted by both the Poisson noise and dark counts, distorted by the perturbation term (9) (while still Ξ SIC and Ξ MUB deliver the same quality. keeping the Poisson noise), then both frames lead to a very similar accuracy. The whole range of the 4. Summary and outlook noise parameter was considered and there is no significant difference between the efficiency of the In this article, two frames have been compared in frames. One can notice that the function Fav() is terms of their applicability in quantum tomography SIC linear, whereas γav() is convex. The plots allow one of qubits. One frame, Ξ , comprised the elements to observe how the quality of state estimation de- of the SIC-POVM and the other, Ξ MUB, contained generates as we increase the amount of noise (dark the vectors from MUBs. Based on the numeri- counts). cal simulations, we have demonstrated that Ξ MUB

167 A. Czerwinski outperforms Ξ SIC only if we consider a measure- [10] A. Czerwinski, J. Phys. A Math. Theor. ment scenario which involves single-photon count- 49, 075301 (2016). ing along with the Poisson noise as the only source [11] A. Czerwinski, Int. J. Theor. Phys. 59, of experimental uncertainty. Then, the overcom- 3646 (2020). plete frame Ξ MUB has an advantage over the min- SIC [12] R.T. Horn, P. Kolenderski, D. Kang et al., imal frame Ξ . However, the improvement in Sci. Rep. 3, 2314 (2013). quality due to the overcomplete frame appears to be rather moderate. [13] R. Horn, T. Jennewein, Opt. Express 27, Furthermore, if we increase the number of pho- 17369 (2019). tons involved in measurements or include dark [14] K. Jaganathan, Y.C. Eldar, B. Hassibi, counts as another source of experimental noise, both “Phase Retrieval: An Overview of Recent frames deliver the same quality of pure state esti- Developments”, in: Optical Compressive mation. In such cases, the four-element minimal Imaging, Ed. A. Stern, CRC Press, Boca frame Ξ SIC which consists of the vectors generating Raton 2016, p. 263. the SIC-POVM, fully suffices for QST of qubits. [15] P.G. Casazza, L.M. Woodland, Contemp. In conclusion, one can agree that an overcomplete Math. 626, 1 (2014). set of measurement operators is advisable only if we [16] R. Balan, P.G. Casazza, D. Edidin, Appl. consider experiments with single photons and the Comput. Harmon. Anal. 20, 345 (2006). perturbation term can be neglected. This outcome [17] A. Jamiolkowski, J. Phys. Conf. Ser. 213, is in line with other results which demonstrate that 012002 (2010). overcomplete measurements can improve the efficiency of QST frameworks [32]. [18] A. Conca, D. Edidin, M. Hering, C. Vin- In the future, a similar approach can be applied zant, Appl. Comput. Harmon. Anal. 38, to investigate the accuracy of different frames in 346 (2015). higher-dimensional cases. In particular, the prob- [19] A.S. Bandeiraa, J. Cahill, D.G. Mixon, lem of reconstructing four-dimensional complex vec- A.A. Nelson, Appl. Comput. Harmon. tors shall be studied since this case includes entan- Anal. 37, 106 (2014). gled photons. [20] C. Vinzant, in: Proc. Int. Conf. on Sampling Theory, Applications (SampTA), Acknowledgments 2015, p. 197 (2015). The author acknowledges financial support from [21] J.M. Renes, R. Blume-Kohout, A.J. Scott, the Foundation for Polish Science (FNP) (the First C.M. Caves, J. Math. Phys. 45, 2171 Team programme co-financed by the European (2004). Union under the European Regional Development [22] W.K. Wootters, B.D. Fields, Ann. Phys. Fund). 191, 363 (1989). References [23] S.W. Hasinoff, in: Computer Vision, Ed. K. Ikeuchi, Springer, Boston (MA) [1] G.G. Stokes, Trans. Cambridge Philos. 2014, p. 608. Soc. 9, 399 (1852). [24] J. Altepeter, E. Jerey, P. Kwiat, Adv. At. [2] M. Kuś, Acta Phys. Pol. A 100, 43 (2001). Mol. Opt. Phys. 52, 105 (2005). [3] A. Beige, B. Englert, Ch. Kurtsiefer, [25] T. Opatrny, D.-G. Welsch, W. Vogel, Phys. H. Weinfurter, Acta Phys. Pol. A 101, Rev. A 56, 1788 (1997). 357 (2002). [26] A. Acharya, T. Kypraios, M. Guţă, [4] G.M. D’Ariano, M.G.A. Paris, M.F. Sac- J. Phys. A Math. Theor. 52, 234001 chi, Adv. Imag. Electron Phys. 128, 205 (2019). (2003). [27] K. Sedziak-Kacprowicz, A. Czerwinski, [5] Quantum State Estimation (Lecture Notes P. Kolenderski, Phys. Rev. A 102, 052420 in Physics), Eds. M.G.A. Paris, J. Ře- (2020). háček, Springer, Berlin 2004. [28] M.A. Nielsen, I.L. Chuang, Quantum Com- [6] D.F.V. James, P.G. Kwiat, W.J. Munro, putation and Quantum Information, Cam- A.G. White, Phys. Rev. A 64, 052312 bridge University Press, Cambridge 2000. (2001). [29] R. Jozsa, J. Mod. Opt. 41, 2315 (1994). [7] W. Wasilewski, P. Kolenderski, [30] J. Řeháček, B.-G. Englert, D. Kasz- R. Frankowski, Phys. Rev. Lett. 99, likowski, Phys. Rev. A 70, 052321 (2004). 123601 (2007). [31] O. Bayraktar, M. Swillo, C. Canalias, [8] P. Kolenderski, W. Wasilewski, Phys. Rev. G. Bjork, Phys. Rev. A 94, 020105(R) A 80, 015801 (2009). (2016). [9] G.M. D’Ariano, L. Maccone, M.G.A. Paris, [32] H. Zhu, Phys. Rev. A 90, 012115 (2014). J. Phys. A Math. Gen. 34, 93 (2001).

168