Max-Planck-Institut

fur Mathematik

in den Naturwissenschaften

Leipzig

Experimental Test of the Relation

between Coherence and Path

Information

byJun Gao, Zi-Qiang Jiao, Cheng-Qiu Hu,

Lu-Feng Qiao, Ruo-Jing Ren, Hao Tang, Zhi-Hao Ma,

Shao-Ming Fei, Vlatko Vedral, and Xian-Min Jin

Preprint no.: 9 2019

Experimental Test of the Relation between Coherence and Path Information

Jun Gao,1,2 Zhi-Qiang Jiao,1,2 Cheng-Qiu Hu,1,2 Lu-Feng Qiao,1,2 Ruo-Jing Ren,1,2

Hao Tang,1,2 Zhi-Hao Ma,3 Shao-Ming Fei,4 Vlatko Vedral,5,6 and Xian-Min Jin1,2,∗1State Key Laboratory of Advanced Optical Communication Systems and Networks,

Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China2Synergetic Innovation Centre of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China

3Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China4School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

5Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore and6Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK

(Dated: November 4, 2018)

Quantum coherence stemming from the su-perposition behaviour of a particle beyondthe classical realm, serves as one of the mostfundamental features in quantum mechanics.The wave-particle duality phenomenon, whichshares the same origin, has a strong relationshipwith quantum coherence. Recently, an elegantrelation between quantum coherence and pathinformation has been theoretically derived.Here, we experimentally test such new dualityby l1-norm measure and the minimum-errorstate discrimination. We prepare three classes oftwo-photon states encoded in polarisation degreeof freedom, with one photon serving as thetarget and the other photon as the detector. Weobserve that wave-particle-like complementarityand Bagan’s equality, defined by the duality re-lation between coherence and path information,is well satisfied. Our results may shed new lighton the original nature of wave-particle dualityand on the applications of quantum coherence asa fundamental resource in quantum technologies.

Introduction

Coherence was recognized early as a superposition of op-tical fields in the theory of electromagnetic waves. To-gether with the energy quantization, a quantum versionof coherence has become one of the most fundamental fea-tures that can mark the departure of quantum mechanicsfrom the classical realm [1–5]. Carrying out general quan-tum operations remotely under local operations and clas-sical communication requires quantum states that con-tain consumable resources. Quantum entanglement isfound to have strong connection with coherence [6–8] andmay even originate from it [9].The development of quan-tum technologies demands a reassessment of fundamental

∗ xianmin.jin@sjtu.edu.cn

resources such as quantum coherence, including relation-s with other quantum physical phenomena and rigorousquantitative description [10–17].

Among these quantum physical phenomena, wave-particle duality has been a unified picture to fully de-scribe the behavior of quantum-scale objects. Quantita-tive characterizations of such wave-particle duality rela-tions have been extensively investigated, aiming to setan upper bound on the sum of the wave behavior andthe particle behavior for a given interferometer [3, 4].Based on the fringe visibility V of the interference pat-tern and the path distinguishability D, a wave-particleduality relation is given by V 2 + D2 ≤ 1, which meansthat full wave-behavior (V = 1) implies no particle be-havior (D = 0) and vice-versa.

Recently, Bagan et.al [18] proposed another elegant re-lation to characterise the new wave-particle-like dualitybased on the coherence Cl1 which quantifies the wave na-ture of a state and the path information which is givenby the minimum-error state discrimination between thedetector state and the target state. In this paper, weexperimentally test such new duality relation by l1-normmeasure and the minimum-error state discrimination. Bypreparing three classes of two-photon states that have d-ifferent upper bounds of quantum coherence, we are ableto investigate the new wave-particle-like complementari-ty in different regions. In every class of state, we contin-uously tune the detector state and observe clear dualitytrade-off as well as the upper bound on the quantum co-herence of the target photon. The Bagan’s equality de-fined by the duality relation between coherence and pathinformation can be well satisfied.

Results

Theoretical description of the duality relation.Consider a particle (target state) entering an N-port in-terferometer via a generalized beam splitter. Once theparticle interacts with the detector (detector state), the

2

FIG. 1. Schematic view of experiment. a. Experimental setup. Pairs of polarisation entangled photons are generatedvia spontaneous parametric downconversion in a 25mm long periodically-poled Potassium Titanyl Phosphate (PPKTP) crystalpumped by a 405nm UV laser with Sagnac loop scheme. After locally rotating the polarisation of one path, the photons areguided into the state generation section. By either inserting different pieces of Brewster window or place a polarisation beamsplitter (PBS), three different classes of states {ψI

i }, {ψIIi } and {ψIII

i } are generated. The states are then analysed by thepolarisation analysis measurement setup composed of a half-wave plate (HWP), a quarter-wave plate (QWP) and a PBS. b.The Brewster window characterization. Transmission rate of fused silica is characterised with different incidence angles andpolarisations. The results suggest 60◦ as a reasonable incidence angle.

state of the entire system is described by

|ψ〉 =

N∑i=1

√pi|i〉|ηi〉, (1)

which is the superposition of the initial target parti-cle state |i〉 and the detector state |ηi〉. The coher-ence of the target state is given by C = Cl1(ρ)/N ,

where ρ = Trdet(|ψ〉〈ψ|) =∑N

i,j=1

√pipj〈ηj |ηi〉|i〉〈j|.

Cl1(ρ) =∑

i6=j |ρij |, where |ρij | is the absolute valueof entry ρij of the density matrix ρ under the referencebasis[10]. Here, C quantifies the wave nature of the tar-get particle.

The detector states are introduced to quantify the pathinformation of the target particle by tracing out the tar-

get particle state, ρdet = Trtar(|ψ〉〈ψ|) =∑N

i=1 piρi,where ρi = |ηi〉〈ηj |. If the detector states ρi are orthog-onal, no coherence property of the target particle will beobtained. To discriminate among the detector states |ηi〉,one employs the minimum-error strategy by using an N -element positive operator valued measure (POVM) withelements {Πi}. Then the average probability of success-

fully identifying the state is Ps =∑N

i=1 pi〈ηj |Πi|ηi〉. It isproved that C and Ps satisfy the following new relationraised by Ref.[18],

(Ps −1

N)2 + C2 ≤ (1− 1

N)2. (2)

This upper bound represents a trade-off between the path

information and the coherence of the target particle. Inour experiment, we consider the case of N = 2. It isworth mentioning that when N = 2, the inequality be-comes an equality which we denote as Bagan’s equali-ty. In this case, Ps can be maximised either by an opti-mised POVM or calculated by an analytic solution[19]. Itshould be noticed that this new definition of path infor-mation is different from Ref.[3]. Here, we use the successprobability in minimum-error state discrimination, whileRef.[3] uses the difference of two probabilities for takingeither one of the two paths.

Experimental test of the duality relation.We ex-perimentally generate the two-photon polarisation en-tangled state via type-II spontaneous parametric down-conversion[20]. As shown in Fig.1a, a 405nm UV laser isfirst coupled into a single mode fiber to acquire a highquality spatial beam profile. After adjusting the polarisa-tion of the pump laser with a combination of a half-waveplate (HWP) and a quarter-wave plate (QWP), the pumplaser enters the Sagnac interferometer and is focused on aperiodically-poled Potassium Titanyl Phosphate (PPK-TP) crystal. The clockwise and the anti-clockwise com-ponents then interfere at the central polarisation beamsplitter (PBS) and generate a superposition state of thesetwo components. The generated singlet state can be writ-ten as |ψ−〉 = 1√

2(|H1V2〉 − |V1H2〉). Two band pass

filters (BPF) centred at the desired wavelength of thedown-converted photons are employed to block the UVlaser. With careful alignment of the Sagnac interferom-

3

FIG. 2. Experimental observation of the new wave-particle-like duality with three different classes of states.From a to c, in each class of states, as we vary the parameter ζ, we experimentally observe this new relation trade-off, namely, acontinuous transition between quantum coherence and path information. The blue diamonds represent the coherence propertieswhile the red dots give the path information. As we vary the states from {ψI

i } to {ψIIIi }, the degree of complementarity also

increases. The solid curves are the theoretical results based on the three different classes of states. Error bars are calculatedwith Monte Carlo simulation with the Poissonian statistics of the detection process taken into account.

eter, we are able to observe the polarisation visibilityas high as 97.7% in H/V basis and 98.3% in D/A basis.Then we further characterise the entanglement quality byconducting state tomography[21]. The measured purityand concurrence of the entangled state is 0.963± 0.0016and 0.964 ± 0.0016 respectively, which suggests that wehave obtained a very pure state with high entanglementquality.

In order to observe the coherence properties of thetarget states, the detector states ηi should be non-orthogonal. In our experiment, we first use a HWP togenerate the following two-qubit state with four compo-nents,

|ψ〉 = α|H1H2〉+ β|H1V2〉+ γ|V1H2〉+ δ|V1V2〉, (3)

where α, β, γ and δ are parameters based on the rotationin Hilbert Space. Note that these parameters are highlycorrelated and can be determined by the rotation angle ofHWP and the normalization condition. Since |ψ−〉 stateis invariant under arbitrary unitary rotation in Hilbert s-pace, polarisation-dependent loss should be additionallyintroduced to generate non-orthogonal basis of the de-tector states. Here, we choose fused silica as Brewsterwindow to introduce this polarisation-dependent loss in-to the above four components.

We experimentally characterise the transmission rateof fused silica with different incidence angles and theresults agree well with the theoretical prediction (seeFig.1b). We choose 60◦ as the incidence angle wherethe transmission rate is 99.7% and 71.9% for horizontaland vertical polarisation respectively (denoted as εh andεv). After passing through this state preparation sectionthe states can be described as

|ψ〉 =|Ht〉(αεnh|Hd〉+ βεnv |Vd〉)+|Vt〉(γεnh|Hd〉+ δεnv |Vd〉)

(4)

where index n represents the number of Brewster win-dows we insert, and subscripts t and d represent the tar-get and detector qubits respectively. We can see that the

detector qubit state (αεnh|Hd〉+ βεnv |Vd〉) is not orthogo-nal to (γεnh|Hd〉 + δεnv |Vd〉), and their inner product canbe tuned with the rotation angle of HWP and the num-ber of Brewster window. We are therefore able to testBagan’s theory in a wide range, a transition from the caseof minimal coherence and maximal path information tothe opposite case.

We then define a simplified parameter ζ as

ζ =

√α2εn2h + β2εn2vγ2εn2h + δ2εn2v

. (5)

From this expression we can see that parameter ζ is deter-mined by the rotation angle of the HWP and the trans-mission rates for different polarisations. Once we havefixed the rotation angle and the number of Brewster win-dow, the state we generated is also determined. To bemore specific, εnh and εnv can set the upper bound on thequantum coherence of the states, and the rotation angleof the HWP (tune α, β, γ and δ) can affect the dualitytrade off. As shown in Fig.1a, by inserting 4 and 6 piecesof Brewster window (affect εnh and εnv ), we are able togenerate two different classes of states which we denoteas {ψI

i } and {ψIIi } respectively. Here subscript “i” in the

states corresponds to different HWP angles. Once εnh andεnv are fixed, then by tuning the angles of the HWP, weare able to observe the complementarity of this new dual-ity relation. As we change the pieces of Brewster windowfrom 4 to 6, the concurrence of the entangled state de-creases from 0.795 ± 0.0039 to 0.650 ± 0.0062 while thepurity of the states remains at a high level, 0.962±0.0035and 0.965± 0.0041 respectively. For each class of states,while we change the parameter ζ (by tuning different H-WP angles), the entanglement and purity are not affect-ed. This can be easily understood since local operationsdo not destroy the entanglement. For an asymptotic lim-it case, we use a PBS to replace the Brewster windowto generate a separable state ({ψIII

i }). With {ψIIIi }, we

can observe full complementarity between quantum co-herence and path information. It should be noticed that

4

FIG. 3. Direct measured results of path information Pand the comparison result. a. Coherence of l1-norm mea-sure and path information given by direct optimised positiveoperator valued measure (POVM) results of the third classstate {ψIII

i }. The solid curves are the theoretical results. b.A comparison of Ps result between the analytic solutions andoptimised POVM. Error bars in Fig.3 are directly calculatedby using the propagation of error formulas.

the three classes of states are not the entire states of thesystem but part of states retrieved in a smaller subspacewith postselection. Such solution has been proven to beexactly equivalent to the ideal one by Sandu Popescu,Lucien Hardy, and Marek Zukowski [22, 23].

These generated states are then analysed by quan-tum state tomography with maximum likelihoodreconstruction[21]. From the tomographic details, we areable to calculate both C and P = Ps− 1

N to estimate thewave and particle nature of the target qubit. We measurethis coherence-path information duality complementari-ty with all three classes of states and plot the results inFig.2. Error bars in Fig.2 are calculated with Monte Car-lo simulation with the Poissonian statistics of the detec-tion process taken into account. For all the three classesof states from Fig.2a to Fig.2c, as we tune the parameterζ, we are able to observe a continuous transition betweenquantum coherence and path information.

We can see that the measured C and P follow the the-oretical curve with some small deviations. From the ana-lytical view, the entanglement quality is reflected by theoff-diagonal term of the two-qubit state density matrix.In practice, the entanglement is prepared to approach theideal value of 0.5. We achieve 0.49 in our experiment. Af-

FIG. 4. The Bagan’s equality defined by the duali-ty relation between coherence and path information.Sum of C square and P square of all generated states are list-ed and compared with the theoretical bound 0.25, where theparts with grey (pink) background are the results obtainedwith state tomography (optimised positive operator valuedmeasure (POVM)). All the states obey this new duality re-lation and Bagan’s equality is well satisfied with acceptableerrors.

ter tracing out the detector qubit, the quantum coherencequantity of the target qubit will reveal the imperfectionwith a small deviation. Another thing we should mentionis that the entanglement quality do affect the degree ofcomplementarity between quantum coherence and pathinformation. From Fig.2a to Fig.2c, the three classes ofstates show apparent difference in degree of complemen-tarity.

5

Apart from direct calculation of P value from the to-mographic data, we also use optimised POVM to directlymeasure the maximum probability of successfully identi-fying the state with {ψIII

i }. We theoretically construct-ed the projective operators according to the formula inRef.[24]. The results are given in Fig.3a and the mea-sured results agree with the theoretical prediction verywell. Error bars in Fig.3a are directly calculated by us-ing the propagation of error formulas. We have also com-pared the mismatch between the P values in Fig.2c andFig.3a, as shown in Fig.3b. Except for the result of ψIII

4 ,the distances between the two different results are nearlyinvisible, which implies that the two methods are equiv-alent in principle.

Finally, we test the upper bound of this new dualityrelation by summing the square of both C and P withour experimental results. As shown in Fig.4, all the mea-sured results meet the Bagan’s equality with acceptableerrors. Each C and P error is estimated via 1, 000 roundsof Monte Carlo simulation based on photon number de-tection governed by Possionian statistics. Considerableerror bars come from the uncertainty when we measurerather small observables. All these experiment resultsfaithfully prove the validation of this new duality rela-tion between quantum coherence and path information.

Discussion

We have experimentally tested the recently derived d-uality relation between quantum coherence and path in-formation with polarisation encoded entangled two-qubitstate. We propose and demonstrate a new way to gener-ate different classes of partial entangled states and there-fore can explore Bagan’s equality for the case of N = 2under different conditions. Note that it is impossible toobtain the same results by using classical light in ourscheme and our setup. We become aware that Yuan etal. also test Bagan’s equality with heralded single pho-ton state[25]. A classical light field (no matter whether aweak laser or thermal light) applied on Yuan et al.’s setupwill generate exactly the same results. In our experimen-t, we design and prepare an exotic two-photon entangledstate specifically for rigorously testing Bagan’s theory inthe quantum regime.

Furthermore, our results show a potential connectionbetween the two quantum resources of coherence and en-tanglement. Although Bagan’s theory doesn’t requireentanglement, with different class of partial entangledquantum states, we are able to test Bagan’s duality trade-off with different degree of complementary. Since the re-

lation between coherence and entanglement remains animportant and open topic[9], our scheme and setup mayserve as a new platform to simultaneously investigate theroles and their relations to coherence, path informationand entanglement.

Last but not least, our scheme can be extended tolarger N for testing multi-path Bagan’s theory and forother quantum tasks. The key operation in generatingour exotic two-photon entangled states is to introducepolarisation-dependent differential transmission rates oneach arm of a Bell state. Such operation can be also ap-plied onto other multi-photon states, such as GHZ states,cluster states and even multi-photon mixed states. Espe-cially applying our operation onto GHZ states will enablea genuine test of multi-path Bagan’s theory, which will gobeyond equality and test the duality between coherenceand path information in an inequality fashion. Therehave been demonstrations of high fidelity multi-photonentanglement state[26, 27], thus it is experimentally fea-sible to generate a multi-path exotic state of our type totest Bagan’s theory and to explore the relation betweencoherence, path information and multi-partite entangle-ment, even though much technical effort still remains tobe made.

The wave-particle duality relations are of significancein quantum mechanics, while the quantum coherenceplays a central role in quantum information processing.Our results may inspire further theoretical and exper-imental investigations on such fundamental researchesand strengthen the prominent role of coherence in quan-tum physics and quantum technologies.

Methods

Experiments. Entangled photon pairs are gener-ated by using type-II spontaneous parametric down-conversion. A 5mW 405nm laser diode pumps a25mm periodically-poled KTiOPO4 (PPKTP) crystal ina Sagnac interferometer. The beam spot size at the cen-ter of the crystal is 220µm. The degenerate downconver-sion photon pairs at 810nm are filtered by 3nm bandpassfilter, then coupled into single mode fibers and guidedto the state preparation stage. The coincidence rate isabout 30 kHz. By either inserting Brewster windows orplace a PBS, three different classes of states can be gen-erated. Photons are analysed by quantum state tomog-raphy and detected by single-photon counting modules(SPCM).

Data availability. The data that support the findingsof this study are available from the corresponding authoron reasonable request.

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Acknowledgments The authors thank J.-W. Pan forhelpful discussions. This research leading to the result-s reported here was supported by the National NaturalScience Foundation of China under Grant No.11374211and No.11675113, the Innovation Program of Shang-hai Municipal Education Commission (No.14ZZ020),Shanghai Science and Technology Development Fund-s (No.15QA1402200), and the open fund from HPCL(No.201511-01). X.-M.J. acknowledges support from theNational Young 1000 Talents Plan. V. V thanks the Ox-ford Martin School, Wolfson College and the Universityof Oxford, the Leverhulme Trust (UK), the John Tem-pleton Foundation, the EU Collaborative Project Ther-MiQ (Grant Agree- 5 ment 618074), the COST ActionMP1209, the EPSRC (UK) and the Ministry of Manpow-er (Singapore). This research is also supported by theNational Research Foundation, Prime Ministers Office,Singapore, under its Competitive Research Programme(CRP Award No. NRF- CRP14-2014-02) and admin-istered by Centre for Quantum Technologies, NationalUniversity of Singapore.

AUTHOR CONTRIBUTIONS

X.-M.J. conceived the project and designed the exper-iment. Z.-H.M., S.-M.F. and V.V. develop the theory.J.G., Z.-Q. J., C.-.H., L.-F.Q, R.-J.R., H.T. and X.-M.J.performed the experiment. J.G., X.-M.J., Z.-H.M. andS.-M.F. analyses the data and wrote the paper.

COMPETING INTERESTS

The authors declare that they have no competing in-terests.