Exploring entanglement with the help of quantum state measurement

Transcription

1 Explorng entangleent wth the help of quantu state easureent E. Dederck and M. Beck a) Departent of Physcs, Whtan College, Walla Walla, Washngton 9936 (Receved 15 August 013; accepted June 014) We have perfored a seres of experents usng a spontaneous paraetrc down-converson source to produce pars of photons n ether entangled or non-entangled polarzaton states. We deterne the full quantu echancal polarzaton state of one photon, condtoned on the results of easureents perfored on the other photon. For non-entangled states, we fnd that the easured state of one photon s ndependent of easureents perfored on the other. However, for entangled states, the easured state does depend on the results of easureents perfored on the other photon. Ths s possble because of the nonlocal nature of entangled states. These experents are sutable for an undergraduate teachng laboratory. VC 014 Aercan Assocaton of Physcs Teachers. [ I. INTRODUCTION In quantu echancs, the nforaton about a quantu syste s contaned n ts state. For a syste prepared n a pure state (a state that s the sae on every tral) the state ay be descrbed by a state vector, or ket, jw. However, n real experents, t s usually the case that the state preparaton procedure s not perfect, so the syste s not always prepared n the sae pure state jw on each tral. In such cases, the syste s sad to be n a xed state, whch can be descrbed n ters of a densty operator ^q. 1 For systes wth a dscrete bass, the densty operator ay be represented as a atrx, often referred to as the densty atrx. The state of a quantu syste encopasses everythng that s knowable about that syste. In ths sense, the state represents ultate knowledge about the syste. If one s able to easure or otherwse deterne the state of a syste, the known state can be used to calculate any quantty of nterest. For ths reason, quantu state easureent has becoe an portant tool for physcsts studyng quantu nforaton. 5 Quantu state easureent s often referred to as a quantu state toography, because the orgnal algorths used to deterne the state were the sae as those used n toographc agng. 6 Classcal partcles ay be correlated n a anner that allows easureents n one locaton to deterne results of easureents n another locaton. For classcal partcles, however, perfect correlatons can exst only n a sngle easureent bass. If easureents are perfored n a dfferent bass the classcal correlatons are reduced. For exaple, f classcal polarzaton easureents are perfectly correlated n the horzontal-vertcal bass, they wll not be correlated n the crcular-polarzaton bass. Quantu partcles can be placed n entangled states that have correlatons that are stronger than those allowed by classcal physcs (for entangled partcles perfect correlatons can rean for easureents perfored n any bass). Entangleent s what leads to unquely quantu echancal phenoena such as volatons of local reals, quantu teleportaton, and quantu coputng. 1,7 Undergraduate experents nvolvng entangled partcles have been prevously reported. 1,8 1 Results of these experents ply that easureents perfored on one partcle can change the state of another partcle, even f the partcles are physcally separated. It s ths property of entangled partcles that so begules physcsts, and that led Ensten to refer to spooky actons at a dstance. 13 However, n these prevous experents, the results are often subtle and not fully apprecated by non-experts. In order to unequvocally deonstrate that easureents perfored on one partcle can actually change the state of the other, t s best to explctly easure the state. Here, we are able to accoplsh ths by easurng the quantu state of one photon, condtoned upon the results of easureents perfored on another photon. II. THEORY Before dscussng the experents, we wll present the theory behnd the. We begn by descrbng the polarzaton states we use n the experents. Then we provde soe background on the densty operator, because ths s what we deterne n the experents. In partcular, we show how the results of easureents affect the densty operator and how one obtans the densty operator of a sngle partcle fro the densty operator of a two-partcle syste. Fnally, we present the theory of quantu state easureent. A. Polarzaton states The polarzaton state of a sngle photon can be wrtten n ters of bass states correspondng to horzontal jh and vertcal jv polarzatons. A general ellptcal polarzaton state s gven by je ¼ajHþbe / jv; (1) where a, b, and / are real nubers, and noralzaton requres a þ b ¼ 1: Iportant specal cases are the 645 lnear polarzaton states jþ45 ¼p 1 ffffff ðjhþjvþ () and j45 ¼p 1 ffffff ðjhjvþ; (3) and the left- and rght-crcular polarzaton states jl ¼p 1 ffffff ðjhþjvþ (4) 96 A. J. Phys. 8 (10), October VC 014 Aercan Assocaton of Physcs Teachers 96

2 and jr ¼p 1 ffffff ðjhjvþ: (5) In our experents, we use a spontaneous paraetrc downconverson (SPDC) source that produces two photons referred to as the sgnal and dler. Our SPDC source s capable of producng pars of photons that are both horzontally polarzed jh; H, both vertcally polarzed jv; V, or any lnear cobnaton of these two possbltes. For the experents nvolvng entangled photons, we adjust our source to produce photons n the Bell state j/ þ ¼p 1 ffffff ðjh; HþjV; VÞ ¼ 1 pffffff jh s jh þjv s jv : (6) Here, we have used two dfferent notatons, the latter of whch explctly labels the polarzaton of each photon. B. The densty operator Here we descrbe, wthout proof, the propertes of the densty operator that are needed for our dscusson of quantu state easureent. More detaled nforaton can be found n Ref. 14 or Copleent 8.A of Ref. 1. The densty operator correspondng to a pure state jw s gven by ^q ¼jwhwj: (7) A syste that s fluctuatng, or that s not always prepared n the sae pure state, s called a xed state. For a xed state n whch each state jw j s prepared wth probablty p j, the densty operator s gven by ^q ¼ X j p j jw j hw j j: (8) The probabltes ust behave lke classcal probabltes, eanng the p j s are real, they all le between zero and one (0 p j 1), and they satsfy the noralzaton condton X p j ¼ 1: (9) j It s portant to note that the states jw j are assued to be noralzed, but they need not be orthogonal nor do they need to for a bass; they are erely states that the syste s prepared n wth soe probablty. The densty atrx s the representaton of the densty operator n a partcular bass. If the states j/ n for an orthonoral bass, the eleents of the densty atrx are gven by q n ¼h/ j^qj/ n : (10) The trace of a atrx s the su of ts dagonal eleents, and we can defne the trace of an operator slarly. The trace of the densty operator s thus Trð^q Þ ¼ X n q nn ¼ X n h/ n j^qj/ n ¼1; (11) where equalty wth 1 s the noralzaton condton for the densty operator, whch s ensured by Eq. (9). The expectaton value of an operator ^O can be found by ultplyng ^O by the densty operator, and then coputng the trace, as n h ^O ¼Tr ^O^q ¼ Tr ^q ^O : (1) The densty atrx descrbng the polarzaton of a photon n the horzontal/vertcal bass s hhj^qjh hhj^qjv ^q ¼ : (13) hvj^qjh hvj^qjv When expressng operators as atrces we wll always use ths bass. The densty atrces of the horzontal and vertcal polarzaton states are thus ^q H ¼ 1 0 ; (14) 0 0 and ^q V ¼ 0 0 : (15) 0 1 Slarly, the densty atrces for the states of Eqs. () (5) are ^q þ45 ¼ ; (16) ^q 45 ¼ ; (17) ^q L ¼ ; (18) and ^q R ¼ 1 1 : (19) 1 For a pure state t can be shown that Tr ^q ¼ 1, whle for any non-pure state Tr ^q < 1. We can thus use Tr ^q as a easure of the purty of a state. The aount of overlap between two states ^q 1 and ^q can be expressed as the fdelty F, defned as 5 qffffffffffffffffffffffffffffffffffffffffffffffffff pffffffffff pffffffffff F ¼ Tr ^q 1 ^q ^q 1 : (0) The fdelty takes on values 0 F 1, and n the case when both states are pure t splfes to F ¼ Trð^q 1^q Þ ¼jhw 1 jw j : (1) C. Projectve easureents Let s assue that our syste conssts of two partcles whose jont quantu state s gven by ^q. A easureent s 963 A. J. Phys., Vol. 8, No. 10, October 014 E. Dederck and M. Beck 963

3 perfored on partcle 1, and ths easureent corresponds to an observable, represented by operator ^A. The postulates of quantu echancs tell us that ths easureent ust return an egenvalue a of ^A, and after the easureent partcle 1 s left n the egenstate ja 1 that corresponds to the easured egenvalue. The queston s, what s the state of partcle, ^q, after ths easureent? To deterne ^q,we frst use the projecton operator ^P a ¼ja 11 haj to project ^q onto the state deterned by the easureent result. The state of partcle s then obtaned by averagng the jont state over the state of partcle 1 (ths operaton s called a partal trace). Snce we want ^q to be noralzed, we then renoralze the resultng state. More detals of ths calculaton are gven n the Appendx, but the result s that the state of partcle s gven by ^q ¼ 1 haj^qja 1 : () Tr 1 haj^qja 1 These concepts are best llustrated wth an exaple. Consder the polarzaton-entangled Bell state j/ þ that we use n the experents, as gven n Eq. (6). The densty operator correspondng to ths state s ^q ¼j/ þ h/ þ j¼ 1 pffffff ðjh; HþjV; VÞ 1 pffffff ðhh; HjþhV; VjÞ ; (3) whch can be expanded as ^q ¼ 1 jh; HhH; HjþjH; HhV; Vj ð þjv; VhH; HjþjV; VhV; VjÞ: (4) Suppose that a easureent s perfored on the dler photon and t s found to be horzontally polarzed. The state of the sgnal photon after ths easureent can be deterned by nsertng Eq. (4) nto Eq. (). The nuerator s gven by hhj^qjh ¼ 1 hhj ð jh; HhH; HjþjH; HhV; Vj þjv; VhH; HjþjV; VhV; VjÞjH ; (5) whch splfes to hhj^qjh ¼ 1 jh ss hhjþ0þ0þ0 1 ¼ jh sshhj: (6) The trace of ths operator [denonator of Eq. ()] s 1/, so the densty atrx of the sgnal state s ^q s ¼jH ss hhj. The sgnal photon s thus n state jh s after the easureent, whch s what we would have ntutvely guessed fro Eq. (6). More generally, assue that the easureent on the dler photon fnds t to be ellptcally polarzed, correspondng to the state je of Eq. (1). In the Appendx, we show that the sgnal photon s then projected nto an ellptcal-polarzaton state that s the coplex conjugate of the dler photon s state: je s ¼ ajh s þ be / jv s : (7) D. Quantu state easureent A sple ethod for deternng the polarzaton state of a bea of photons, assung the state s pure, s descrbed n Copleent 5.A of Ref. 1. However, n general, the state wll not be pure so we wll need to deterne the densty operator. To do ths, we wll use the ethod descrbed by Altepeter and coworkers n Ref. 5. Consder a bea of photons prepared n state ^q. We can perfor a easureent of the polarzaton of ths bea n the horzontal-vertcal bass (HV-bass) by usng a polarzaton analyzer PA (e.g., a bea-dsplacng polarzer) that splts the bea nto ts horzontal and vertcal coponents (Fg. 1). If we assgn the egenvalue þ1 to photons easured to be horzontally polarzed, and 1 to photons easured to be vertcally polarzed, we can construct a Hertan polarzaton operator ^S 1 n ters of the projecton operators onto the jh and jv states as ^S 1 ¼ ðþ1þjhhhj þð1þjvhvj ¼ 1 0 : (8) 0 1 We can perfor polarzaton easureents n other bases as well. To perfor easureents n the 645 lnear polarzaton bass, we nsert a half-wave plate whose fast axs s orented at.5 wth respect to the horzontal before the PA n Fg. 1. Ths wave plate wll rotate lnear polarzaton by 45, wth the netresultthatþ45 polarzed photons (egenvalue þ1) wll be detected at one detector and 45 polarzed photons (egenvalue 1) wll be detected at the other. The polarzaton operator correspondng to easureents n ths bass s ^S ¼ ðþ1þjþ45hþ45j þð1þj45h45j; (9) whch can be expressed as a atrx n the horzontal-vertcal bass as ^S ¼ : (30) To perfor easureents n the crcular polarzaton bass, we nsert a quarter-wave plate whose fast axs s orented at 45 wth respect to the horzontal before the PA n Fg. 1. Ths wave plate converts lnear polarzaton to crcular polarzaton (and vce versa), wth the net result that left-crcular polarzed photons (egenvalue þ1) wll be detected at one detector and rght-crcular polarzed photons (egenvalue 1) wll be detected at the other. The operator correspondng to these easureents s ^S 3 ¼ ðþ1þjlhlj þ 1 ð ÞjRhRj ¼ 0 0 : (31) Fg. 1. A polarzaton analyzer (PA) that splts a bea nto ts horzontal and vertcal coponents. 964 A. J. Phys., Vol. 8, No. 10, October 014 E. Dederck and M. Beck 964

4 Fnally, we wll fnd t convenent to express the dentty operator as ^S 0 : ^S 0 ¼ 1 0 : (3) 0 1 Experentally, the expectaton values h^s j can be calculated by sply averagng the easured values. Theoretcally, the expectaton values can be coputed fro the densty operator usng Eq. (1),or h^s j ¼Tr ^S j^q : (33) Note that the expectaton value of the dentty operator s unty: h^s 0 ¼Tr ^S 0^q ¼ Trð^q Þ ¼ 1: (34) Those falar wth the theory of polarzaton of a classcal electroagnetc wave ay recognze the expectaton values h ^S j as the (noralzed) Stokes paraeters of the bea. 15 The Stokes paraeters specfy the polarzaton of a fluctuatng classcal electroagnetc wave. It turns out that they also specfy the quantu polarzaton state of a bea of photons, as t can be shown that the densty operator can be wrtten as 5 ^q ¼ 1 X 3 j¼0 h ^S j ^S j : (35) Thus, the procedure for easurng the densty operator descrbng the polarzaton state of a bea of photons s as follows. Frst, perfor easureents of the polarzaton n the HV-bass usng the apparatus of Fg. 1, and fro these easureents deterne h ^S 1. Next, nsert a half-wave plate nto the apparatus n order to perfor easureents n the 645 -bass and deterne h^s. Then replace the half-wave plate wth a quarter-wave plate n order to perfor easureents n the crcular polarzaton bass and deterne h ^S 3. Fnally, usng the atrx representatons of the operators ^S j, Eqs. (8) (3), and fact that h^s 0 ¼1, the densty atrx can be calculated usng Eq. (35). It s portant to note that n order for ths state deternaton technque to work properly, the syste ust be prepared n the sae state for all of the easureents. III. EXPERIMENTS A. The experental apparatus The experental apparatus s shown n Fg.. A 100- W, 405-n laser dode pups a par of Type-I beta-baru borate (BBO) down-converson crystals, whose axes are orented at rght angles wth respect to each other. Downconverted photons ake angles of 3 wth respect to the pup bea and have a wavelength of approxately 810 n. As descrbed above, the source produces pars of photons n an arbtrary lnear cobnaton of the jh; H and jv; V polarzaton states. The relatve apltude and phase of the states are vared by adjustng the half-wave plate and the brefrngent plate that the pup laser passes through before strkng the down-converson crystals. More detals Fg.. A scheatc of the experental apparatus. Here k/ denotes a halfwave plate, BP denotes a brefrngent plate, DC denotes the downconverson crystals, WP denotes an optonal wave plate, and RP denotes a Rochon polarzer. Lenses at A, A 0, B and B 0 focus the down-converted beas nto ultode optcal fbers, whch drect the lght to sngle photon countng odules (SPCMs). regardng the experental apparatus can be found n Refs. 1 and 16. The dler bea passes through a Rochon polarzer (RP) that transts horzontally polarzed photons to detector A and deflects vertcally polarzed photons to detector A 0.A half- or quarter-wave plate ay be nserted n front of the RP n order to project dagonally or crcularly polarzed lght onto the two detectors. The sgnal bea passes through an optonal half- or quarter-wave plate that can be used to odfy the state of the sgnal bea. Then an RP and wave plate s used to allow the B and B 0 detectors to perfor polarzaton easureents n the three dfferent bases needed to reconstruct the densty operator of the sgnal bea, as descrbed n Sec. II D. To deterne the state of the sgnal bea condtoned on the detecton of an dler photon at A, we easure the nuber of concdence counts n a fxed te nterval between the A and B detectors (N AB ) and the A and B 0 detectors (N AB 0). For easureents perfored n the horzontal-vertcal bass, we can obtan the frst Stokes paraeter as h ^S 1 ¼ N AB N AB 0 : (36) N AB þ N AB 0 By perforng easureents n the two other requred bases, as descrbed above, we can deterne the state of the sgnal bea. In a slar anner, we can deterne the state of the sgnal bea condtoned on the detecton of an dler photon at A 0 by easurng the concdences N A 0 B and N A 0 B 0. In order to easure the state of the sgnal bea condtoned on the presence of an dler photon (.e., detecton at ether A or A 0 ) we can sply add the A and A 0 concdences and use N AB þ N A 0 B and N AB 0 þ N A 0 B0 to deterne the state. 965 A. J. Phys., Vol. 8, No. 10, October 014 E. Dederck and M. Beck 965

5 B. Non-entangled state Frst we adjust the source to produce down-converted photons n the non-entangled state jh; H, n whch both sgnal and dler photons are horzontally polarzed. We nsert a quarter-wave plate nto the sgnal bea n order to change the state of ths bea to jl s. We nsert a quarter-wave plate nto the dler bea so that left-crcularly polarzed photons are detected at A, and rght-crcularly polarzed photons are detected at A 0. (The reason for nsertng ths quarter-wave plate s to obtan roughly equal counts at the A and A 0 detectors. Wthout ths wave plate, the horzontally polarzed photons produced at the source would be detected at A, and very few photons would be detected at A 0. The state reconstructon condtoned on detecton at A 0 would be unrelable under these crcustances.) The reconstructed states of the sgnal bea are shown n Fg. 3. Fgure 3(a) shows the state for dler detectons at A. Coparng the easured state to ^q L [Eq. (18)], we fnd that the fdelty s F ¼ 0:96. (Recall that the fdelty s a easure of the aount of overlap of two states, so by ths easure there s 96% overlap between our easured state and ^q L.) We also fnd that the easured state s farly pure, wth Tr ^q ¼ 0:98. As such, we can say that the easured state closely approxates the state jl s, whch s the state that we prepared the sgnal bea n and that we expected to easure. Fgure 3(b) shows the easured state of the sgnal bea for dler photons detected at A 0, and Fg. 3(c) shows the state for detectons at ether A or A 0. The ost portant thng to note s that all three easured states are nearly the sae. All three easured states have Tr ^q 0:94, and copared to ^q L they have F 0:93. We have also repeated ths experent, replacng the quarter-wave plate n the dler bea wth a half-wave plate and projectng þ45 polarzed photons onto detector A, and 45 polarzed photons onto A 0. We agan fnd that the sgnal bea s well descrbed by the state jl s. All of these easureents taken together ndcate that for photons prepared n the state jh; H, the state of the sgnal bea s ndependent of easureents perfored on the dler bea. The purtes and fdeltes of our easured states are lower than the deal values of 1. Ths s ost lkely due to perfectons n our state preparaton procedure, whch eans that we are not perfectly producng the state jh; H. For exaple, any depolarzaton of the pup laser would ean that we would be producng soe vertcally polarzed sgnal and dler photons. Also, f the two down-converson crystals are not perfectly orthogonal, soe non-horzontally polarzed photons wll be produced. We can prove the qualty of our state producton by nsertng a lnear polarzer orented along the horzontal drecton nto the sgnal bea n order to better defne the polarzaton of the sgnal photon. Ths polarzer s nserted after the down-converson crystal (but before the quarter wave plate), whch converts the sgnal bea nto state jl s. If we do ths we fnd that our purtes and fdeltes are proved. We have repeated the state easureents descrbed above (analogous to those presented n Fg. 3) wth ths polarzer n place and fnd that for all three easured states Tr ^q 0:998 and F 0:998. These states are extreely pure and are well descrbed by the state jl s. Fg. 3. The real and agnary parts of the densty atrx for the sgnal bea prepared n state jl s. The easureents were condtoned on dler photons detected to be: (a) left-crcularly polarzed at A, (b) rght-crcularly polarzed at A 0, (c) present at ether A or A 0. Dark boxes correspond to postve values whle lght boxes correspond to negatve values. 966 A. J. Phys., Vol. 8, No. 10, October 014 E. Dederck and M. Beck 966

6 C. Entangled state Now we adjust the source (as descrbed n Refs. 1, 9, and 10) to produce photons n the state j/ þ of Eq. (6). We nsert a quarter-wave plate nto the dler bea, so that leftcrcularly polarzed photons are detected at A and rghtcrcularly polarzed photons are detected at A 0. We do not nsert a polarzer or wave plate nto the sgnal bea to prepare t n any partcular state. Instead, we sply easure the state of the sgnal photon, condtoned on an dler photon detecton at A and/or A 0. The easured state of the sgnal bea s shown n Fg. 4. In Fg. 4(a), the state of the sgnal photons s found to closely approxate the rght-crcular state ^q R (F ¼ 0:87), condtoned upon the dler photons havng left-crcular polarzaton (the state of the sgnal s the coplex conjugate of that of the dler). Ths s what we would expect because, as dscussed n Sec. II C and the Appendx, for source photons n state j/ þ, f the dler photon s projected onto the ellptcal polarzaton state je the sgnal photon s projected nto the coplex-conjugate state je s. Fgure 4(b) confrs ths behavor. There we see that f the dler photon s easured to be rght-crcularly polarzed, the sgnal photon s n a state that s well descrbed by the left-crcular polarzaton state ^q L (F ¼ 0:93). In Fg. 4(c), the sgnal bea state easureent s condtoned on the detecton of photons of ether polarzaton n the dler bea. The easured state s found to be xed Tr ^q ¼ 0:5 and to closely reseble the state ^q s ¼ 1 jh ss hhjþjv ss hvj 1 ¼ ; (37) whch corresponds to the sgnal photon beng horzontally polarzed half of the te and vertcally polarzed the other half of the te. As descrbed n the Appendx, ths s the result we expect for ths easureent. It s nterestng to note that the data used to deterne the states dsplayed n Fg. 4 were all easured n the sae experent at the sae te, but that we are able to easure three dfferent states. The three states are all condtoned on dfferent detecton events n the dler bea, and such condtonng (or post-selectng ) s done by sortng the data after t has been acqured. Note that we would have easured essentally the sae result as that depcted n Fg. 4(c) f we had reoved the polarzer fro the dler bea and condtoned the easureents on the detecton of a photon n ths bea. The fact that easureents perfored n one place affect the results of easureents perfored n another place (as depcted n Fg. 4) s a consequence of the nonlocal character of the entangled state j/ þ. However, the results shown n Fg. 4 are not suffcent to prove that the source produces photons n an entangled state. These results would be essentally dentcal f the source produced photons n the classcal xed state ^q x ¼ 1 ðjl; RhL; RjþjR; LhR; LjÞ; (38) Fg. 4. The real and agnary parts of the densty atrx for the sgnal bea, wth source photons prepared n the entangled state j/ þ. The easureents were condtoned on dler photons detected to be: (a) left-crcularly polarzed at A, (b) rght-crcularly polarzed at A 0, (c) present at ether A or A 0. Dark boxes correspond to postve values, whle lght boxes correspond to negatve values. 967 A. J. Phys., Vol. 8, No. 10, October 014 E. Dederck and M. Beck 967

7 whch corresponds to the photons beng prepared n the state jl; R half of the te, and the state jr; L the other half of the te. In any sngle bass, t s possble for a classcal xed state to have perfect correlatons between easureents perfored n two locatons, and thus c the behavor of an entangled state (e.g., the state of Eq. (38) can c the behavor of j/ þ for easureents n the crcular polarzaton bass). However, ths s only true n a sngle bass. If easureents are perfored n a dfferent bass the classcal correlatons wll no longer be perfect, whereas the quantu correlatons of an entangled state persst for easureents n any bass. To verfy that we are seeng true quantu correlatons, we have perfored easureents n other bases as well. In Fg. 5(a), we show the easured state of the sgnal photon condtoned upon the easureent of an dler photon beng horzontally polarzed. We see that the state s well descrbed by ^q H (F ¼ 0:97), whch s what we would expect for photons prepared n the entangled state j/ þ (as descrbed n Sec. II C). If nstead the photons had been prepared n the state ^q x of Eq. (38), the easured state would have been gven by the xed state of Eq. (37); ths can be seen by usng Eq. () and nsertng ^q x for ^q and jh for ja 1. In Fg. 5(b), we show the easured state of the sgnal photon condtoned upon the easureent of an dler photon beng lnearly polarzed along 45. The state s well descrbed by ^q 45 (F ¼ 0:89), whch s what we would expect for photons prepared n the entangled state j/ þ. Once agan, photons prepared n ^q x would have yelded a sgnal state gven by Eq. (37) for ths easureent. The echanss descrbed at the end of Sec. III B also contrbute to the lack of purty of the entangled state j/ þ. Addtonally, there are other factors that can degrade the purty of an entangled state. For the entangled state to be pure, the photons produced n the two crystals ust be copletely ndstngushable. Any nforaton that ght n prncple allow one to deterne the polarzaton of a photon would collapse ts polarzaton state and destroy the entangleent. For exaple, f one was able to separately age the two down-converson crystals and deterne whch crystal a photon was produced n, one would know the photon polarzaton and the entangleent would be destroyed. Partal nforaton reduces the purty of an entangled state, and hence the fdelty of the easureents, wthout copletely destroyng the entangleent. In practce, t s dffcult to produce photons that are absolutely ndstngushable. To suarze our easureents for the entangled state, we have easured the state of the sgnal bea, condtoned on projectve easureents perfored on the dler bea. These easureents, shown n Fgs. 4 and 5, dsplay correlatons between the sgnal and dler beas that are consstent wth an entangled state, but not wth a classcal xed state. It s possble to agne a classcal state that could fully descrbe the results of Fg. 4 and Eq. (38), but ths state could Fg. 5. The real and agnary parts of the densty atrx for the sgnal bea, wth source photons prepared n the entangled state j/ þ. The easureents were condtoned on dler photons detected to be: (a) horzontally polarzed at A, (b) lnearly polarzed along 45 at A. Dark boxes correspond to postve values whle lght boxes correspond to negatve values. 968 A. J. Phys., Vol. 8, No. 10, October 014 E. Dederck and M. Beck 968

8 not also explan the results shown n Fg. 5. Ths classcal state would yeld copletely xed states for easureents correspondng to those presented n Fg These results deonstrate that the correlatons we observe cannot be explaned classcally and are due to the nonlocal character of the entangled state j/ þ. IV. CONCLUSIONS We have perfored quantu state easureents of one photon of a two-photon par produced by spontaneous paraetrc down-converson. Ths state easureent s condtoned on the results of easureents perfored on the other photon. We have also presented theoretcal results that allow us to descrbe how the easureents perfored on the second photon wll affect the state of the frst photon. When the two photons are produced n a non-entangled state, the easured state of the frst photon s ndependent of easureents perfored on the second photon (we always easure the sae state for the frst photon). Ths s deonstrated n Fg. 3 by observng that all three easured states are slar. However, when the photons are produced n an entangled state, the easured state of the frst photon does depend on the results of easureents perfored on the second photon. Ths s deonstrated n Fg. 4 by observng that the three easured states are very dfferent. Measureents perfored on the second photon change the state of the two-photon syste, whch projects the frst photon nto dfferent states. Note that the state of the frst photon depends on easureents perfored on the second, no atter what bass s used for the easureents. Snce strong classcal correlatons should exst only for easureents perfored n one bass, we conclude that our photons are prepared n an entangled state. It s the nonlocal nature of entangleent that allows the results of easureents perfored n one place to depend on the results of easureents perfored soewhere else. ACKNOWLEDGMENTS Ths work was supported by Whtan College. APPENDIX Assue that a syste s prepared n state ^q and that a projectve easureent s perfored on the syste. The state of the syste after the easureent ^q 0 s gven by 18 ^q 0 ¼ ^P^q ^P ; (A1) Tr ^P^q ^P where ^P s an operator that projects onto the state correspondng to the results of the easureent. We are nterested a two-partcle syste, and we wsh to deterne the state of partcle after a projectve easureent s perfored on partcle 1. Thus, the projecton operator operates n the subspace of partcle 1 and can be wrtten as ^P 1 ¼ jw n 11 hw n j ; (A) X n where the states jw n 1 are egenstates of the observable correspondng to the easureent. These states are orthogonal, but they need not for a coplete set of states for partcle 1. The su n Eq. (A) s over the states correspondng to the possble easureent results recorded by the detector. Note that f we square Eq. (A) we obtan X ^P 1 ¼ jw n 11 hw n j jw 11 hw j ¼ X n X n X jw n 11 hw n jw 11 hw j ; (A3) whch splfes to ^P 1 ¼ X n jw n 11 hw n j¼ ^P 1 : (A4) Thus, the square of a projecton operator s equal to the orgnal projecton operator. Now, we wll assue that the states ja 1 for a bass for partcle 1, the states jb n for a bass for partcle, and the states ja ; b n for a bass for the two-partcle syste. We wsh to deterne the state of partcle, ^q, f the state of the two-partcle syste s gven by ^q 0 n Eq. (A1). The state ^q s soetes referred to as the partal densty operator, or the reduced densty operator, and t s obtaned fro ^q 0 by perforng a partal trace over the state space of partcle 1: 14 ^q ¼ Tr 1 ^q 0 ¼ X The atrx eleents of ^q are gven by hb nj^q jb n 0 ¼ X 1 ha j^q 0 ja 1 : (A5) ha ; b n j^q 0 ja ; b n 0: (A6) Note that the denonator n Eq. (A1) s present to noralze the densty operator, so for the oent we wll concern ourselves only wth the nuerator and wll noralze our results at the end. The partal trace of the nuerator n Eq. (A1) s gven by X Tr 1 ^P 1^q ^P 1 ¼ 1 ha j ^P 1^q ^P 1 ja 1 ¼ X 1 ha j ^P 1^1 1^q ^P 1 ja 1 : (A7) If we express the dentty operator for partcle 1, ^1 1, as a su over projectors onto a coplete set of states, we can wrte ths as XX Tr 1 ^P 1^q ^P 1 ¼ 1 ha j ^P 1 ja 11 ha j^q ^P 1 ja 1 : (A8) We now rearrange the ters n the su, whch yelds Tr 1 ^P 1^q ^P 1 XX ¼ ¼ X and further splfes to 1 ha j^q ^P 1 ja 11 ha j ^P 1 ja 1 1 ha j^q ^P 1^1 1 ^P 1 ja 1 ; (A9) 969 A. J. Phys., Vol. 8, No. 10, October 014 E. Dederck and M. Beck 969

9 X Tr 1 ^P 1^q ^P 1 ¼ 1 ha j^q ^P 1 ja 1 ¼ X 1 ha j^q ^P 1 ja 1 : (A10) To obtan the fnal expresson n ths equaton, we have used Eq. (A4). We wll now consder two specal cases. In the frst we assue that ^P 1 projects onto a sngle state ja 1 so that ^P 1 ¼ja 11 haj. In ths case, Eq. (A10) becoes X Tr 1 ^P 1^q ^P 1 ¼ 1 ha j^qja 11 haja 1 ¼ X 1 haja 11 ha j^qja 1 ; (A11) whch splfes to Tr 1 ^P 1^q ^P 1 ¼ 1 haj^1 1^qja 1 ¼ 1 haj^qja 1 : (A1) We can now noralze ths expresson to obtan ^q ¼ 1 haj^qja 1 : (A13) Tr 1 haj^qja 1 Ths s the result gven n Eq. (). As an exaple, consder a source that produces photons n the entangled polarzaton state j/ þ whose densty operator s gven n Eq. (4). A polarzaton easureent s perfored on the dler photon and t s found to be ellptcally polarzed wth a correspondng polarzaton state je gven by Eq. (1). To calculate the state of the sgnal photon after ths easureent, we begn by calculatng the nuerator of Eq. (A13) and fnd hej^qje ¼ 1 ða hhjþbe/ hvjþ ð jh; HhH; Hj Expandng ths, we see that þjh; HhV; VjþjV; VhH; Hj þjv; VhV; VjÞðajH þ be / jv Þ : (A14) hej^qje ¼ 1 ðajh s hh; HjþajH shv; Vj þ be / jv s hh; Hjþbe / jv s hv; VjÞ ajh þ be / jv ; (A15) whch splfes to hej^qje ¼ 1 a jh ss hhjþabe / jh ss hvj þabe / jv ss hhjþb jv ss hvj : (A16) Sung the coeffcents of the dagonal ters, we fnd that the trace of ths operator s 1 Tr hej^qje ¼ ð a þ b Þ ¼ 1 : (A17) The densty operator of the sgnal bea photon s thus ^q s ¼ða jh ss hhjþabe / jh ss hvjþabe / jv ss hhj þb jv ss hvj : ða18þ Ths can be wrtten ore sply as ^q s ¼ðajH s þ be / jv s Þða s hhjþbe / s hvjþ ¼je ss he j; (A19) and we see that the easureent on the dler photon projects the sgnal photon nto the coplex conjugate state je s of Eq. (7). The second case we are nterested n s one n whch we sply regster the presence of partcle 1. We are effectvely projectng onto all possble states, and hence the projecton operator s the dentty operator: ^P 1 ¼ ^1 1 : (A0) In ths case, Eq. (A10) reduces to the partal trace of ^q: Tr 1 ^P 1^q ^P 1 ¼ Tr1 ð^q Þ ¼ X 1 ha j^qja 1 : (A1) To fnd ^q, we noralze ths result by tracng t over the state space of partcle : Tr ½Tr 1 ð^q ÞŠ¼ X ;j 1 ha ;b j j^qja ;b j 1 ¼ Trð^q Þ¼ 1; (A3) whch eans that t s already noralzed. In ths case, the state of partcles s sply gven by ^q ¼ Tr 1 ð^q Þ: (A4) Once agan, consder the exaple of photons n the entangled polarzaton state j/ þ, whose densty operator s gven by Eq. (4). If we sply detect the presence of an dler photon, the sgnal photon s projected onto the state ^q s ¼ Tr ð^q Þ ¼ hhj^qjh þ hvj^qjv : Expandng, we fnd that ^q s ¼ 1 ½ hhjðjh; HhH; HjþjH; HhV; Vj þjv; VhH; HjþjV; VhV; VjÞjH þ hvjðjh; HhH; HjþjH; HhV; Vj þjv; VhH; HjþjV; VhV; VjÞjV Š; (A5) (A6) whch then splfes to ^q s ¼ 1 jh ss hhjþjv ss hvj : (A7) Physcally, ths represents a classcal xed state n whch the polarzaton of the sgnal photon s rando; half of the te a photon wll be found to be horzontally polarzed and half of the te t wll be found to be vertcally polarzed. Fnally, we note that the randoness of the polarzaton for a photon n the state of Eq. (A7) s not lted to easureents perfored n the horzontal-vertcal bass. For ths state, the polarzaton wll be found to be rando for easureents perfored n any bass. 970 A. J. Phys., Vol. 8, No. 10, October 014 E. Dederck and M. Beck 970

10 a) Electronc al: M. Beck, Quantu Mechancs: Theory and Experent (Oxford U.P., Oxford, 01). M. G. Rayer, Measurng the quantu echancal wave functon, Contep. Phys. 38, (1997). 3 Ulf Leonhardt, Measurng the Quantu State of Lght (Cabrdge U.P., Cabrdge, UK, 1997). 4 Quantu State Estaton, edted by M. G. A. Pars and J. Rehacek (Sprnger-Verlag, Berln, 004). 5 J. B. Altepeter, E. R. Jeffrey, and P. G. Kwat, Photonc state toography, n Advances n Atoc, Molecular and Optcal Physcs, edted by P. R. Beran and C. C. Ln (Elsever, Asterda, 006), Vol. 5, pp D. T. Sthey, M. Beck, M. G. Rayer, and A. Fardan, Measureent of the Wgner dstrbuton and the densty atrx of a lght ode usng optcal hoodyne toography: Applcaton to squeezed states and the vacuu, Phys. Rev. Lett. 70, (1993). 7 M. A. Nelsen and I. L. Chuang, Quantu Coputaton and Quantu Inforaton (Cabrdge U.P., Cabrdge, 000). 8 D. Dehlnger and M. W. Mtchell, Entangled photon apparatus for the undergraduate laboratory, A. J. Phys. 70, (00). 9 D. Dehlnger and M. W. Mtchell, Entangled photons, nonlocalty, and Bell nequaltes n the undergraduate laboratory, A. J. Phys. 70, (00) A. J. Phys., Vol. 8, No. 10, October J. A. Carlson, M. D. Olstead, and M. Beck, Quantu ysteres tested: An experent pleentng Hardy s test of local reals, A. J. Phys. 74, (006). 11 E. J. Galvez, Qubt quantu echancs wth correlated-photon experents, A. J. Phys. 78, (010). 1 J. Carvoto-Lagos, G. Arendarz, V. Velazquez, E. Lopez-Moreno, M. Grether, and E. J. Galvez, The Hong-Ou-Mandel nterferoeter n the undergraduate laboratory, Eur. J. Phys. 33, (01). 13 Letter fro Albert Ensten to Max Born, 3 March 1947, The Born-Ensten Letters; Correspondence between Albert Ensten and Max and Hedwg Born fro 1916 to 1955, Translated by Irene Born (Walker, New York, 1971). 14 C. Cohen-Tannoudj, B. Du, and F. Lalo e, Quantu Mechancs (John Wley and Sons, New York, 1977), pp E. Hecht, Optcs, 4th ed. (Addson-Wesley, San Francsco, 00), pp Modern Quantu Mechancs Experents for Undergraduates, < Indeed, ths s why we have chosen to perfor easureents n the bases that we have (horzontal-vertcal, 645, and left- rght-crcular). Classcally, perfect correlatons n one of these bases lead to no correlaton n the others; for a quantu echancal entangled state the correlatons are antaned. 18 Lesle E. Ballentne, Quantu Mechancs: A Modern Developent (World Scentfc, Sngapore, 1998), pp E. Dederck and M. Beck 971