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1 Supplementary Informaton Quantum correlatons wth no causal order Ognyan Oreshkov 2 Fabo Costa Časlav Brukner 3 Faculty of Physcs Unversty of Venna Boltzmanngasse 5 A-090 Venna Austra. 2 QuIC Ecole Polytechnque CP 65 Unversté Lbre de Bruxelles 050 Brussels Belgum. 3 Insttute of Quantum Optcs and Quantum Informaton Austran Academy of Scences Boltzmanngasse 3 A-090 Venna Austra. SUPPLEMENTARY FIGURES Supplmentary Fgure S: Nonlnear model of closed tme-lke curve. In the model of closed tme-lke curves consdered n Refs. [34 36] a chronology-respectng system A ntally n a stateσ nteracts wth a second system A whch travels back n tme accordng to a untary U. Ths model can be represented n our formalsm by an unphyscal process matrx.e. one for whch probabltes do not sum up to one.

2 2 SUPPLEMENTARY METHODS Formal dervaton of the causal nequalty A causal structure for nstance space-tme) s a set of event locatons equpped wth a partal orderng relaton that defnes the possble causal relatons between events at these locatons. If A and B are two such locatons A B reads A s n the causal past of B or equvalently B s n the causal future of A e.g. f A and B are space-tme ponts A B corresponds to A beng n the past lght cone of B). Operatonally f A B an agent at A can sgnal to an agent at B by encodng nformaton n events at A that get correlated wth events at B whch the other agent can observe. Formally sgnallng from A to B s the exstence of statstcal correlatons between a random varable at A whch can be chosen freely and another random varable at B. By defnton a freely chosen varable s one that can be correlated only wth varables n ts causal future. Note that a freely chosen varable s an dealzaton snce the result of a con toss or any other canddate for a freely chosen varable may be correlated wth ntal condtons n the past or wth space-lke separated events but these correlatons are gnored as not relevant to the varables of nterest.) The fact that the relaton s a partal order means that t satsfes the followng condtons: ) A A reflexvty); 2) f A B and B C then A Ctranstvty); and 3) f A B and B A then A B antsymmetry). The last condton says that f A and B are two dfferent locatons there can ether be sgnallng from A to B or vce versa but no sgnallng n both drectons s possble.e. there are no causal loops). If A s not n the causal past of B we wll wrte A B. Note that n a causal structure both A B and B A may hold as n the case when A and B are space-lke separated) and at least one of the two must hold for A B. We wll denote the stuaton where both A B and B A hold by A B. Snce every event specfes an event locaton we wll use the same notaton drectly for events. For nstance f X and Y are two events such that the locaton of X s n the causal past of the locaton of Y we wll wrte X Y smlarly for and ). The man events n our communcaton task are the systems enterng Alce s and Bob s laboratores whch we wll denote by A and B respectvely and the partes producng the bts a b b x and y whch we wll denote by the same letters as the correspondng bts. The fact that Alce generates the bt a and produces her guess x after the system enters her laboratory means that A a y. Smlarly we have B b b y. The assumptons behnd the causal nequalty are: Causal structure CS) The events A B a b b x y are localzed n a causal structure. Free choce FC) Each of the bts a b and b can only be correlated wth events n ts causal future ths concerns only events relevant to the task). We assume also that each of them takes values 0 or wth probablty /2. Closed laboratores CL) x can be correlated wth b only f b A and y can be correlated wth a only f a B. We want to show that these assumptons mply p succ 2 pxb b 0) 2 pya b ) 3 4 S) for the success probablty that Alce and Bob can acheve n ther task. Frst notce that assumpton FC mples that the bts a b and b are ndependent of each other CS s assumed throughout). Indeed there are two general ways n whch the three bts could be correlated two of them are correlated wth each other whle the thrd one s ndependent or each of them s correlated wth the other two. In the frst case the free-choce assumpton mples that the two correlated bts would have to be n each other s causal pasts whch s mpossble. In the second case each of the bts would have to be n the causal past of the other two whch s agan mpossble. Hence the bts are uncorrelated. Next consder the followng three possbltes that can be realzed n a causal structure CS s assumed throughout): A B B A A B. Snce these possbltes are mutually exclusve and exhaustve ther probabltes satsfy pa B ) pb A ) pa B ). From assumpton FC t follows that the bts a b and b are ndependent of whch of these possbltes s realzed. To see ths consder for nstance b. Snce B b we have that b must be ndependent of whether A takes place n the causal past of B or not.e. pa B b ) pa B ). Smlarly b must be ndependent of whether A takes place n the larger regon whch s a complement of the causal future of B whch mples pb A b ) pb A ). But pb A b ) pa B b ) pa B b ) pa B ) pa B b ) whle pb A ) pa B ) pa B ) whch mples pa B b ) pa B ). Fnally snce pa B b ) pa B b ) pb A b ) pa B ) pa B ) pb A b ) pa B ) pa B ) pb A ) we have pb A b ) pb A ). An analogous argument shows that a and b are also ndependent of the causal relaton between A and B.

3 3 Usng the above the success probablty can be wrtten p succ 2 pxb b 0) 2 pya b ) 2 pxb b 0; A B )pa B ) 2 pxb b 0; B A )pb A ) 2 pxb b 0; A B )pa B ) 2 pya b ; A B )pa B ) 2 pya b ; B A )pb A ) 2 pya b ; A B )pa B ) 2 pxb b 0; A B ) ) 2 pya b ; A B ) pa B ) 2 pxb b 0; B A ) ) 2 pya b ; B A ) pb A ) 2 pxb b 0; A B ) ) 2 pya b ; A B ) pa B ). S2) If A B whch mples B A ) from the transtvty of partal order t follows that A b and thus b A ). From assumpton CL x can only be correlated wth b f b s n the causal past of A thus pb x; A B ) pb A B ) 2 [the last equalty follows from the ndependence of b from the causal relatons between A and B together wth assumpton FC]. Usng also that b and b are ndependent we thus obtan pxb b 0; A B ) pb0; x0 b 0; A B ) pb x b 0; A B ) pb0 x0; b 0; A B )px0 b 0; A B ) pb x; b 0; A B )px b 0; A B ) 2 px0 b 0; A B ) 2 px b 0; A B ) 2. If B A whch mples A B ) by an analogous argument we obtan pya b ; B A ) 2. Fnally f A B we have both pya b ; A B ) 2 and pxb b 0; A B ) 2. Substtutng ths n Eq. S2) we obtan p succ 4 ) 2 pya b ; A B ) pa B ) 2 pxb b 0; B A ) ) pb A ) 4 4 ) pa B ) pa B ) 3 4 pb A ) 3 4 pa B ) 3 4. S3) Ths completes the proof. Characterzaton of process matrces Here we derve necessary and suffcent condtons for a matrx to satsfy 0 [non-negatve probabltes] S4) and Tr [ M A A 2 M B B 2 )] M A A 2 M B B 2 0 Tr 2 M A A 2 A Tr 2 M B B 2 B S5) [probabltes sum up to ] n terms of an expanson of the matrx n a Hlbert-Schmdt bass. A Hlbert-Schmdt bass oflh X ) s gven by a set of matrces {σµ} X d2 X µ0 wthσx 0 X Trσµσ X ν X d X δ µν and Trσ X 0 for... dx 2. A general element oflh A H A 2 H B H B 2 ) can be expressed as w µνλγ σ A µσ A 2 νσ B λ σb 2 γ w µνλγ C S6) µνλγ we omt tensor products and dentty matrces whenever there s no rsk of confuson). Snce a process matrx has to be Hermtan we consder only the cases w µνλγ R. S7)

4 We wll refer to terms of the formσ A rest ) as of the type A terms such asσ A σ A 2 rest ) as of the type A A 2 and so on. The propertes of a process matrx can be analysed wth respect to the terms t contans. For example terms of the type A B produce non-sgnallng correlatons between the measurements terms such as A 2 B correlate Alce s outputs wth Bob s nputs yeldng sgnallng from Alce to Bob etc. as llustrated n Fg. 3. Note that not all terms are compatble wth the condton S5). We wll prove that a matrx W satsfes condton S5) f and only f t only contans the terms lsted n Fg. 3. The CJ matrx of a local operaton can be smlarly wrtten M X X 2 µν r µν σ X µσ X 2 ν r µν R. The condton Tr X2 M X X 2 X s equvalent to the requrement r 00 d X2 r 0 0 for >0. Thus CJ matrces correspondng to CPTP maps have the form M X X 2 d X2 a σ X 2 t σ X σ X 2 S8) >0 a t R. Let us consder frst the case of a sngle party say Alce. Snce the set of matrces M A A 2 0 s a substantal set condton S5) can be equvalently mposed on arbtrary matrces of the form S8) and for a sngle party t can be rewrtten as Tr d A2 WA A 2 a σ A 2 t σ A σ A 2 >0 a t R. Usng an expanson of the process matrx n the same bass n a smlar way W A A 2 µν w µν σ A µσ A 2 ν w µν R the above condton becomes d A w 00 w 0 a w t >0 >0 a t R and one obtans w 00 d A w 0 w 0 for >0. Thus the most general process matrx observed by a sngle party has the form W A A 2 d A >0 >0 >0 v R W A A 2 0 whch can be recognzed as a state. Ths result that all probabltes a sngle agent can observe are descrbed by quantum states s an extenson of Gleason s theorem from POVMs [52 53] to CP maps note that here the lnear structure of quantum operatons s assumed whle n Gleason s theorem for POVMs t s derved from dfferent hypotheses. However by a smlar argument one could derve lnearty for CP maps too). Let us now consder a bpartte process matrx µνλγ w µνλγ σ A µσ A 2 νσ B λ σb 2 γ w µνλγ R. We have to mpose S5) for arbtrary matrces M A A 2 M B B 2 of the form S8). Frst f we fx M B B 2 B B 2 we obtan v σ A w 0000 w 000 a w 00 t >0 a t R whch mposes w 0000 and w 000 w 00 0 for >0. Smlarly by fxng M A A 2 A A2 d A2 we can derve w 000 w 00 0 for >0. Fnally mposng S5) for arbtrary M A A 2 a σ A 2 t σ A σ A 2 d A2 M B B 2 d B2 >0 k>0 b k σ B 2 k >0 >0 d B2 s kl σ B kl>0 k σb 2 l 4 S9)

5 5 we obtan w 00k a b k w 0kl a s kl k>0 kl>0 w 0k t b k w kl t s kl 0 k>0 kl>0 a t b k s kl R from whch we conclude that the most general matrx that satsfes S5) has the form σ B A σ A B σ A B) σ B A : c σ A σ B 2 d k σ A σ B σ B 2 k >0 >0 k>0 σ A B : e σ A 2 σ B σ A B : v σ A >0 >0 k>0 x σ B where c d k e f k g v x R. f k σ A σ A 2 σ B k >0 g σ A σ B Ths form together wth the condton 0 completely characterzes the most general bpartte process matrx. Terms not appearng n process matrces The not-allowed terms are lsted n Fg. 4 along wth possble nterpretatons. Partcularly nterestng are the cases nvolvng terms of the type A A 2. These would correlate Alce s output wth her nput and not gve unt probabltes for some CPTP maps that she can choose to perform. Ths knd of correlatons resemble a backward n tme transmsson of nformaton: one can magne that they can be generated by a quantum channel n the nverse order from the output A 2 to the nput A. It s worth notng that a recently proposed model of closed tme-lke curves [34 36] can be expressed precsely n ths way. Usng our termnology such a model consders an agent recevng two quantum systems n her laboratory: a chronology-respectng system A and a second system A whch after leavng the laboratory s sent back n tme to the laboratory s entrance see Supplementary Fg. S). Ths can be descrbed by the process matrx W A A A 2A 2 σ A A 2 U φ φ A A 2U ) whereσ A s the state of the chronology-respectng system when t enters the laboratory and U φ φ A A 2U ) s a process matrx correspondng to a untary U from A 2 to A descrbng the evoluton back n tme of the chronology-volatng system. The labels A A represent the two systems enterng the laboratory whle A 2 A 2 represent the systems gong out. Note that here the two systems belong to the same laboratory and they can undergo any ont operaton.) In ths model probabltes have to be renormalzed n order to sum up to one whch ntroduces a non-lnearty that volates our orgnal assumptons n partcular as opposed to quantum mechancs probabltes are contextual n ths model snce t s necessary to specfy the events that dd not occur n order to perform the renormalzaton step). The same can be sad for Deutsch s model of closed tme-lke curves [33] whch s also non-lnear although t uses a dfferent mechansm to obtan well-defned probabltes) and thus volates our premse that ordnary quantum mechancs holds locally n each laboratory. Casual order n the classcal lmt Let us now show that n the classcal lmt all correlatons are causally ordered. Classcal operatons can be descrbed by transton matrces M k) Pk ) where Pk ) s the condtonal probablty that the measurement outcome s observed and the classcal output state k s prepared gven that the nput state s. They can be expressed n the quantum formalsm as CP maps dagonal n a fxed ponter ) bass and the correspondng CJ matrces are M k M k) A k k A 2. In order to express arbtrary bpartte probabltes of classcal operatons t s suffcent to consder process matrces of the standard form σ B A σ A B) S0)

6 6 whereσ B A andσ A B are dagonal n the ponter bass. Probabltes are stll gven by P ) [ )] M A MB Tr W A A 2 B B 2 M A A 2 M B B 2. S) We wll show that any such dagonal process matrx can be wrtten n the form ρ A A 2 B ρ A B B 2 ) S2) whereρ A A 2 B andρ A B B 2 are postve semdefnte matrces. Ths s suffcent to conclude that s causally separable. Indeed f could be wrtten n the form S2) we know thatρ A A 2 B would not contan Hlbert-Schmdt terms of the types A A 2 or A 2 whch are not allowed n a process matrx) snce by assumpton these terms are not part of. Therefore the matrx W B A ρa A 2 B Trρ A A 2 B d A2 d B2 S3) whch s postve semdefnte has trace d A2 d B2 and contans only terms of the allowed types would be a vald process matrx wth no sgnallng from B to A. Smlarly W A B ρa B B 2 Trρ A B B 2 d A2 d B2 S4) would be a vald process matrx wth no sgnallng from A to B. The whole process matrx could then be wrtten n the causally separable form qw B A q)w A B S5) where q TrρA A 2 B d A d A2 d B d B2. S6) Note that 0 q snceρ A A 2 B andρ A B B 2 n Eq. S2) are postve semdefnte and Tr d A2 d B2. To prove Eq. S2) we wll constructρ A A 2 B andρ A B B 2 from the general form n Eq. S0). Let the mnmum egenvalue ofσ B A σ A B be m. Snce s postve semdefnte andσ B A σ A B s traceless we have m [ 0]. Defne the matrces κ A A 2 B mσ B A κ A B B 2 σ A B. S7) S8) The full process matrx can then be wrtten m)κ A A 2 B κ A B B 2 ) S9) whereκ A A 2 B κ A B B 2 s postve semdefnte. We are now gong to modfyκ A A 2 B andκ A B B 2 by addng matrces of the formκ A B toκ A A 2 B and subtractng them from κ A B B 2 therefore leavngκ A A 2 B κ A B B 2 unchanged) untl we transform bothκ A A 2 B andκ A B B 2 n Eq. S9) nto postve semdefnte matrces. Denote the ponter bass of system X by X... d X XA A 2 B B 2. All matrces we consder are dagonal n the bass{ A A 2 k B l B 2 }. Let m k l) denote the egenvalues ofκ A A 2 B correspondng to the egenvectors A A 2 k B l B 2 and let m 2 k l) be the egenvalues ofκ A B B 2 correspondng to the same vectors. For every and k we do the followng. Defne m k)mn l m k l) S20) m 2 k)mn l m 2 k l). S2)

7 Note that m k l) do not depend on l snceκ A A 2 B acts trvally on B 2 and smlarly m 2 k l) do not depend on. Ths means that for gven and k the mnmum of the egenvalues ofκ A A 2 B κ A B B 2 for all egenvectors of the type A A 2 k B l B 2 s equal to m k) m 2 k). But by constructonκ A A 2 B κ A B B 2 s postve semdefnte so we have 7 m k) m 2 k) 0. S22) Now f both m k) and m 2 k)} are non-negatve we wll not modfyκ A A 2 B andκ A B B 2. However f one of these numbers s negatve say m k)<0 both cannot be negatve due to S22)) we wll add the term m k) A A 2 k k B B 2 toκ A A 2 B and subtract the same term fromκ A B B 2. After ths step the modfedκ A A 2 B s such that the egenvalues m k l) have been changed to m k l) m k) m k) m k)0.e.κ A A 2 B does not have any more negatve egenvalues m k l) for the gven and k. The same holds forκ A B B 2 snce the egenvalues m 2 k l) change to m 2 k l) m k) m 2 k) m k) 0. In other words the egenvalues of the modfedκ A A 2 B andκ A B B 2 satsfy m k l) m 2 k l) 0 l. S23) By performng ths procedure for all and k we eventually transformκ A A 2 B andκ A B B 2 nto matrces all of whose egenvalues are non-negatve. Denote the resultant postve semdefnte matrces by κ A A 2 B and κ A B B 2. We can now add the term m) n Eq. S9) for nstance to κ A A 2 B recall that m [ 0]) defnng the postve semdefnte matrces ρ A A 2 B m) κ A A 2 B ρ A B B 2 κ A B B 2. S24) S25) We thus arrve at the desred form S2) whch mples S5) as argued above. SUPPLEMENTARY REFERENCES 52. Gleason A. M. Measures on the closed subspaces of a Hlbert space. J. Math. Mech ). 53. Caves C. M. Fuchs C. A. Manne K. K. & Renes J. M. Gleason-Type Dervatons of the Quantum Probablty Rule for Generalzed Measurements. Found. Phys ).