Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 1 / 16
Non-Signling Boxes The primry lesson from lst lecture is tht pir of input/output oxes cn e used for communiction if nd only if they violte the non-signling condition. Non-Signling Condition Two oxes with inputs elonging to set X = {1,, X }, outputs elong to set B = {, 1,, B } nd trnsition proilities {p( x)} B,x X re clled non-signling if p( x) = p( x ) B, x, x X. In this lecture we will return to the quntum setting nd see if quntum entnglement cn e used fster-thn-light communiction. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 2 / 16
Quntum Boxes Suppose Alice nd Bo shre some iprtite stte ρ AB. They hve red populr science mgzine climing tht quntum entnglement llows for spooky ction t distnce. They get excited nd think this mens their entnglement cn e used for instntneous communiction. Alice: I will perform one of two quntum opertions on my system. This will cuse n instntneous chnge on your system so tht y mesuring your system you will e le to determine which ction I performed. Then if I wnt to send messge 0, I will perform opertion {A (0) 0, A(0) 1 {A (1) 0, A(1) 1 }. } nd if I wnt to send messge 1, I will perform ction Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 3 / 16
Quntum Boxes Bo: Gret ide! But wht mesurement should I use to recover your messge? Let me choose etween two of them - {B (0) 0, B(0) 1 } or {B (1) } - nd see which one works etter. 0, B(1) 1 The mesurements {B (0) 0, B(0) 1 } nd {B(1) 0, B(1) 1 } re two decoding opertions for Alice s messge. He mesures his system nd clims tht Alice sent vlue x if he otins mesurement outcome x. Alice nd Bo s strtegy cn e descried within the lck ox frmework. These oxes re more sophisticted since they hve pirs of inputs nd pirs of inputs. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 4 / 16
Quntum Boxes The trnsition proilities re given y p(, x, y) = tr[(a (x) Do these oxes llow for communiction? B (y) )ρab (A (x) B (y) ) ],, x, y {0, 1}. There re two possiilities now: communiction from Alice to Bo or communiction from Bo to Alice. For Alice to Bo, we look t Bo s output distriution p( x, y). For ech of his inputs y, this genertes n input/output ox p y ( x) := p( x, y) where p( x, y) = p(, x, y). Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 5 / 16
Quntum Boxes No signling from Alice to Bo mens tht p y ( x) = p y ( x ); in other words p(, x, y) = p(, x, y ), x, y, y. No signling from Bo to Alice mens tht p(, x, y) = p(, x, y), y, x, x. Do quntum oxes stisfy the non-signling conditions with trnsition proilities given y p(, x, y) = tr[(a (x) B (y) )ρab (A (x) B (y) ) ]? Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 6 / 16
Quntum Boxes Check: p(, x, y) = = tr[(a (x) tr[(a (x) ) A (x) B (y) )ρab (A (x) B (y) ) ] (B (y) ) B (y) ρab ] = tr[i (B (y) ) B (y) ρab ] = tr[(b (y) ) B (y) ρb ]. This is just the proility tht Bo otins outcome when he performs mesurement {B (y) } on his reduced stte ρb. It is independent of oth Alice s input nd output p( x, y) = p( x, y) It stisfies the non-signling conditions. Quntum entnglement does not llow instntneous communiction! Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 7 / 16
Quntum Boxes The trnsition proilities {p(, x, y)},,x,y generted y quntum oxes (i.e. generted y mesuring n entngled quntum stte) do not llow for communiction etween two sptil positions X 1 nd X 2 in time intervl fster thn c X (i.e. Specil Reltivity is not violted). Does this men tht sptilly seprted clssicl systems cn lso generte these trnsition proilities {p(, x, y)}? To properly nswer this question, let us develop generl frmework of input/output oxes tht will llow us to distinguish etween clssicl nd quntum oxes. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 8 / 16
The Theory of Correlted Boxes Definition Let A, B, X, nd Y e ritrry finite sets of integers. A set of correltions is collection of non-negtive numers {p(, x, y)} A, B,x X,y Y such tht A, B p(, x, y) = 1 for ll (x, y) X Y. The correltions re clled non-signling (NS) if p(, x, y) = p(, x, y ), x, y, y p(, x, y) = p(, x, y), y, x, x. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 9 / 16
The Theory of Correlted Boxes The correltions re clled quntum correltions (QC) if there exists iprtite quntum stte ρ AB s well s quntum opertions {A (x) } A nd {B (y) } B for ll (x, y) X Y such tht p(, x, y) = tr[(a (x) B (y) )ρab (A (x) B (y) ) ]. Wht re the type of correltions tht cn e generted y clssicl oxes? Clssicl systems re ssumed to stisfy property known s loclity. The following mening of loclity will suffice for our purposes. Two oxes re sid to function loclly if their correltions tke the form p(, x, y) = p( x)p( y). In other words, for every input nd output of one ox, the input/output sttistics of the other ox re not chnged (i.e. p(, x, y) = p(, x, y ) nd p(, x, y) = p(, x, y); compre with NS oxes). Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 10 / 16
The Theory of Correlted Boxes However, clssicl physics clerly does not require tht every pir of oxes ehve loclly. Clssicl physics llows for interction etween systems, nd this interction estlishes correltion. Consider drwer of full of socks nd ech sock hs two properties: (i) Color {Blck, White} nd (ii) Fric {Cotton, Nylon}. For simplicity lso ssume tht ech sock either elongs to right or left foot. Ech pir in the drwer consists of right nd left sock, ut the color nd fric properties re not mtched! Ech pir is descried y list λ CL F L C R F R = (Color C L, Fric F L, Color C R, Fric F R ), where C L /F L is the color/fric of the left sock nd C R /F R is the color/fric of the right sock. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 11 / 16
The Theory of Correlted Boxes Suppose tht the pirs of socks re distriuted in the drwer ccording to proilities p(λ CL F L C R F R ) with C L/R {B, W } nd F L/R {C, N}. I rech into the drwer nd rndomly select pir. I then put the left sock in lck ox nd give tht to Alice, nd I put the right sock in lck ox nd give tht to Bo. Ech oxes re designed so tht it revels only the Color or the Fric of the sock it holds, ut Alice nd Bo cn choose which property is reveled. Let x {C, F } e the property tht property tht Alice chooses to lern, nd y {C, F } the property tht Bo chooses to lern. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 12 / 16
The Theory of Correlted Boxes Their oxes re descried y the correltions p(x L, y L x, y). The outputs (x L, y L ) will depend not only on the choices (x, y) ut lso which socks were drwn from the drwer. To compute p(x L, y L x, y) we need to sum over ll the socks in the drwer: p(x L, y L x, y) = C L F L C R F R p(x L, y L, λ CL F L C R F R x, y) = C L F L C R F R p(x L, y L x, y, λ CL F L C R F R )p(λ CL F L C R F R ). Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 13 / 16
The Theory of Correlted Boxes We oviously hve tht p(x L, y L x, y, λ CL F L C R F R ) = p(x L x, λ CL F L C R F R )p(y L y, λ CL F L C R F R ) since λ CL F L C R F R specifies the vlue of x L nd y L for every choice of x nd y. Thus, Alice nd Bo s function loclly given the vlue of λ CL F L C R F R. Their correltions hve the form p(x L, y L x, y) = p(x L x, λ CL F L C R F R )p(y L y, λ CL F L C R F R )p(λ CL F L C R F R ). C L F L C R F R Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 14 / 16
The Theory of Correlted Boxes Alice nd Bo s oxes re not locl since they do not hve the form p(x L, y L x, y) = p(x L x)p(y L y). However their oxes re locl given the vlue of some hidden vrile λ CL F L C R F R. The oxes ecme dependent on the hidden vrile λ CL F L C R F R y some interction in the pst. Specificlly, this interction ws when I distriuted the pir of socks to ech ox. The oxes cme together t sometime in the pst when I distriuted the socks. This mde them correlted for ll future times, regrdless of how fr prt they were seprted. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 15 / 16
Locl Hidden Vriles In generl, set of correltions {p(, x, y)} re clled locl hidden vrile (LHV) correltions if they cn e written s p(, x, y) = λ p(λ)p A ( x, λ)p B ( y, λ),, x, y for some vrile λ with distriution p(λ). Here p A ( x, λ)p B ( y, λ) is product of locl proility distriutions. Wht is the reltionship etween the vrious correltions LHV, QC, nd NS? We hve lredy shown tht QC NS. To conclude tody s lecture, we will mke two simple oservtions: (i) LHV CQ nd (ii) The quntum correltions generted y every seprle stte elong to LHV. Quntum Nonloclity Pt. 2: No-Signling nd Locl Hidden Vriles My 1, 2018 16 / 16