Ph 219b/CS 219b. Exercises Due: Wednesday 9 March 2016

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1 1 Ph 219b/CS 219b Eercises Due: Wednesdy 9 Mrch Positivity of quntum reltive entropy ) Show tht ln 1 for ll positive rel, with equlity iff = 1. b) The (clssicl) reltive entropy of probbility distribution {p()} reltive to {q()} is defined s D(p q) p() (logp() logq()). (1) Show tht D(p q) 0, (2) with equlity iff the probbility distributions re identicl. Hint: pply the inequlity from () to ln(q()/p()). c) The quntum reltive entropy of the density opertor ρ with respect to σ is defined s D(ρ σ) = tr ρ(logρ logσ). (3) Let {p i } denote the eigenvlues of ρ nd {q } denote the eigenvlues of σ. Show tht D(ρ σ) = ( p i log p i ) D i log q, (4) i where D i is doubly stochstic mtri. Epress D i in terms of the eigensttes of ρ nd σ. ( mtri is doubly stochstic if its entries re nonnegtive rel numbers, where ech row nd ech column sums to one.) d) Show tht if D i is doubly stochstic, then (for ech i) ( ) log D i q D i logq, (5) with equlity only if D i = 1 for some.

2 2 e) Show tht where r i = D iq. D(ρ σ) H(p r), (6) f) Show tht D(ρ σ) 0, with equlity iff ρ = σ. 3.2 Properties of Von Neumnn entropy ) Use nonnegtivity of quntum reltive entropy to prove the subdditivity of Von Neumnn entropy H(ρ B ) H(ρ ) + H(ρ B ), (7) with equlity iff ρ B = ρ ρ B. Hint: Consider the reltive entropy of ρ B nd ρ ρ B. b) Use subdditivity to prove the concvity of the Von Neumnn entropy: H( p ρ ) p H(ρ ). (8) Hint: Consider ρ B = p (ρ ) ( ) B, (9) where the sttes { B } re mutully orthogonl. c) Use the condition H(ρ B ) = H(ρ ) + H(ρ B ) iff ρ B = ρ ρ B (10) to show tht, if ll p s re nonzero, ( ) H p ρ = iff ll the ρ s re identicl. d) Use subdditivity to prove the tringle inequlity: p H(ρ ) (11) H(ρ B ) H(ρ ) H(ρ B ). (12) Hint: Construct purifiction of ρ B introduce third system C nd consider Φ BC such tht tr C ( Φ Φ ) = ρ B ; (13) then use the subdditivity reltions H(ρ BC ) H(ρ B ) + H(ρ C ) nd H(ρ C ) H(ρ ) + H(ρ C ).

3 3 3.3 Seprbility nd mjoriztion The hllmrk of entnglement is tht in n entngled stte the whole is less rndom thn its prts. But in seprble stte the correltions re essentilly clssicl nd so re epected to dhere to the clssicl principle tht the prts re less disordered thn the whole. The objective of this problem is to mke this epecttion precise by showing tht if the biprtite (mied) stte ρ B is seprble, then λ(ρ B ) λ(ρ ), λ(ρ B ) λ(ρ B ). (14) Here λ(ρ) denotes the vector of eigenvlues of ρ, nd denotes mjoriztion. seprble stte cn be relized s n ensemble of pure product sttes, so tht if ρ B is seprble, it my be epressed s ρ B = p ψ ψ ϕ ϕ. (15) We cn lso digonlize ρ B, epressing it s ρ B = j r j e j e j, (16) where { e j } denotes n orthonorml bsis for B; then by the HJW theorem, there is unitry mtri V such tht rj e j = V j p ψ ϕ. (17) lso note tht ρ cn be digonlized, so tht ρ = p ψ ψ = µ s µ f µ f µ ; (18) here { f µ } denotes n orthonorml bsis for, nd by the HJW theorem, there is unitry mtri U such tht p ψ = µ U µ sµ f µ. (19) Now show tht there is doubly stochstic mtri D such tht r j = µ D jµ s µ. (20)

4 4 Tht is, you must check tht the entries of D jµ re rel nd nonnegtive, nd tht j D jµ = 1 = µ D jµ. Thus we conclude tht λ(ρ B ) λ(ρ ). Just by interchnging nd B, the sme rgument lso shows tht λ(ρ B ) λ(ρ B ). Remrk: Note tht it follows from the Schur concvity of Shnnon entropy tht, if ρ B is seprble, then the von Neumnn entropy hs the properties H(B) H() nd H(B) H(B). Thus, for seprble sttes, conditionl entropy is nonnegtive: H( B) = H(B) H(B) 0 nd H(B ) = H(B) H() 0. In contrst, if the stte of B is n entngled pure stte, then H(B) = 0 nd H(B ) = H( B) < Entnglement of typicl biprtite pure sttes Suppose tht pure stte is chosen t rndom on the biprtite system B, where /d B 1. Then with high probbility the density opertor on will be very nerly mimlly mied. The purpose of this problem is to derive this property. To begin with, we will clculte the vlue of tr ρ 2, where denotes the verge over ll pure sttes { ϕ } of B, nd ρ = tr B ( ϕ ϕ ). ) It is convenient to evlute tr ρ 2 using trick. Imgine introducing copy B of the system B. Show tht tr ρ 2 = tr B B [(S I BB )( ϕ ϕ ) B ϕ ϕ B )], (21) where S denotes the swp opertor S : ϕ ψ ψ ϕ. (22) b) We wish to verge the epression found in () over ll pure sttes ϕ. Rther thn go into the detils of how such n verge is defined, I will simply ssert tht ϕ ϕ ϕ ϕ = C Π, (23) where C is constnt nd Π denotes the projector onto the subspce of tht is symmetric under interchnge of nd. Eq. (23) cn be proved using invrince properties of the verge nd some group representtion theory, but I hope you

5 5 will regrd it s obvious. The stte being verged is symmetric, nd the verge should not distinguish ny symmetric stte from ny other symmetric stte. Epress the constnt C in terms of the dimension d =. c) Use the property Π = 2 1 (I + S ) to evlute the epression found in (). Show tht tr ρ 2 = + d B d B + 1. (24) d) Now estimte the verge L 2 distnce of ρ from the mimlly 1 mied density opertor I, where M 2 = trm M; show tht ρ 1 I 2 1. (25) db Hints: First estimte ρ d 1 I 2 2 using eq. (24) nd the obvious property ρ = d 1 I. Then show tht for ny nonnegtive function f, it follows from the Cuchy-Schwrz inequlity tht f f, nd use this property to estimte ρ 1 I 2. e) Finlly, estimte the verge L 1 distnce of ρ from the mimlly mied density opertor, where M 1 = tr M M. Use the Cuchy-Schwrz inequlity to show tht M 1 d M 2, if M is d d mtri, nd tht therefore ρ 1 I 1 d d B. (26) It follows from (d) tht the verge entnglement entropy of nd B is close to miml for /d B 1: H() log 2 /2d B ln2, though you re not sked to prove this bound. Thus, if for emple is 50 qubits nd B is 100 qubits, the typicl entropy devites from miml by only bout