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The classical entropy of quantum states1 Jan Philip Solovej Department of Mathematics University of Copenhagen Spectral Days, CIRM June 10, 2014 1 Joint work with Elliott Lieb The classical entropy of quantum states Jan Philip Solovej Slide 1/13 Abstract of Talk • To quantify the inherent uncertainty of quantum states Wehrl • • • • (’79) suggested a definition of their classical entropy based on the coherent state transform. He conjectured that this classical entropy is minimized by states that also minimize the Heisenberg uncertainty inequality, i.e., Gaussian coherent states. Lieb (’78) proved this conjecture and conjectured that the same holds when Euclidean Glauber coherent states are replaced by SU(2) Bloch coherent states. This conjecture was settled last year in joint work with Lieb. Recently we simplified the proof and generalized it to SU (N ) for general N . I will present this here. In proving the conjecture we study the quantum channels known as Universal Quantum Cloning Machines and determine their minimal output entropy. Thanks to K. Bradler for pointing this out. The classical entropy of quantum states Jan Philip Solovej Slide 2/13 Outline of Talk 1 Coherent states and quantization 2 States of minimal classical entropy 3 SU (N )-coherent states 4 Classical SU (N ) entropy inequality 5 Generalization to Quantum Channels 6 Formulation in terms of majorization 7 Using bosonic 2nd quantization 8 A normal ordering formula 9 The classical limit (if time permits) The classical entropy of quantum states Jan Philip Solovej Slide 3/13 Quantization • Classical phase space: M = R2n position and momentum (q, p). • Quantum description: Hilbert space H = L2 (Rn ). • Quantization: Function A on M to operator Op(A) on H. • Pure states2 : Described by normalized ψ ∈ L2 (Rn ) gives “distribution” on phase space Φψ such that ZZ −n hψ, Op(A)ψi = (2π) Φψ (q, p)A(q, p)dqdp • Weyl quantization leads to Φψ (q, p) Wigner distribution, which is not necessarily positive. • Better to use Wick or coherent state quantization 2 The state is really represented by the 1-dim projection |ψihψ|. More general non-pure states represented by density matrices (operators): 0 ≤ ρ, Tr ρ = 1. The classical entropy of quantum states Jan Philip Solovej Slide 4/13 Coherent state quantization Coherent states, i.e., states of minimal Heisenberg uncertainty fq,p (x) = π −n/4 exp(−(x − q)2 /2 + ipx) ∈ L2 (Rn ) satisfy (x + ∇)fq,p = (q + ip)fq,p . They define quantization map ZZ −n Op(A) = (2π) A(q, p)|fq,p ihfq,p |dqdp. leads to lower or covariant symbol or Husimi Q-function Φψ (q, p) = |hfq,p |ψi|2 . RR Then 0 ≤ Φψ (q, p) ≤ 1 and (2π)−n Φψ (q, p)dqdp = 1. Wehrl classical entropy: ZZ cl −n S (ψ) = (2π) −Φψ (q, p) log (Φψ (q, p)) dqdp. The classical entropy of quantum states Jan Philip Solovej Slide 5/13 States of minimal entropy Theorem (Lieb ’78, Conjectured by Wehrl) States of minimal entropy are states of minimal Heisenberg uncertainty, i.e., for all ψ and all q, p S cl (ψ) ≥ S cl (fq,p ). Proof based on sharp Young and Hausdorff-Young inequalities3 . Carlen ’91 proved “uniqueness” of minimizers. In fact, −t log(t) may be replaced by any concave function: Theorem (Lieb-Solovej ’12) For all continuous concave f : [0, 1] → R, f (0) = 0 ZZ ZZ 0 0 0 0 f Φψ (q , p ) dq dp ≥ f Φfq,p (q 0 , p0 ) dq 0 dp0 3 Note that both Y and HY inequalities are optimized by Gaussians The classical entropy of quantum states Jan Philip Solovej Slide 6/13 SU (N ) coherent states • Consider the Hilbert space HM = NM SYM C N, i.e., the space of M Bosons with N degrees of freedom. • SU (N ) acts irreducibly on HM (not all irr. repr. unless N = 2). • Special states on HM , coherent vectors, highest weight vectors, pure condensates: ⊗M u, u ∈ CN . • The state | ⊗M uih⊗M u| depends only on the unit vector u ∈ CN modulo a phase, i.e., really u ∈ CPN −1 • CPN −1 is a classical phase space and HM is a quantization. Quantization map: Z Op(A) = dim HM A(u)|⊗M uih⊗M u|du, Op(1) = I CPN −1 du is SU (N ) invariant (Liouville) measure on CPN −1 . • Husimi function for general state ρ on HM Φ∞ (ρ)(u) = h⊗M u|ρ| ⊗M ui, The classical entropy of quantum states Jan Philip Solovej Slide 7/13 Classical SU (N ) entropy inequality Theorem (Classical “entropy” inequality, Lieb-Solovej ’13) For all integers M, N , all concave f : [0, 1] → R, all states ρ on HM , and all v ∈ CPN −1 Z Z f (Φ∞ (ρ)(u)) du ≥ f Φ∞ (| ⊗M vih⊗M v|)(u) du CPN −1 CPN −1 • For N = 2, i.e., SU (2), CPN −1 = S2 is the Bloch sphere. In this case and for the entropy function f (t) = −t log(t) the result was conjectured by Lieb 1978. A proof of this case is to appear soon and is on the archive. • Special cases of M for N = 2 and the entropy function had been considered by Schupp ’99, Scutaru ’02 • For N > 2 the compact manifold CPN −1 is not a sphere. The classical entropy of quantum states Jan Philip Solovej Slide 8/13 Generalization to Quantum channels Φ∞ is a map from a quantum state to a classical prob. distribution. We generalize to completely positive trace preserving maps, i.e., quantum channels Φk from operators on HM to operators on HM +k , Φk (ρ) = CM,N.k Psym (ρ ⊗ I⊗k CN )Psym . The value of the normalization constant CM,N.k is not important here. The channels Φk are known as universal quantum cloners. We determine their minimal output entropy. Theorem (Lieb-Solovej ’13) For all M , all k, all concave f : [0, 1] → R, all states ρ on HM , and all v ∈ CPN −1 Tr HM +k f Φk (ρ) ≥ Tr HM +k f Φk (| ⊗M vih⊗M v|) The classical entropy of quantum states Jan Philip Solovej Slide 9/13 Formulation in terms of majorization The classical entropy inequality follows from the classical limit: dim HM +k k 1 Tr HM +k f Φ (ρ) lim k→∞ dim HM +k dim HM Z = f (Φ∞ (ρ)(u)) du. CPN −1 Alternatively to using traces of concave functions the previous theorem may be equivalently (Karamata’s Theorem) rephrased as Theorem For all states ρ on HM and all v ∈ CPN −1 the ordered eigenvalues of Φk (| ⊗M vih⊗M v|) majorizes the ordered eigenvalues of Φk (ρ). Def. a1 ≥ a2 ≥ · · · ≥ aJ majorizes b1 ≥ b2 ≥ · · · ≥ bJ if m X j=1 aj ≥ m X bj , m ≤ J − 1, j=1 The classical entropy of quantum states and J X j=1 Jan Philip Solovej aj = J X j=1 Slide 10/13 bj . Using bosonic 2nd quantization We introduce the Bosonic annihilation operators ai , i = 1, . . . , N (indexing a basis ei of CN ) and their adjoints the creation operators a∗i : a∗i : ∞ M HM → M =0 ∞ M HM , a∗i (HM ) ⊆ HM +1 M =0 a∗i φ √ = M + 1Psym (ei ⊗ φ) for φ ∈ HM . Then in Kraus form 0 Φk (ρ) = CM,N,k X a∗i1 · · · a∗ik ρaik · · · ai1 i1 ,...,ik Two observations: • Ordered eigenvalue sums are convex: may assume ρ = |ψihψ|. • The non-zero eigenvalues of Φk (|ψihψ|) equal the non-zero eigenvalues (counting multiplicities) of the matrix 0 CM,N,k hψ|aik · · · ai1 a∗j1 · · · a∗jk |ψi. The classical entropy of quantum states Jan Philip Solovej Slide 11/13 A normal ordering formula The matrix (the outcome of the transpose channel to Φk ) Γi1 ,...,ik ;j1 ,...jk = hψ|aik · · · ai1 a∗j1 · · · a∗jk |ψi. represents an operator Γ on Hk . It is the anti-normal ordering of the matrix elements of the reduced k-particle density matrix (γψ )i1 ,...,ik ;j1 ,...jk = hψ|a∗j1 · · · a∗jk aik · · · ai1 |ψi. In fact, normal ordering gives (See also Chiribella ’10) Γ= k X (k−`) C` Φ` (γψ ) `=0 for coefficients C` > 0. The majorization theorem follows by induction on k: Induction start: Φ0 = Id. Induction step: (k−`) Φ` (γ⊗M v ) = cM,k,` Φ` (| ⊗k−` vih⊗k−` v|), (k−`) majorizes Φ` (γψ (k−`) (cM,k,` = Tr γψ ) for all ` < k, but ` = k obvious. The classical entropy of quantum states Jan Philip Solovej Slide 12/13 ) The classical limit (only one sided inequality) Will show a version of the Berezin-Lieb inequality: For f concave Z 1 dim HM +k k Tr HM +k f Φ (ρ) ≤ f (Φ∞ (ρ)(u)) du dim HM +k dim HM CPN −1 If ρ = | ⊗M vih⊗M v| right side explicitly limit k → ∞ of left side: That is all we need! Jensen’s inequality implies Berezin-Lieb-inequality: 1 dim HM +k k Tr HM +k f Φ (ρ) dim HM +k dim HM Z M +k dim HM +k k M +k = ⊗ u f Φ (ρ) ⊗ u du dim HM CPN −1 Z dim HM +k D M +k k M +k E ≤ f ⊗ u Φ (ρ) ⊗ u du dim HM CPN −1 Z Z M M = f h⊗ u|ρ| ⊗ ui du = f (Φ∞ (ρ)(u)) du. CPN −1 The classical entropy of quantum states CPN −1 Jan Philip Solovej Slide 13/13