A Violation Of The Area Law For Fermionic Entanglement Entropy

Transcription

1 A Violation Of The Area Law For Fermionic Entanglement Entropy or: How much can you say in a phone call? Robert Helling 1 Hajo Leschke 2 and Wolfgang Spitzer 2 1 Arnold Sommerfeld Center Ludwig-Maximilians-Universität München 2 Institut für Theoretische Physik Universität Erlangen-Nürnberg Nordic String Meeting, March 27, 2009

2 Outline Entanglement Entropy Entanglement Entropy and the Area Law Applications of Entanglement Entropy Fermionic Entanglement Entropy Not the Strict Area Law Ingredients of a Proof The One-Dimensional Case Summary

3 Entanglement Entropy the price of ignorance Take your favourite QFT model in R n and put it in its ground state Ψ. Restrict yourself to measurements outside a compact black-box Ω R n Outside the black-box the system is described by a density matrix ϱ Ω = tr Hblack box Ψ Ψ In general, ϱ Ω is a mixed state with entanglement entropy S(Ω) = tr(ϱ Ω log ϱ Ω ) > 0.

4 Entanglement Entropy the price of ignorance Take your favourite QFT model in R n and put it in its ground state Ψ. Restrict yourself to measurements outside a compact black-box Ω R n Outside the black-box the system is described by a density matrix ϱ Ω = tr Hblack box Ψ Ψ In general, ϱ Ω is a mixed state with entanglement entropy S(Ω) = tr(ϱ Ω log ϱ Ω ) > 0.

5 Entanglement Entropy the price of ignorance Take your favourite QFT model in R n and put it in its ground state Ψ. Restrict yourself to measurements outside a compact black-box Ω R n Outside the black-box the system is described by a density matrix ϱ Ω = tr Hblack box Ψ Ψ In general, ϱ Ω is a mixed state with entanglement entropy S(Ω) = tr(ϱ Ω log ϱ Ω ) > 0.

6 Entanglement Entropy the price of ignorance Take your favourite QFT model in R n and put it in its ground state Ψ. Restrict yourself to measurements outside a compact black-box Ω R n Outside the black-box the system is described by a density matrix ϱ Ω = tr Hblack box Ψ Ψ In general, ϱ Ω is a mixed state with entanglement entropy S(Ω) = tr(ϱ Ω log ϱ Ω ) > 0.

7 Entanglement Entropy and the Area Law S(Ω) is a complicated function of Ω and the QFT model. Semi-classical asymptotics: Blow up Ω by a factor of R. A universal area law S(RΩ) R n 1 was observed for many models: Bosons, Lattice models, CFT s,... (NB: read R 0 as log R). Intuition: Entanglement originates from correlations across the boundary Ω in a layer of width ξ. (but: CFT). Gioev and Klich first observerved that for free fermions S(RΩ) O(R n 1 log R). We improve this including an area type geometric coefficient by proving a special case of Widom s conjecture.

8 Entanglement Entropy and the Area Law S(Ω) is a complicated function of Ω and the QFT model. Semi-classical asymptotics: Blow up Ω by a factor of R. A universal area law S(RΩ) R n 1 was observed for many models: Bosons, Lattice models, CFT s,... (NB: read R 0 as log R). Intuition: Entanglement originates from correlations across the boundary Ω in a layer of width ξ. (but: CFT). Gioev and Klich first observerved that for free fermions S(RΩ) O(R n 1 log R). We improve this including an area type geometric coefficient by proving a special case of Widom s conjecture.

9 Entanglement Entropy and the Area Law S(Ω) is a complicated function of Ω and the QFT model. Semi-classical asymptotics: Blow up Ω by a factor of R. A universal area law S(RΩ) R n 1 was observed for many models: Bosons, Lattice models, CFT s,... (NB: read R 0 as log R). Intuition: Entanglement originates from correlations across the boundary Ω in a layer of width ξ. (but: CFT). Gioev and Klich first observerved that for free fermions S(RΩ) O(R n 1 log R). We improve this including an area type geometric coefficient by proving a special case of Widom s conjecture.

10 Entanglement Entropy and the Area Law S(Ω) is a complicated function of Ω and the QFT model. Semi-classical asymptotics: Blow up Ω by a factor of R. A universal area law S(RΩ) R n 1 was observed for many models: Bosons, Lattice models, CFT s,... (NB: read R 0 as log R). Intuition: Entanglement originates from correlations across the boundary Ω in a layer of width ξ. (but: CFT). Gioev and Klich first observerved that for free fermions S(RΩ) O(R n 1 log R). We improve this including an area type geometric coefficient by proving a special case of Widom s conjecture.

11 Entanglement Entropy and the Area Law S(Ω) is a complicated function of Ω and the QFT model. Semi-classical asymptotics: Blow up Ω by a factor of R. A universal area law S(RΩ) R n 1 was observed for many models: Bosons, Lattice models, CFT s,... (NB: read R 0 as log R). Intuition: Entanglement originates from correlations across the boundary Ω in a layer of width ξ. (but: CFT). Gioev and Klich first observerved that for free fermions S(RΩ) O(R n 1 log R). We improve this including an area type geometric coefficient by proving a special case of Widom s conjecture.

12 Entanglement Entropy and the Area Law S(Ω) is a complicated function of Ω and the QFT model. Semi-classical asymptotics: Blow up Ω by a factor of R. A universal area law S(RΩ) R n 1 was observed for many models: Bosons, Lattice models, CFT s,... (NB: read R 0 as log R). Intuition: Entanglement originates from correlations across the boundary Ω in a layer of width ξ. (but: CFT). Gioev and Klich first observerved that for free fermions S(RΩ) O(R n 1 log R). We improve this including an area type geometric coefficient by proving a special case of Widom s conjecture.

13 Applications of Entanglement Entropy Formal similarity to Beckenstein-Hawking entropy of black holes. Could be mechanism of entropy associated to horizons. Quantum information theory (measures quantumness of a state) Condensed matter: Measure for convergence of density matrix renormalization group method (DMRG)

14 Applications of Entanglement Entropy Formal similarity to Beckenstein-Hawking entropy of black holes. Could be mechanism of entropy associated to horizons. Quantum information theory (measures quantumness of a state) Condensed matter: Measure for convergence of density matrix renormalization group method (DMRG)

15 Applications of Entanglement Entropy Formal similarity to Beckenstein-Hawking entropy of black holes. Could be mechanism of entropy associated to horizons. Quantum information theory (measures quantumness of a state) Condensed matter: Measure for convergence of density matrix renormalization group method (DMRG)

16 Applications of Entanglement Entropy Formal similarity to Beckenstein-Hawking entropy of black holes. Could be mechanism of entropy associated to horizons. Quantum information theory (measures quantumness of a state) Condensed matter: Measure for convergence of density matrix renormalization group method (DMRG)

17 Fermionic Entanglement Entropy Quasi-free fermions at finite density are characterized by the Fermi sea Γ R n of forbidden states with E < ɛ F in momentum space. At zero temperature, all states with E < ɛ F are occupied and the system is in the state ϱ(x, y) = P Γ = χ Γ (x y). One-particle Hilbert space H = L 2 (Ω) L 2 (R n \ Ω), Fock spaces F(H) = F(L 2 (Ω)) F(L 2 (R n \ Ω)) Reduced many particle state ϱ Ω is characterised by one-particle state ρ = Q Ω P Γ Q Ω with projection Q Ω given by multiplication by characteristic function χ Ω (x). Entanglement entropy S(Ω, Γ) = tr(ρ log ρ + (1 ρ) log(1 ρ)).

18 Fermionic Entanglement Entropy Quasi-free fermions at finite density are characterized by the Fermi sea Γ R n of forbidden states with E < ɛ F in momentum space. At zero temperature, all states with E < ɛ F are occupied and the system is in the state ϱ(x, y) = P Γ = χ Γ (x y). One-particle Hilbert space H = L 2 (Ω) L 2 (R n \ Ω), Fock spaces F(H) = F(L 2 (Ω)) F(L 2 (R n \ Ω)) Reduced many particle state ϱ Ω is characterised by one-particle state ρ = Q Ω P Γ Q Ω with projection Q Ω given by multiplication by characteristic function χ Ω (x). Entanglement entropy S(Ω, Γ) = tr(ρ log ρ + (1 ρ) log(1 ρ)).

19 Fermionic Entanglement Entropy Quasi-free fermions at finite density are characterized by the Fermi sea Γ R n of forbidden states with E < ɛ F in momentum space. At zero temperature, all states with E < ɛ F are occupied and the system is in the state ϱ(x, y) = P Γ = χ Γ (x y). One-particle Hilbert space H = L 2 (Ω) L 2 (R n \ Ω), Fock spaces F(H) = F(L 2 (Ω)) F(L 2 (R n \ Ω)) Reduced many particle state ϱ Ω is characterised by one-particle state ρ = Q Ω P Γ Q Ω with projection Q Ω given by multiplication by characteristic function χ Ω (x). Entanglement entropy S(Ω, Γ) = tr(ρ log ρ + (1 ρ) log(1 ρ)).

20 Fermionic Entanglement Entropy Quasi-free fermions at finite density are characterized by the Fermi sea Γ R n of forbidden states with E < ɛ F in momentum space. At zero temperature, all states with E < ɛ F are occupied and the system is in the state ϱ(x, y) = P Γ = χ Γ (x y). One-particle Hilbert space H = L 2 (Ω) L 2 (R n \ Ω), Fock spaces F(H) = F(L 2 (Ω)) F(L 2 (R n \ Ω)) Reduced many particle state ϱ Ω is characterised by one-particle state ρ = Q Ω P Γ Q Ω with projection Q Ω given by multiplication by characteristic function χ Ω (x). Entanglement entropy S(Ω, Γ) = tr(ρ log ρ + (1 ρ) log(1 ρ)).

21 Fermionic Entanglement Entropy Quasi-free fermions at finite density are characterized by the Fermi sea Γ R n of forbidden states with E < ɛ F in momentum space. At zero temperature, all states with E < ɛ F are occupied and the system is in the state ϱ(x, y) = P Γ = χ Γ (x y). One-particle Hilbert space H = L 2 (Ω) L 2 (R n \ Ω), Fock spaces F(H) = F(L 2 (Ω)) F(L 2 (R n \ Ω)) Reduced many particle state ϱ Ω is characterised by one-particle state ρ = Q Ω P Γ Q Ω with projection Q Ω given by multiplication by characteristic function χ Ω (x). Entanglement entropy S(Ω, Γ) = tr(ρ log ρ + (1 ρ) log(1 ρ)).

22 Not the Strict Area Law We find (up to lower order terms in R) S(RΩ, Γ) This is our main result. ( ) R n 1 ln(r) 2π 4π 2 da(x)da(p) n x n p. Ω Γ There are strong indications that this inequality is in fact the correct asymptotic scaling Violation of area law but simple surface expression, e.g. no curvature terms.

23 Not the Strict Area Law We find (up to lower order terms in R) S(RΩ, Γ) This is our main result. ( ) R n 1 ln(r) 2π 4π 2 da(x)da(p) n x n p. Ω Γ There are strong indications that this inequality is in fact the correct asymptotic scaling Violation of area law but simple surface expression, e.g. no curvature terms.

24 Not the Strict Area Law We find (up to lower order terms in R) S(RΩ, Γ) This is our main result. ( ) R n 1 ln(r) 2π 4π 2 da(x)da(p) n x n p. Ω Γ There are strong indications that this inequality is in fact the correct asymptotic scaling Violation of area law but simple surface expression, e.g. no curvature terms.

25 Not the Strict Area Law We find (up to lower order terms in R) S(RΩ, Γ) This is our main result. ( ) R n 1 ln(r) 2π 4π 2 da(x)da(p) n x n p. Ω Γ There are strong indications that this inequality is in fact the correct asymptotic scaling Violation of area law but simple surface expression, e.g. no curvature terms.

26 Ingredients of a Proof We wish to compute tr(q Ω P Γ Q Ω ) k = dx 1 dx k χ Γ (x 1 x 2 ) χ Γ (x 2 x 3 ) χ Γ (x k x 1 ) Ω with χ Γ (x) = dp Γ e ix p. (2π) n/2 Asymptotic integrals using steepest descent Cancelations and localiazations like R 1 dx eiαx x Ω = δ α,0 ln(r) + O(1) Crucial observation: for p Γ and v R n, asymptotically dxθ(x v) χ Γ ( x)e ix p = i(2π) d/2 1 sgn(v n p ) ln(r) RΩ independent of Ω or Γ.

27 Ingredients of a Proof We wish to compute tr(q Ω P Γ Q Ω ) k = dx 1 dx k χ Γ (x 1 x 2 ) χ Γ (x 2 x 3 ) χ Γ (x k x 1 ) Ω with χ Γ (x) = dp Γ e ix p. (2π) n/2 Asymptotic integrals using steepest descent Cancelations and localiazations like R 1 dx eiαx x Ω = δ α,0 ln(r) + O(1) Crucial observation: for p Γ and v R n, asymptotically dxθ(x v) χ Γ ( x)e ix p = i(2π) d/2 1 sgn(v n p ) ln(r) RΩ independent of Ω or Γ.

28 Ingredients of a Proof We wish to compute tr(q Ω P Γ Q Ω ) k = dx 1 dx k χ Γ (x 1 x 2 ) χ Γ (x 2 x 3 ) χ Γ (x k x 1 ) Ω with χ Γ (x) = dp Γ e ix p. (2π) n/2 Asymptotic integrals using steepest descent Cancelations and localiazations like R 1 dx eiαx x Ω = δ α,0 ln(r) + O(1) Crucial observation: for p Γ and v R n, asymptotically dxθ(x v) χ Γ ( x)e ix p = i(2π) d/2 1 sgn(v n p ) ln(r) RΩ independent of Ω or Γ.

29 Ingredients of a Proof We wish to compute tr(q Ω P Γ Q Ω ) k = dx 1 dx k χ Γ (x 1 x 2 ) χ Γ (x 2 x 3 ) χ Γ (x k x 1 ) Ω with χ Γ (x) = dp Γ e ix p. (2π) n/2 Asymptotic integrals using steepest descent Cancelations and localiazations like R 1 dx eiαx x Ω = δ α,0 ln(r) + O(1) Crucial observation: for p Γ and v R n, asymptotically dxθ(x v) χ Γ ( x)e ix p = i(2π) d/2 1 sgn(v n p ) ln(r) RΩ independent of Ω or Γ.

30 The One-Dimensional Case In n = 1 dimension, Ω and Γ are the unions of m and l intervals respectively. Direct computation leads to asymptotically S(RΩ, Γ) = log(r)ml/π 2 independent of the size and relative positions of the intervals (as long as they do not touch). This measures the amount of information that can be transmitted in a phone call over time interval Ω and frequency band Γ.

31 The One-Dimensional Case In n = 1 dimension, Ω and Γ are the unions of m and l intervals respectively. Direct computation leads to asymptotically S(RΩ, Γ) = log(r)ml/π 2 independent of the size and relative positions of the intervals (as long as they do not touch). This measures the amount of information that can be transmitted in a phone call over time interval Ω and frequency band Γ.

32 The One-Dimensional Case In n = 1 dimension, Ω and Γ are the unions of m and l intervals respectively. Direct computation leads to asymptotically S(RΩ, Γ) = log(r)ml/π 2 independent of the size and relative positions of the intervals (as long as they do not touch). This measures the amount of information that can be transmitted in a phone call over time interval Ω and frequency band Γ.

33 Summary S(RΩ, Γ) ( ) R n 1 ln(r) 2π 4π 2 da(x)da(p) n x n p. Ω Γ