Entanglement fluctuations

Entanglement fluctuations

Esko Keski-Vakkuri

University of Helsinki

Mat. Phys. Seminar, Jan/25/2017

ref. J. de Boer, J. Järvelä, E. K-V., arXiv:170?.????

(+ see the refs. in the preprint to appear )

Thoughts:

How to characterizequantum fluctuationsin quantum states ρ ? Observables -> probability density function,moments and cumulants

von Neumann entropy S as a random variable How to compute hSⁿi , hSⁿi_c?

Specifically: consider a subregion A and entanglement entropy Are therespecial relationsamong the moments?

If agravity dual exists, is there a telltale special relation?

Results:

Rényi entropy as a generating function for moments Focus onvarianceof entropy, ∆S²

Equals to aheat capacity Upper boundon ∆S²

Subregion A and entanglement entropy:

For (2d) CFTs:∆S_EE² = S_EE(leading terms)

Holds even throughout quenches, non-equilibrium to thermality Violatedwhen deforming away from criticality

Density matrix written with the modular Hamiltonian K (canonical normalization)

ρ = e^−K von Neumann entropy = hKi:

S_V = −Trρ log ρ = −hlog ρi = hKi Moments of S = moments of K:

S_Vⁿ = Tr[ρ(− log ρ)ⁿ] = hKⁿi Rényi entropies:

Sα= 1

1 − αlog Trρ^α Well known how to obtain SV:

S_V = lim

α→1S_α

But haven’t seen this (?): Rényi entropy as a generating function for the moments:

k(α) = Tr (ρ^α) = (1 − α)Sα

S_Vⁿ = hKⁿi = (−1)ⁿdⁿk(α) dαⁿ |α=1

The logarithm is then a generating function for the cumulants:

˜k(α) = log k(α)

hKⁿic= (−1)ⁿdⁿ˜k(α) dαⁿ |α=1. In particular, the variance

∆S²_V = hK²ic= (−1)²k˜⁽²⁾(α = 1) = k⁰⁰(1)k(1) − (k⁰(1))² (k(1))²

= hK²i · 1 − hKi²

1 = S_V² − (SV)²

Condensed matter theory: interest in the full eigenvalue spectrum, the eigenvalues {λm= e^−εn} of the density matrix ρ (or the eigenvalues εn

of the modular Hamiltonian) and their multiplicities g_m. Partition function k(α) =X

m

g_mλ^α_m=X

m

g_me^−αεm = Z(β_α)

with ‘temperature’

T_α= 1/β_α= 1/α . Thus,

eigenvalue spectrum ⇔ moments/cumulants of entropy Different ways to package the same information, but may highlight different features of the system.

A variant of Rényi entropy: modular entropy (Xi Dong):

S˜_α≡ α²∂_α α − 1 α S_α



Reduces to von Neumann entropy as α = 1. Satisfies thermodynamic relation:

S˜α= −∂Fα

∂Tα

, Define then

Eα= ∂

∂β_α(βαFα) . and a α-heat capacity

Cα≡ ∂Eα

∂T_α

Of particular interest is α = 1, where modular entropy = von Neumann entropy. Define the heat capacity (related work: Nakaguchi, Nishioka)

C = lim

α=1C_α

We show the heat capacity satisfies the usual thermodynamic relation C = ∆S²_V .

Specifically when ρ is a reduced density matrix, starting from a CFT vacuum, with half-line subregion, can map to a thermal CFT on hyperbolic cylinder (Hung, Myers, Smolkin, Yale). Then:

T_α=1→ Tthermal

SV = SEE→ Sthermal

C → Cthermal

C = ∆²S_EE → C_thermal= ∆S_thermal²

So map to the natural thermodynamic counterparts.

Example 1: Two-qubit system,

|θ, φi = cos(θ/2)|10i + e^iφsin(θ/2)|01i trace over one spin:

0.2 0.4 0.6 0.8 1.0x

0.1 0.2 0.3 0.4 0.5 0.6 0.7

svar(x) s(x)

(cf. Feld’man, Yurishchev 2009)

Example 2: Eigenvalue spectrum with N Gaussian distributed eigenvalues λiof ρ:

Entropy and variance: (σ = 0: pure state, σ = ∞: max mixed)

Upper bound on variance? (Warning: not yet triple-checked!) Consider N eigenvalues λi. ConstraintP

iλi= 1. Maximize ∆S². Find: can reduce to n1, n₂eigenvalues λ1, λ₂, rest =0.

Then,

∆S²≤ ∆Smax= 1

4(log N )²=1 4S_max²

where S_max= log N for a max mixed state. Different state gives ∆S_max² ! Anything special about it?

(For max mixed state: moments Sⁿ = (ln N )ⁿ; cumulants vanish for n > 1. )

How to measure entanglement? (Klich, Refael, Silva, Levitov,...) Quantum many-body system in its ground state |ψi

Fluctuating number of (quasi)particles (or U (1) charge) in region A, hN_Ai = Tr(ρ_AN ).ˆ

Cumulants Cnof the number distribution.[H.F.Song et. al. PRB.85.035409]

Entanglement associated w/ (entangled) particles in/out of A.

The Rényi entropy of subregion A related to cumulants of particle number,

S_α(A) =

∞

X

k=1

s^(α)_k C_2k

Simplest case systems ∼ free fermions (Song et al, Calabrese et al.).

Find:

S_EE(A) =

∞

X

k=1

(2π)^2k|B2k|

(2k)! C_2k= π²

3 C₂+π⁴

45C₄+ · · ·

∆S_EE² (A) = π² 3 C₂+

∞

X

k=2

2(2π)^2k|B_2k|

(2k − 1)! C_2k= π²

3 C₂+8π⁴

45 C₄+ · · · ,

In the limit of large particle number (density), C2term dominates, so

∆S_EE² = SEE ! (N → ∞) (∗)

Speculation:assumethe system has a gravity dual. Typically invoke large N limit of particle species, thus average particle number

N_A= 1 N

N

X

i=1

N_A⁽ⁱ⁾,

in the large N limit central limit theorem -> Gaussian N_A-> (∗)!

Caveats:

Strong coupling

Non-local observables (e.g. Wilson loops in gauge sector)

1+1 CFTs: General analytical results for the Rényi entropies exist, Tr(ρ^α_A) = cαe^−12c(^α−_α¹)^WA

implying (leading terms)

∆S_EE² = S_EE = c

6W_A+ · · · Examples:

WA=



























2 log ^l_

infinite system, T = 0 2 log _π^L sin^lπ_L

finite system, size L, T = 0 2 log_β

πsinh^lπ_β

infinite system, T > 0 ; log_β

πcosh(2πt/β)

global quench at t = 0 log

t²+λ²

λ/2



local quench at t = 0 ,

In higher dimensions, can derive

∆S_EE² ∼ SEE

but more subtleties; associated with sensitivity to different regularization schemes

What happens away from criticality / conformal symmetry?

Consider the Heisenberg quantum XY spin chain

H = −

∞

X

j=−∞

(1 + γ)σ_j^xσ_j+1^x + (1 − γ)σ_j^yσ^y_j+1+ hσ_j^z ,

Franchini, Its, Korepin; arXiv:0707.2534

Franchini, Its, Korepin computed the Rényi entropies. Derive from that SEE− ∆S_EE² .

Entropy, variance deviate deforming away from critical lines:

(A) (B)

Kuva: Samples of the difference SEE− ∆S_EE² near the critical line hc= 2 (A) and near the critical line γ = 0 (B) .