Mutual Information in Conformal Field Theories in Higher Dimensions - PowerPoint PPT Presentation

Mutual Information in Conformal Field Theories in Higher Dimensions John Cardy University of Oxford Conference on Mathematical Statistical Physics Kyoto 2013 arXiv:1304.7985; J. Phys. A: Math. Theor. 46 (2013) 285402

Outline ◮ Quantum entanglement in general and its quantification ◮ Path integral approach ◮ Area law in higher dimensions ◮ Mutual information for a general CFT ◮ Results for a gaussian free field ◮ Universal logarithmic corrections

Quantum Entanglement (Bipartite, Pure State) ◮ quantum system in a pure state | Ψ � , density matrix ρ = | Ψ �� Ψ | ◮ H = H A ⊗ H B ◮ Alice can make unitary transformations and measurements only in A , Bob only in the complement B ◮ in general Alice’s measurements are entangled with those of Bob ◮ example: two spin- 1 2 degrees of freedom | ψ � = cos θ | ↑� A | ↓� B + sin θ | ↓� A | ↑� B

Measuring bipartite entanglement in pure states ◮ Schmidt decomposition: � | Ψ � = c j | ψ j � A ⊗ | ψ j � B j j c 2 with c j ≥ 0, � j = 1. ◮ one quantifier of the amount of entanglement is the entropy | c j | 2 log | c j | 2 = S B � S A ≡ − j ◮ if c 1 = 1, rest zero, S = 0 and | Ψ � is unentangled ◮ if all c j equal, S ∼ log min ( dim H A , dim H B ) – maximal entanglement

◮ equivalently, in terms of Alice’s reduced density matrix: ρ A ≡ Tr B | Ψ �� Ψ | S A = − Tr A ρ A log ρ A = S B ◮ the von Neumann entropy: similar information is contained in the Rényi entropies ( n ) = ( 1 − n ) − 1 log Tr A ρ A n S A ◮ S A = lim n → 1 S A ( n )

◮ other measures of entanglement exist, but entropy has several nice properties: additivity, convexity, . . . ◮ it increases under Local Operations and Classical Communication (LOCC) ◮ it gives the amount of classical information required to specify ρ A (important for numerical computations) ◮ it gives a basis-independent way of identifying and characterising quantum phase transitions ◮ in a relativistic theory the entanglement in the vacuum encodes all the data of the theory (spectrum, anomalous dimensions, . . . )

Entanglement entropy in a (lattice) QFT In this talk we consider the case when: ◮ the degrees of freedom are those of a local relativistic QFT in large region R in R d ◮ the whole system is in the vacuum state | 0 � ◮ A is the set of degrees of freedom in some large (compact) subset of R , so we can decompose the Hilbert space as H = H A ⊗ H B ◮ in fact this makes sense only in a cut-off QFT (e.g. a lattice), and some of the results will in fact be cut-off dependent ◮ How does S A depend on the size and geometry of A and the universal data of the QFT?

Rényi entropies from the path integral ( d = 1) A B ������������������ ������������������ 0 ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ τ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ _ ������������������ ������������������ 8 ������������������ ������������������ ◮ wave functional Ψ( { a } , { b } ) is proportional to the conditioned path integral in imaginary time from τ = −∞ to τ = 0: � Ψ( { a } , { b } ) = Z − 1 / 2 [ da ( τ )][ db ( τ )] e − ( 1 / � ) S [ { a ( τ ) } , { b ( τ ) } ] 1 a ( 0 )= a , b ( 0 )= b � 0 � � where S = −∞ L a ( τ ) , b ( τ ) d τ ◮ similarly Ψ ∗ ( { a } , { b } ) is given by the path integral from τ = 0 to + ∞

Example: n = 2 � db Ψ( a 1 , b )Ψ ∗ ( a 2 , b ) ρ A ( a 1 , a 2 ) = � Tr A ρ 2 da 1 da 2 db 1 db 2 Ψ( a 1 , b 1 )Ψ ∗ ( a 2 , b 1 )Ψ( a 2 , b 2 )Ψ ∗ ( a 1 , b 2 ) A = A B 2 = Z ( C ( 2 ) ) / Z 2 Tr A ρ A 1 where Z ( C ( 2 ) ) is the euclidean path integral (partition function) on an 2-sheeted conifold C ( 2 )

◮ in general n = Z ( C ( n ) ) / Z n Tr A ρ A 1 where the half-spaces are connected as A B to form C ( n ) . ◮ conical singularity of opening angle 2 π n at the boundary of A and B on τ = 0 ◮ in 1+1 dimensions many results are known, e.g for a single interval of length ℓ in a CFT (Holzhey et al., Calabrese-JC) S ( n ) ∼ ( c / 6 )( 1 + n − 1 ) log ( ℓ/ǫ ) A

Higher dimensions d > 1 � B A ◮ the conifold C ( n ) is now locally { 2 d conifold } × R d − 1 , A formed by sewing together n copies of { τ > 0 } × R d − 1 to n copies of { τ < 0 } × R d − 1 along τ = 0, so that copy j is sewn to j + 1 for r ∈ A , and j to j for r ∈ B S ( n ) ∝ log ( Z ( C ( n ) A ) / Z n ) ∼ Vol ( ∂ A ) · ǫ − ( d − 1 ) A ◮ this is the ‘area law’ in 3+1 dimensions [Srednicki 1992] ◮ coefficient is non-universal

Mutual Information of multiple regions A B 2 A 1 ◮ the non-universal ‘area’ terms cancel in I ( n ) ( A 1 , A 2 ) = S ( n ) A 1 + S ( n ) A 2 − S ( n ) A 1 ∪ A 2 ◮ this mutual Rényi information is expected to be universal depending only on the geometry and the data of the CFT ◮ however this dependence is very difficult to compute, even in 1+1 dimensions (Calabrese-JC-Tonni)

Operator Expansion Method For any region X Z ( C ( n ) � � X ) S ( n ) = ( 1 − n ) − 1 log X Z n So    Z ( C ( n ) A 1 ∪ A 2 ) Z n A 1 ∪ A 2 = ( n − 1 ) − 1 log I ( n ) ( A 1 , A 2 ) ≡ S ( n ) A 1 + S ( n ) A 2 − S ( n )  Z ( C ( n ) A 1 ) Z ( C ( n ) A 2 ) Write Z ( C ( n ) A 1 ∪ A 2 ) = � Σ ( n ) A 1 Σ ( n ) A 2 � ( R d + 1 ) n Z n where = Z ( C ( n ) n − 1 A ) Σ ( n ) Φ k j ( r ( j ) � C A � A ) A { k j } Z n { k j } j = 0

Z ( C ( n ) Z ( C ( n ) Z ( C ( n ) n − 1 A 1 ∪ A 2 ) A 1 ) A 2 ) C A 1 { k j } C A 2 � Φ k j ( r ( j ) j ( r ( j ) � � = A 1 )Φ k ′ A 2 ) � Z n Z n Z n { k ′ j } { k j } , { k ′ j } j = 0 Z ( C ( n ) Z ( C ( n ) A 1 ) A 2 ) { k j } r − 2 � j x kj � C A 1 { k j } C A 2 = Z n Z n { k j } ◮ last equation flows from orthonormality of 2-point functions, valid in any CFT ◮ this gives an expansion of I ( n ) ( A 1 , A 2 ) in increasing powers of 1 / r , valid for large r ◮ first term comes from the identity operator with x k j = 0 ∀ j , but this cancels in I ( n ) ( A 1 , A 2 ) ◮ leading terms come from taking either 1 or 2 of the x k j � = 0

The coefficients C A { k j } B A These may be computed by inserting a complete set of operators on a single conifold C ( n ) A : �    n − 1 � � j ′ ( r ( j ′ ) ) � C ( n ) � j ′ ( r ( j ′ ) ) � C A � Φ k j ( r ( j ) ) � Φ k ′ = Φ k ′  { k j }  A j ′ j ′ { k j } j = 0 ( R d + 1 ) n Using orthonormality � j x kj � | r ( j ) | � Φ k j ( r ( j ) ) � C ( n ) C A { k j } = lim { r ( j ) }→∞ j A j

� j x kj by dimensional analysis ◮ note that C A { k j } ∝ R A ◮ the 1- and 2-point functions on C ( n ) are still very hard to A compute, and we have succeeded only for a free field theory

Free scalar field theory (gaussian free field) � ( ∂φ ) 2 d d + 1 x , and we normalise so Action is proportional to 2-point function in R d + 1 is � φ ( x ) φ ( x ′ ) � ≡ G 0 ( x − x ′ ) = | x − x ′ | − ( d − 1 ) . We need to compute x , x ′ →∞ ( xx ′ ) d − 1 � φ j ( x ) φ j ′ ( x ′ ) � C ( n ) ( j � = j ′ ) lim A x →∞ x 2 ( d − 1 ) � : φ 2 lim j ( x ): � C ( n ) A where φ j ( x , 0 − ) = φ j + 1 ( x , 0 +) for x ∈ A , and φ j ( x , 0 − ) = φ j ( x , 0 +) for x / ∈ A . These can be though of as the potential at x ′ on copy j ′ due to a unit charge at x on copy j , and the self-energy of a unit charge at x .

The case n = 2 Define φ ± = 2 − 1 / 2 ( φ 0 ± φ 1 ) ◮ φ + is continuous everywhere and so � φ + ( x ) φ + ( x ′ ) � = G 0 ( x − x ′ ) ◮ φ − changes sign across A ∩ { τ = 0 } ; on the other hand, if the source x lies on τ = 0 then � φ − ( x ) φ − ( x ′ ) � must be symmetric under τ ′ → − τ ′ , so it vanishes on A ∩ { τ = 0 } x 1 x 2 A ◮ � φ − ( x ) φ − ( x ′ ) � is the potential at x ′ due to a unit charge at x in the presence of a conductor held at zero potential at A ∩ { τ = 0 }