31st Jerusalem Winter School in Theoretical Physics: Problem Set 2

Transcription

1 31st Jerusalem Winter School in Theoretical Physics: Problem Set Contents Frank Verstraete: Quantum Information and Quantum Matter : 3 : Solution to Problem 9 7 Daniel Harlow: Black Holes and Quantum Information (Problems in the appendix of his lecture notes) 7 Ady Stern: Topological Quantum Computing (Problems posted separately) 7 1

2 Frank Verstraete Quantum Information and Quantum Matter Frank Verstraete: Quantum Information and Quantum Matter FV1. The marginal problem is concerned with the identification of the convex set of all possible reduced density matrices of a quantum many-body system. Given the fact that quantum many-body Hamiltonians are local, which means that they only act on e.g. at most 4 parties, what is special to the marginals corresponding to ground states of such Hamiltonians? What is the relevance of the so-called monogamy of entanglement in discussing such ground states? FV. Consider a quantum many-body system on a lattice. Then the natural language to describe the system is Fock space (second quantization). There are however many different ways of choosing a basis in second quantization, with the different bases related by canonical transformations. For what kind of Hamiltonians is it more natural to work in a real-space or in a momentum space basis? FV3. In strongly correlated quantum systems, there is always a huge emphasis on the ground state properties of the Hamiltonian under study. This is very strange as it is impossible to cool systems to absolute zero temperature. Can you come up with reasons why the ground state still plays a central role, even in a setting of finite (but low) temperature?

3 : RM1. Complete the calculation of the entanglement entropy in the ground state of two coupled simple harmonic oscillators, discussed in the first lecture. Recall that Hamiltonian was given by H = 1 p 1 + p +! (x 1 + x )+ (x 1 x ) and the ground state wavefunction could be written as (x 1,x )= (! +! ) 1/4 p apple 1 exp 4! +(x 1 + x ) 1 4! (x 1 x ) where! + =! and! =(! + ) 1/. This calculation can be found in Srednicki s paper [1]. For students unfamiliar with the solution of the simple harmonic oscillator in quantum mechanics, they might consult: harmonic oscillator RM. In the first lecture, I mentioned that the previous calculation is readily extended to a system of many coupled oscillators, e.g., a lattice description of a free scalar field theory. This calculation was first discussed by Bombelli et al [] and then again by Srednicki [1]. However, I will refer you to a nice review of the calculation is section. of [3]. There the Hamiltonian is written as H = 1 NX p i + 1 i=1 NX i,j=1 i K ij j. (Actually they replace p i with i, as is standard in Lagrangian mechanics. Note that the mass of the particles is 1.) The ground state wavefunction is then written as where W = p K. ( i )= det W 1/4 apple 1 exp Show that this formalism can be applied to describe the previous calculation for two coupled oscillators. Show that your previous result for the entanglement entropy matches the general expression in eq. (4) of [3]. T W RM3. Consider the scalar field action I = 1 Z d d x p g g + m + R(g) where R(g) is the Ricci scalar of the background metric g µ. Show that the action is invariant (up to total derivatives, i.e., boundary terms) under local Weyl rescalings g µ! e!(x) g µ,! e!(x) with = d if m =0and = d 4(d 3). 3

4 The stress tensor is defined as T µ = I p g g µ. Show that in the case where the mass and the coupling have been tuned as above, that T µ µ = g µ T µ =0 if we use the equations of motion of the scalar field. The solution for the first part of this problem can be found at: Students unfamiliar with the concepts of the spacetime metric or the stress-energy tensor might consult: %8general relativity%9 tensor %8general relativity%9 RM4. Given the expression for the Rényi entropies S n = 1 1 n log Tr [ n A], verify that one recovers the standard von Neumann entropy in the limit n! 1, i.e., S 1 =lim n!1 S n = Tr ( A log A ). RM5. Consider a d-dimensional CFT in its vacuum state on R S d 1. Further let us choose the region A to be the ball on the t E =0slice enclosed by (d )-dimensional sphere with an angular width of 0. Verify that the universal contribution to the entanglement entropy is given by S univ =( ) d R 1 4 A log sin 0, where A is the central charge appearing in front of the Euler density in the trace anomaly. In solving this problem, you may wish to consult section 5 of [4]. RM6. Killing vectors are used describe the symmetries of a geometry. The fancy way of describing such a symmetry is to say that the Lie derivative of the metric with respect to a Killing vector vanishes L K g ab = r a K b + r b K a =0. An equivalent approach is to consider an infinitesimal coordinate transformation x a! x a = x a + "K a, where " is an infinitesimal parameter and K a is a Killing vector, then the line element should transform as g ab dx a dx b! g ab d x a d x b = g ab d x a d x b + O(" ). (This transformation is nicely described on the second webpage below.) One extra comment here is that it is convenient and common notation to represent any vectors using the notation V = V a, where as usual there is a summation over the repeated index a from 0 to d 1, as usual. Consider the following three vectors: K 1 = b@ x (where b is a constant), K = x@ y y@ x, K 3 = x@ t + t@ x. 4

5 Show that these are each Killing vectors of the flat space metric ab. For each of these vectors, in the (x, y)- or (t, x)-plane, draw a small vector at various points indicating the direction and magnitude of the corresponding vector. The latter exercise should allow you to see that K 1, K and K 3 generate a translation along the x-axis, a rotation in the (x, y)-plane and a boost in the (t, x)-plane, respectively. If one replaces t = it E, show that the flat space metric becomes ds E = dt E + dx + dy + and the boost Killing vector becomes K 3 = i K E,3 where That is, it becomes a rotation in the (t E,x)-plane. K E,3 = x@ te t x. Students who want to get a few more pointers about Killing vectors might consult: vector field RM7. In the second lecture, I introduced the idea of a modular or entanglement Hamiltonian to represent a density matrix as an operator A = e H /Tr(e H ). Now in general, H is a complicated nonlocal operator, however, as discussed in the lecture, it is a relatively simple object when considering the Minkowski vacuum of a QFT reduced to the Rindler wedge. The key to this simplification is that the global state (i.e., the vacuum) and the background geometry (i.e., flat space) are invariant under a Killing vector that leaves the entangling surface invariant (i.e., the boost vector K 3 in the previous problem 1 ). In this case, the entanglement Hamiltonian can be written in a more covariant form with Z H = d µ T µ K A where d µ is the covariant volume element on the spatial slice that is being integrated over, e.g., d µ = d d 1 y p gn µ where n µ is a unit vector normal to the surface. Show that in the case of the Rindler wedge, this expression reduces to the same expression given in class if one integrates over the surface t =0. Given the above covariant form one can consider evaluating this expression on other Cauchy surfaces that end of the same entangling surface. Hence try evaluating this expression for the Rindler wedge but on some other interesting Cauchy surface, e.g., t = bxwhere b < 1. Show that in fact H is a conserved charge. That is, it will yield the same result when evaluated on any Cauchy surface ending on the same entangling surface. Hint: This follows from conservation of the stress tensor, i.e., r µ T µ =0, and the fact that K µ is a Killing vector, i.e., r µ K + r K µ =0. If the QFT being considered is in fact a conformal Killing vector, then the above simplications also apply when K µ is a conformal Killing vector satisfying r µ K + r K µ = d r K g µ. In terms of an infinitesimal coordinate transformation, x a! x a = x a + "K a, a conformal Killing vector yields g µ dx µ dx! g µ d x µ d x = ( x) g µ d x µ d x + O(" ). That is, rather than remaining invariant, the metric transforms by a Weyl rescaling. Show that for a CFT and K µ being a conformal Killing vector, the H given above is a still conserved charge, i.e., it yields the 1 Implicitly, here as in the lecture, the entangling surface was chosen as x =0in the t =0time slice. 5

6 same result when evaluated on any Cauchy surface ending on the same entangling surface. Hint: Use the fact that the stress tensor of a CFT is traceless, i.e., T µ µ =0. RM8. Verify the details of the construction of an entropic c-theorem for RG flows of two-dimensional quantum field theories (after it is discussed in the third lecture). In flat space in higher dimensions, one could could construct an entangling surface consisting of the two flat parallell walls, e.g., the surfaces x = ±` in a constant time surface. One could again apply the same construction as above for this situation in higher dimensions. What are the implications of the resulting inequality? The original source for the entropic c-theorem in two dimensions is [5]. A discussion of the the second part of the question appears in [6] see the discussion beginning around eq. (7.). RM9. Consider (d + 1)-dimensional anti-de Sitter space with metric ds = L z dz dt + d~x which describes the boundary CFT in flat space, i.e., the corresponding boundary metric is just ds bdy = dt + d~x. Consider a spherical entangling surface in the boundary theory given by ~x = R on the t =0 slice. Evaluate the holographic entanglement entropy using the Ryu-Takayanagi formula. In particular, what is the universal contribution to the entanglement entropy? Hint: Show that the extremal surface in the bulk is given by the hemisphere : z + ~x = R. This problem was first discussed in [7]. RM10. Consider the following vector in d-dimensional flat space = R@ t 1 R [t + ~x ]@ t R i. Show that this vector is a conformal Killing vector in flat space. Describe the orbits of in particular, show that the entangling surface in the previous problem is left invariant. Construct the modular Hamiltonian for the boundary CFT in the previous problem (in terms of the stress tensor T ab ). RM11. Consider the following vector in (d + 1)-dimensional AdS space = 1 R [R z t ~x t R t [z@ z + x i ]. Show that this vector is a Killing vector in AdS space. Show that reduces to from the previous problem on the boundary of AdS space, i.e., in the limit z! 0. Describe the orbits of in particular, show that the extremal bulk surface in problem 7 is left invariant. The vectors considered in the previous two problems and related topics are discussed in [8, 9]. RM1. Using holographic techniques, evaluate the Rényi entropies for the case described in problem 7. The solution of this problem may be found in [10]. 6

7 References [1] M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/ ]. [] L. Bombelli, R. K. Koul, J. Lee and R. D. Sorkin, A Quantum Source of Entropy for Black Holes, Phys. Rev. D 34 (1986) 373. [3] H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 4 (009) [arxiv: [hep-th]]. [4] R. C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 1101 (011) 15 [arxiv: [hep-th]]. [5] H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (004) 14 [hep-th/ ]. [6] R. C. Myers and A. Singh, Comments on Holographic Entanglement Entropy and RG Flows, JHEP 104 (01) 1 [arxiv: [hep-th]]. [7] S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 0608 (006) 045 [hep-th/ ]. [8] H. Casini, M. Huerta and R. C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 1105 (011) 036 [arxiv: [hep-th]]. [9] T. Faulkner, M. Guica, T. Hartman, R. C. Myers and M. Van Raamsdonk, Gravitation from Entanglement in Holographic CFTs, arxiv: [hep-th]. [10] L.-Y. Hung, R. C. Myers, M. Smolkin and A. Yale, Holographic Calculations of Renyi Entropy, JHEP 111 (011) 047 [arxiv: [hep-th]]. RM 7

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