Bit Threads and Holographic Entanglement

Transcription

1 it Threads and Holographic Entanglement Matthew Headrick randeis Uniersity ased on arxi: [hep-th], with Michael Freedman Entropy and area In semiclassical graity, entropies are related to surface areas General surface: S area 4G N Special surfaces (horizon, minimal surface, extremal surface): S = area 4G N Ryu-Takayanagi [ 06]: S() = 4G N area(m()) m() = minimal surface homologous to (From now on, we set 4G N = ln = ) m()

2 Interpretation Why? Naie answer: Microstate bits of ρ lie on m(), one bit per 4 Planck areas onfusions: Under continuous changes in boundary region, minimal surface can jump, for example: m() =m() [ m() m() 6= m() [ m() (Note: ρ does not jump; not a conentional exchange-of-dominance phase transition [Headrick 3]) Important quantities, like condiitonal entropy H( ) = S() S(), mutual information conditional mutual information I( : ) = S() + S() S(), I( : ) = S() + S() S() S(), are gien by differences of areas of surfaces passing through different regions of bulk Let s recall their information-theoretic meaning lassical: H( ) = # of (independent) bits belonging purely to I( : ) = # shared with

3 H( ) conditional S() entropies S() S() H( ) S() I( S() : ) Quantum: Entangled pair H( ) of bits contributes I( : to ) I( : ), H( ) to H( ) an lead to H( ) < 0 p ( i + i) 0 0 = p ( i + 0 i) I( : ) S() I( : ) H( ) What do differences between areas of surfaces, passing through different parts of bulk, hae to do with redundancy, entanglement, etc between bits of and? What does holographic proof of strong subadditiity hae to do with monotonicity of correlations? To answer these questions, I will present a new formulation of RT Does not refer to minimal = p surfaces; ( i + these i) are demoted to a calculational deice S() S() S() H( ) I( : ) H( ) 0 0 Suggests a new way to think about the connection between spacetime geometry and information Max flow-min cut (Originally on graphs, in context of network theory; continuous ersion [Federer 74, Strang 83, Nozawa 90]) onsider a Riemannian manifold with boundary Define a flow as a ector field st = 0, Let be a subset of boundary For any surface m homologous to, Strongest bound is min cut: = m area(m) max min area(m) m 3

4 Max flow-min cut theorem says there are no other obstructions to increasing flux: max = min area(m) m () := maximizer Notes: () m() On m(), () = unit normal ector; elsewhere, () is highly non-unique Unlike min cut, max flow is a linear programming problem flow can be thought of as a set of oriented threads (flow lines) with transerse density = RT ersion 0: S() = max = max # of threads coming out of m() () Each thread has cross section of 4 Planck areas utomatically incorporates homology condition & global minimization Threads can end on c or horizon Each thread carries one independent bit of ρ, either entangled with c or in a mixed state Threads can also return to, but those don t count Minimal surface does not play a fundamental role, but acts as bottleneck limiting number of threads Naturally implements holographic principle: entropy is area because bits are carried by one-dimensional objects Threads & information Now we address conceptual puzzles with RT raised before First, it can be shown that () changes continuously with, een when m() jumps Now consider two regions, We can maximize flux through or If S() < S() + S(), then we cannot simultaneously maximize through both ut we can always maximize through and (nesting property) all such a flow (, ) () (, ) () 4

5 Example : S() = S() =, S() = 3 I( : ) =, H( ) = Maximizing on, we can also maximize on either or (, ) (,) Lesson : Threads that are stuck on represent bits unique to Threads that can be moed between & represent correlated pairs of bits Example : S() = S() =, S() = I( : ) = 3, H( ) = entanglement! One thread leaing must go to, and ice ersa (, ) (,) Lesson : Threads connecting & (and switch orientation) represent entangled pairs of bits onditional entropy: H( ) = S() S() = (, ) = (, ) (, ) Mutual information: I( : ) = S() H( ) = (, ) (, ) Subadditiity is clear Max flows can be defined without regulator, as flows that cannot be augmented 5

6 Regulator-free definition of mutual information: I( : ) = Define flow ( : ) = ((, ) (, )) ((, ) (, )) which goes from to through entanglement wedge r() Implies I( : ) area (r() bottleneck) ( : ) Gien lessons aboe, freedom to moe threads around on indicates correlations with ; freedom to add loops that begin and end on indicates entanglement within onditional mutual information: I( : ) = H( ) H( ) = (,, ) (,, ) = (max on ) (min on ), while maximizing on & = moeable between &, while maximizing on & = (moeable between & ) (moeable between & ) = I( : ) I( : ) Strong subadditiity is clear Exercise for reader: Find flow-based proof of monogamy of mutual information inequality [Hayden-Hedarick- Maloney ], I( : ) I( : ) + I( : ) and higher inequalities [ao et al 5] annot be proed using just nesting property (see eg 4-party GHZ); indicates some new property of flows Do bit threads indicate that bipartite entanglement is priileged? Possible to represent 3-party GHZ state 6

7 oariant flows To appear (with Veronika Hubeny) Flow ersion of Hubeny-Rangamani-Takayanagi [ 07] coariant entanglement entropy formula: Define a flow as a ector field (in the full spacetime) st = 0 and integrated norm bound: timelike cure and unit normal ector field u on it, ds u In other words, any obserer sees oer their lifetime a total of at most thread per 4 Planck areas HRT ersion 0: S() = max (D() = boundary causal domain of ) D() Max flow () stays inside entanglement wedge, squeezes out through extremal surface it threads can lie on a common auchy slice (equialent to maximin [Wall ]), or spread out in time 7