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The information theoretic interpretation of the length of a curve

Czech, Bartlomiej, Hayden, Patrick, Lashkari, Nima, Swingle, Brian·2015·Journal of High Energy Physics
DOI 10.1007/jhep06(2015)157
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Figure 1 a) Geodesic g I , which subtends a boundary interval I (black) and geodesics, which are tangent to g I on the boundary along with the corresponding boundary intervals I ( x ) (color). The dashed geodesics contribute zero to integral (2.3). b) a I ( x ), the linear size of the interval I ( x ) centered at x .
Figure 1 a) Geodesic g I , which subtends a boundary interval I (black) and geodesics, which are tangent to g I on the boundary along with the corresponding boundary intervals I ( x ) (color). The dashed geodesics contribute zero to integral (2.3). b) a I ( x ), the linear size of the interval I ( x ) centered at x .CC-BY · From the PDF
Figure 2 a) A curve, which asymptotes to the geodesic g I . We have marked in color the geodesics tangent to the curve and the boundary intervals J ( x ), which they subtend. b) a plot of a J ( x ) -the length of the intervals J ( x ) as a function of the centerpoint x . If the curve asymptotes to the geodesic g I then a J ( x ) must agree with a I ( x ) (shown for comparison in dashed gray) outside some interval ( x L , x R ) [25].
Figure 2 a) A curve, which asymptotes to the geodesic g I . We have marked in color the geodesics tangent to the curve and the boundary intervals J ( x ), which they subtend. b) a plot of a J ( x ) -the length of the intervals J ( x ) as a function of the centerpoint x . If the curve asymptotes to the geodesic g I then a J ( x ) must agree with a I ( x ) (shown for comparison in dashed gray) outside some interval ( x L , x R ) [25].CC-BY · From the PDF
Figure 3 The way to read off ¯ a I,J ( r ) from the plot of a I,J ( x ) / 2.
Figure 3 The way to read off ¯ a I,J ( r ) from the plot of a I,J ( x ) / 2.CC-BY · From the PDF
Figure 4 a) The intervals J ( θ ) defined by a closed, convex curve in global AdS 3 . b) The intervals (2.16)-(2.17) initially held by Alice and Bob. We have indicated J ( θ N ) and J ( θ 2 N ) and the bulk axis, which joins their centers.
Figure 4 a) The intervals J ( θ ) defined by a closed, convex curve in global AdS 3 . b) The intervals (2.16)-(2.17) initially held by Alice and Bob. We have indicated J ( θ N ) and J ( θ 2 N ) and the bulk axis, which joins their centers.CC-BY · From the PDF
Figure 5 Intermediate stage in a constrained merging protocol. The top row depicts the interval I = [ L, R ] with endpoints L and R . Step x of the protocol acts on interval I ( x ) = [ ℓ ( x ) , r ( x )], drawn in the second row. Because r ( x ) is non-decreasing with x , the sites marked by the orange bar have not yet been acted upon. The third row depicts the interval I ( x +1) = [ ℓ ( x +1) , r ( x +1)]. Once step x is complete, none of the sites indicated by the green bar will ever be acted upon again since ℓ ( x ) is non-decreasing with x . Therefore, after step x the reduced density operators corresponding to the green marked sites on B and the orange marked sites on A should approximate the reduced density operator of the target state | ψ 〉 .
Figure 5 Intermediate stage in a constrained merging protocol. The top row depicts the interval I = [ L, R ] with endpoints L and R . Step x of the protocol acts on interval I ( x ) = [ ℓ ( x ) , r ( x )], drawn in the second row. Because r ( x ) is non-decreasing with x , the sites marked by the orange bar have not yet been acted upon. The third row depicts the interval I ( x +1) = [ ℓ ( x +1) , r ( x +1)]. Once step x is complete, none of the sites indicated by the green bar will ever be acted upon again since ℓ ( x ) is non-decreasing with x . Therefore, after step x the reduced density operators corresponding to the green marked sites on B and the orange marked sites on A should approximate the reduced density operator of the target state | ψ 〉 .CC-BY · From the PDF
Figure 6 Sequential merging error. The top line depicts the interval I = [ L, R ] and a set of merging constraints in the form of intervals I ( x ) depicted as green ovals. The state to be merged is a maximally entangled state shared between L and R , as illustrated in the second line. One way to merge this state is for Bob to prepare a maximally entangled state between L and L +1 in the first sequential merging step and then, in the subsequent steps, simply swap the information to the right until a maximally entangled state is established between L and R . This procedure doesn't require any Alice-Bob entanglement and yet produces long-range entanglement in Bob's lab. It is prohibited by the sequential merging error condition because, while the final state is correct, the intermediate state fails the sequential merging condition. The joint state of Bob's L and Alice's R is not correct at the intermediate merging steps after L has been completed and before R has been reached; they should be maximally entangled but instead they are product.
Figure 6 Sequential merging error. The top line depicts the interval I = [ L, R ] and a set of merging constraints in the form of intervals I ( x ) depicted as green ovals. The state to be merged is a maximally entangled state shared between L and R , as illustrated in the second line. One way to merge this state is for Bob to prepare a maximally entangled state between L and L +1 in the first sequential merging step and then, in the subsequent steps, simply swap the information to the right until a maximally entangled state is established between L and R . This procedure doesn't require any Alice-Bob entanglement and yet produces long-range entanglement in Bob's lab. It is prohibited by the sequential merging error condition because, while the final state is correct, the intermediate state fails the sequential merging condition. The joint state of Bob's L and Alice's R is not correct at the intermediate merging steps after L has been completed and before R has been reached; they should be maximally entangled but instead they are product.CC-BY · From the PDF
Figure 7 a) A surface in Poincar´ e AdS 4 built up from minimal surfaces anchored on a sequence of boundary circles. Its area computes the entanglement cost of a constrained merging protocol. b) A surface built up from a cycle of minimal surfaces. Its area computes the entanglement cost of a constrained swapping protocol.
Figure 7 a) A surface in Poincar´ e AdS 4 built up from minimal surfaces anchored on a sequence of boundary circles. Its area computes the entanglement cost of a constrained merging protocol. b) A surface built up from a cycle of minimal surfaces. Its area computes the entanglement cost of a constrained swapping protocol.CC-BY · From the PDF

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Metadata

Authors
Czech, B., Hayden, P., Lashkari, N., & Swingle, B.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2015-06
Volume
2015
Issue
6
ISSN
1029-8479
DOI
10.1007/jhep06(2015)157
Cited by
45
References
5
Language
en

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Czech, B., Hayden, P., Lashkari, N., & Swingle, B. (2015). The information theoretic interpretation of the length of a curve. Journal of High Energy Physics. https://doi.org/10.1007/jhep06(2015)157