Peter shor quantum error correction how to#

And the entanglement wedge reconstruction predicts that can be represented as an integral of local boundary operators on AC if and only if is inside the entanglement wedge of AC. The entanglement wedge of AC is a bulk region enclosed by this global geodesic line of AC. Then the geodesic lines for A and C do not form the geodesic line for the union of A and C because we can draw shorter arcs by connecting endpoints of A and C, which form the global geodesic line. Imagine we split the boundary into four pieces A,B,C,D such that A,C are larger than B,D. Recently, causal wedge reconstruction has been further generalized to the notion of entanglement wedge reconstruction. It seems that a quantum secret is shared among three parties, and the AdS/CFT correspondence somehow realizes the three-party quantum secret-sharing code! It can be accessed only if multiple parties cooperate and perform joint measurements.

The quantum information at the center of the bulk cannot be accessed from any single party A, B or C because does not have representation on A, B, or C. This situation resembles the so-called quantum secret-sharing code. Even though, , are different boundary operators, they may be equivalent inside a certain low energy subspace on the boundary. Instead of corresponding to a single boundary operator, may correspond to different operators in different regions, say, , living in AB, BC, CA respectively. They proposed that the AdS/CFT correspondence can be viewed as a quantum error-correcting code. Does this mean that the bulk operator corresponds to an identity operator on the boundary?Īlmheiri, Dong and Harlow have recently proposed an intriguing way of reconciling this paradox with the AdS/CFT correspondence. The bulk operator at the center is contained inside causal wedges of BC, AB, AC. Of course, there are multiple regions A,B,C,… whose causal wedges contain, and the reconstruction should work for any such region. The AdS-Rindler reconstruction says that can be represented by some integral of local boundary operators supported on A if and only if is contained inside the causal wedge of A. On a fixed time-slice, the causal wedge of A is a bulk region enclosed by the geodesic line of A (a curve with a minimal length). Consider a bulk operator and a boundary region A on a hyperbolic space (in other words, a negatively-curved plane).

Peter shor quantum error correction how to#

The AdS-Rindler reconstruction tells us how to “reconstruct” a bulk operator from boundary operators. The AdS/CFT correspondence says that there is some kind of correspondence between quantum gravity on (d+1)-dimensional asymptotically-AdS space and d-dimensional conformal field theory on its boundary. This post may be a bit technical compared to other recent posts, but anyway, let me give it a try…īulk locality paradox and quantum error-correction In this post, I will try to extract some key features of the AdS/CFT correspondence and construct a toy model which captures these features. My personal philosophy is that a toy model must be as simple as possible while capturing key properties of the system of interest.

This is certainly a challenging task because I need to make it accessible to everyone while explaining real physics behind the paper. In this post, I hope to write an introduction which may serve as a reader’s guide to our paper, explaining why I’m so fascinated by the beauty of the toy model. Fernando has already written how this research project started after a fateful visit by Daniel to Caltech and John’s remarkable prediction in 1999. In a recent paper with Daniel Harlow, Fernando Pastawski and John Preskill, we have proposed a toy model of the AdS/CFT correspondence based on quantum error-correcting codes. Why don’t we use more modern concepts, such as the theory of quantum error-correcting codes? But, many of the concepts utilized so far rely on entanglement entropy and its generalizations, quantities developed by Von Neumann more than 60 years ago. The lessons we learned from the Ryu-Takayanagi formula, the firewall paradox and the ER=EPR conjecture have convinced us that quantum information theory can become a powerful tool to sharpen our understanding of various problems in high-energy physics.