Interpreting discord through state merging

Vaibhav Madhok Animesh Datta

Tuesday, September 7, 2010 Correlations: Classical

X and Y are two random variables H(X) = p(x) log p(x) is the Shannon entropy of X − x !

I(X : Y ) = H(X) + H(Y ) H(X, Y ) − J(X : Y ) = H(X) H(X Y ) − | For classical probability distributions I(X : Y ) = J(X : Y )

Tuesday, September 7, 2010 Correlations: Quantum

Quantum mutual information. ρAB : Bipartite quantum state

I(A : B) = S(A) + S(B) S(A, B) − S(ρ) = T r(ρ logρ) is the von Nuemann entropy. − Quantum analogue of conditional entropy ˜ S Πi (A B) piS(ρA i) and thus, { } | ≡ | i ! ˜ Π (ρAB) = S(ρA) S Π (A B), J{ i} − { i} |

Tuesday, September 7, 2010 Quantum Discord

In general, for quantum systems

I(A : B) = J(A : B) ! Discord can be defined as

(ρ ) = I(ρ ) (ρ ) D AB AB − J AB ˜ = S(ρA) S(ρAB) + min S Πi (A B). − Πi { } | { }

Tuesday, September 7, 2010 Discord: As a measure of Non classical correlations

(ρ ) = I(ρ ) (ρ ) D AB AB − J AB ˜ = S(ρA) S(ρAB) + min S Πi (A B). − Πi { } | { }

(ρAB) = S(ρA) piS(ρA i) J − | i ! ----is a measure of classical correlations ----is equal to the distillable common randomness

Tuesday, September 7, 2010 Discord: As a measure of Non classical correlations

(ρ ) = I(ρ ) (ρ ) D AB AB − J AB ˜ = S(ρA) S(ρAB) + min S Πi (A B). − Πi { } | { }

Thus, discord is the difference between, the quantum mutual information and the classical correlations

Tuesday, September 7, 2010 Operational interpretation of discord

Operational interpretation give us a way to characterize a mathematical quantity as a physical resource

Gives additional insights, and makes connections between various processes more clear.

Tuesday, September 7, 2010 Operational interpretation : Earlier attempts Discord as work deficit:

Search for an operational interpretation related to an information theoretic task.

Tuesday, September 7, 2010 Discord and quantum state merging: Motivation

Positivity of quantum discord SSA of Von Nuemann entropy

Quantum State Merging Operational proof of SSA

Does state merging give an operational interpretation of discord ?

Tuesday, September 7, 2010 Classical Slepian-Wolf protocol Alice and Bob have two random variables X and Y respectively

If Bob needs to learn X fully, how much does Alice need to send?

H(X|Y) will suffice

Tuesday, September 7, 2010 Classical Slepian-Wolf

Achievability region for the rates

Tuesday, September 7, 2010 Quantum state merging Extension of classical state merging protocol to the quantum domain

The amount of quantum communication needed by Alice to merge her state with Bob = S(A B) |

Tuesday, September 7, 2010 Quantum state merging

When S(A B) < 0 | Bob can get the full state with LOCC. In addition, Alice and Bob can distill -S(A|B) ebits for future use. Tuesday, September 7, 2010 Interpreting SSA through state merging Strong sub additivity:

S(A, B, C) + S(B) S(A, B) + S(B, C) ≤ or alternatively,

S(A BC) S(A B) | ≤ | Having more prior information makes state merging cheaper!

Tuesday, September 7, 2010 Interpreting discord through state merging S(A BC) S(A B) | ≤ | Discarding relevant prior information makes state merging more expensive. Operational Interpretation of discord: Discord is the mark up in the cost of quantum communication in the process of state merging, if one discards relevant prior information

Tuesday, September 7, 2010 Interpreting discord through state merging Consider a bipartite quantum state AB, and an arbitrary quantum operation, , on B. E C is the ancilla with B, initially in a pure state, 0 | ! S(A, B) = S(A, BC) A A’ I(A : BC) = I(A! : B!C!) B B’ U C C’ S(A B) = S(A) I(A : B) | − = S(A) I(A : BC) = S(A BC). − | I(A! : B!) I(A! : B!C!) ≤

S(A! B!) S(A! B!C!) = S(A B). | ≥ | |

Tuesday, September 7, 2010 Interpreting discord through state merging

S(A B) = S(A) I(A : B) | − = S(A) I(A : BC) = S(A BC). − | I(A! : B!) I(A! : B!C!) ≤ S(A! B!) S(A! B!C!) = S(A B). | ≥ | | A A’ B B’ U C C’ Define D as: D = I(A : B) I(A! : B!) − D reduces to discord when quantum operations on B are measurements and we perform maximization. Tuesday, September 7, 2010 Interpreting discord through state merging Define D as:

D = I(A : B) I(A! : B!) − D reduces to discord when quantum operations on B are measurements and we perform maximization. After measurement on B,

ρAB! = pjρA j πj, ρA! = pjρA j = ρA, ρB! = pjπj. | | ⊗ j j j ! ! I(A! : B!)!= S(A!) + S(B!) S(A!, B!), − = S(A!) + H(p) H(p) + pjS(ρA j) , − | j ! " # = S(A) pjS(ρA j). equals to J, after maximization − | Tuesday, September 7, 2010 j " Discord as a mark up price

Operational Interpretation of discord:

Discord is the mark up in the cost of quantum communication in the process of state merging, if one discards relevant prior information, through quantum measurements on one of the subsystems.

Tuesday, September 7, 2010 Properties of discord: Positivity

Measurement on B will always result in discarding information or at best preserving the original correlations. Therefore, we always have a mark up and hence positive discord,

Tuesday, September 7, 2010 Properties of discord: Condition of Nullity

Zero discord implies zero mark up: For zero discord, density matrix is of the form:

ρAB = piρA i λi λi | ⊗ | "# | i ! λi is the eigen basis of ρB | ! Measurement on B with projectors, λ λ gives: | i!" i| M ρAB = PjρABPj = ρAB therefore zero mark up! i !

Tuesday, September 7, 2010 Properties of discord: Condition of Nullity Zero mark up implies zero discord:

D = I(A : B) I(A! : B!) − From SSA: Equality of I(A:B) and I(A’:B’) is the condition for equality condition in SSA, i.e.

S(A B, C) = S(A B). | | which is exactly the condition for zero discord!

Tuesday, September 7, 2010 Properties of discord: Upper bound Discord is equal to the loss of information at B’s end.

D = I(A : B) I(A! : B!) − B cannot loose more information than there is at his disposal. Therefore, the upper bound is : S(B). The von Nuemann entropy of the measured subsystem.

Tuesday, September 7, 2010 Summary

We have presented an interpretation of discord through the state merging protocol

Similar concepts can be used to define new measures of non classical correlations, and also give an operational interpretation for other known measures.

Tuesday, September 7, 2010