Distance Measures for Quantum Information

Distance Measures for Quantum Information

Aaryan Gupta and Siddhant Midha

Last updated on August 27, 2022

Aaryan Gupta and Siddhant Midha Distance Measures for Quantum Information

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Classical Measures of Distance

We have two different measures of similarity or distance between two

probability distributions/states {px } and {qx } over the same index set x. They are -

1

Trace

Distance

=

D(px ,

qx )

=

1 2

x |px - qx |

Properties:

It is a metric on probability distributions.

D(px , qx ) = maxS | xS px - xS qx | over all subsets S of the index set {x}, relation to distinguishability. 2 Fidelity = F (px , qx ) = x px qx Properties:

It is not a metric but cos-1(F ) is a metric Iutncitanvebcetogrseompextraicnadllyiqnxterpreted as the inner product between the

Note that these are static measures of distance between probability

distributions.

Aaryan Gupta and Siddhant Midha Distance Measures for Quantum Information

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Classical Measures of Distance

Now let us consider a dynamic measure of distance which encapuslates how well information is preserved by a physical process i.e. noise.

Suppose you have the state X and you subject it to a Markov process to get Y . A natural measure would be P(X = Y ). Now make a perfectly correlated copy X^ of X . It turns out trace has an intimate relation with this dynamic measure of distance which is

D(X , X^) = P(X = Y )

Aaryan Gupta and Siddhant Midha Distance Measures for Quantum Information

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Quantum Trace Distance

Define the quantum trace distance between two density operators as 1

D(, ) = Tr (| - |) 2

where |A| = AT A (positive square root) We have,

The tuple (Dens(H), D) is a metric space. For commuting states,

D(, ) = D(i , ?i )

Aaryan Gupta and Siddhant Midha Distance Measures for Quantum Information

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Quantum Trace Distance

Bloch sphere. Let r , s. Then, |r - s|

D(, ) = 2

1 Converts to euclidean distance. 2 Hints towards rotation invariance. 3 Helpful visualization. Relation to distinguishability via measurement.

D(, ) = maxP:PI tr (P( - )) Proof. Key step, - Q - S.

Aaryan Gupta and Siddhant Midha Distance Measures for Quantum Information

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