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Observational Entropy with General Quantum Priors

Ge Bai, Dominik Šafránek, Joseph Schindler, Francesco Buscemi, Valerio Scarani

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arXiv:2308.08763

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Von Neumann Entropy

  • For state
  • Invariant under unitary evolution

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  • However, intuitively in this case the entropy should increase

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Observer is macroscopic

The entropy should also depend on their coarse-graining

John von Neumann

Microstate�probability

Unitary

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Observational Entropy (OE) [Šafránek, Deutsch, Aguirre 2019]

  • A coarse-grained von Neumann entropy

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Observational entropy

State

Statistics

Measurement�POVM

The “volume” of POVM element

The probability of outcome

Observer is macroscopic

The entropy should also depend on their coarse-graining

John von Neumann

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Observational Entropy (OE) [Šafránek, Deutsch, Aguirre 2019]

  • A coarse-grained von Neumann entropy

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  • = intrinsic uncertainty + uncertainty due to coarse-graining

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  • OE equal to von Neumann if measurement is the state eigenbasis

State

Statistics

(always )

Von Neumann

Observational entropy

Measurement�POVM

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Summary

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Observational entropy

Interpretation #1: statistical deficiency

Interpretation #2: irretrodictability

Unification in classical case

Mismatch in quantum case

Observational entropy with general quantum priors

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Relative Entropy as Pseudo-Distance

  • Kullback-Leibler (KL) divergence

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  • Umegaki relative entropy: a quantum generalization of KL divergence

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    • Equals to KL divergence between eigenvalues if they commute

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  • KL divergence (relative entropy)

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for states

for distributions

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Interpretation 1: OE as Statistical Deficiency

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Interpretation #1: Observational Entropy as Statistical Deficiency

(uniform distribution)

Umegaki relative entropy

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Interpretation 1: OE as Statistical Deficiency

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KL divergence

Interpretation #1: Observational Entropy as Statistical Deficiency

(uniform distribution)

Umegaki relative entropy

#1: Statistical Deficiency

uncertainty due to coarse-graining

intrinsic uncertainty

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Retrodiction: Inferring the Cause

  • Sometimes we want to answer questions that are not accessible
    • What did dinosaurs look like?
    • I get a negative result in a medical test. Am I sick?

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Unknown true input,

what we want to know

Known observed output

A stochastic process that we can characterize

Healthy

Sick

Positive

Negative

Input

Output

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Retrodiction with Bayes Rule

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Forward stochastic process

Unknown true�input distribution

Retrodiction process

Updated belief of�input distribution

Known transition probabilities

Reference input distribution�A belief of input distribution since the true one is unknown

Reference output distribution

Bayesian Inference

Healthy

Sick

Positive

Negative

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Measurement as a Classical Process

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Forward input-output relation

True input distribution

Input-output joint distribution

Born rule

  • Regarding as a probabilistic preparation of its eigenstates

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  • Forward process: prepare with probability , and measure with

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Prepare

Measurement

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Retrodiction of Measurement

  • Reverse process: inferring input distribution of �with Bayes rule given observed output distribution
  • Assuming the eigenbasis is known
    • This assumption to be lifted

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Reverse input-output relation

from Bayes inference rule

Reference input distribution

Reference output

distribution

Observed output

distribution

Forward input-output relation

Depends on

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  •  

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Interpretation #2: Observational Entropy as Irretrodictability

Reverse input-output relation

from Bayes inference rule

#2: Irretrodictability

Forward input-output relation

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Unification and Mismatch

  • Two interpretations always match if reference state is uniform

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#2: Irretrodictability

#1: Statistical Deficiency

 

  • Can we replace this uniform prior with a general quantum state ?
    • Defining observational entropy with general quantum priors

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Unification and Mismatch

  • Two interpretations always match if reference state is uniform

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#2: Irretrodictability

#1: Statistical Deficiency

 

#1: Statistical Deficiency

  • A quantum generalization of Bayesian retrodiction is needed

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A Quantum Bayes Rule

  • Petz map is always a valid CPTP map
  • It reduces to classical Bayes rule if all states commute

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Forward quantum process

Petz map: a quantum Bayes rule

Retrodiction process

Known quantum process

Unknown true�input state

Reference input state

Reference output state

Updated belief of input state

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Unification and Mismatch

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#2: Irretrodictability

#1: Statistical Deficiency

 

#2: Irretrodictability

#1: Statistical Deficiency

 

  • Mismatch after replacing with general reference state
  • Representations of the forward (measurement) and reverse (Petz map) processes
  • is a state instead of a probability distribution

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#3: Belavkin-Staszewski Relative Entropy

Observational Entropy with General Quantum Priors

Belavkin-Staszewski relative entropy

Equal to Umegaki relative entropy iff

Statistical deficiency

Irretrodictability

Some definition of “joint distribution” different from previous slide

  • The more the measurement confuses states, the harder it is to infer the original state from measurement statistics

KL divergence

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Using Umegaki relative entropy, two

interpretations #1, #2 do not match

#1 Statistical Deficiency

#2 Irretrodictablity

#3 BS Relative Entropy

#1 and #3 match if

All equal to von Neumann

entropy when can be

perfectly recovered

Perfectly recovered by Petz recovery

Different values �of

Von Neumann Entropy

Using Umegaki relative entropy

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Summary

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Observational entropy

Interpretation #1: statistical deficiency

Interpretation #2: irretrodictability

Unification in classical case

Mismatch in quantum case

Observational entropy with general quantum priors

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Discussion

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Statistical deficiency

Irretrodictability

Observational entropy with general quantum priors

Unification?

Measurement

General quantum processes

  • Measures the performance of recovery maps [Junge, et al. 2018; Buscemi et al. 2022]
  • (Quantum?) entropy production

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  • Representations of quantum processes, states over time

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Q&A

Full paper on arXiv:2308.08763

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