Coherent Premeasurement and Correlation Tax

Abstract

A coherent premeasurement retains the input entropy and divides system-record mutual information into diagonal entropy and relative entropy of coherence.

The copying isometry sends basis vector i to the joint basis vector (i,i). The joint state is V rho V*, including the input’s off-diagonal entries. Both marginals are obtained by partial trace.

Theorem 1.1 (Coherence survives on the correlated subspace).

$$\forall \rho ,i,j,\mathrm{jointEntry}\left(\rho ,i,j\right)=\mathrm{inputEntry}\left(\rho ,i,j\right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/CoherentCopyCorrelationTax.coherentCopyState_correlated_entry (✓ std3). ∎

Citation. John Watrous (2018). The Theory of Quantum Information — spectral calculus, reductions and entropy. URL: https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf.

Commentary.

For every input density state and every pair of indices, the joint matrix entry at (i,i),(j,j) equals the input entry at i,j.

Theorem 1.2 (The system marginal is the pinched input).

$$\forall \rho ,\mathrm{marginalRight}\left(\mathrm{coherentCopyState}\left(\rho \right)\right)=\mathrm{basisPinchingState}\left(\rho \right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/CoherentCopyCorrelationTax.marginalRight_coherentCopyState (✓ std3). ∎

Citation. John Watrous (2018). The Theory of Quantum Information — spectral calculus, reductions and entropy. URL: https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf.

Commentary.

Tracing out the record leaves precisely the diagonal of the input.

Theorem 1.3 (The record marginal has the Born weights).

$$\forall \rho ,\mathrm{marginalLeft}\left(\mathrm{coherentCopyState}\left(\rho \right)\right)=\mathrm{basisPinchingState}\left(\rho \right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/CoherentCopyCorrelationTax.marginalLeft_coherentCopyState (✓ std3). ∎

Citation. John Watrous (2018). The Theory of Quantum Information — spectral calculus, reductions and entropy. URL: https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf.

Commentary.

Tracing out the system produces the same diagonal state: its ith diagonal entry is rho(i,i), and its off-diagonal entries vanish.

Theorem 1.4 (Coherent copying preserves entropy).

$$\forall \rho ,\mathrm{vonNeumannEntropy}\left(\mathrm{coherentCopyState}\left(\rho \right)\right)=\mathrm{vonNeumannEntropy}\left(\rho \right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/CoherentCopyCorrelationTax.vonNeumannEntropy_coherentCopyState (✓ std3). ∎

Citation. John Watrous (2018). The Theory of Quantum Information — spectral calculus, reductions and entropy. URL: https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf.

Commentary.

The copying map preserves multiplication, adjoints and trace. Functional calculus transports the logarithm through this map, including zero eigenvalues, so the entropy trace is unchanged.

Theorem 1.5 (Correlation equals record entropy plus coherence tax).

$$\forall \rho ,\mathrm{quantumMutualInformation}\left(\mathrm{coherentCopyState}\left(\rho \right)\right)=\mathrm{vonNeumannEntropy}\left(\mathrm{basisPinchingState}\left(\rho \right)\right)+\mathrm{quantumRelativeEntropy}\left(\rho ,\mathrm{basisPinchingState}\left(\rho \right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/CoherentCopyCorrelationTax.coherent_copy_correlation_tax (✓ std3). ∎

Citation. T. Baumgratz, M. Cramer, M. B. Plenio (2014). Quantifying Coherence. URL: https://arxiv.org/abs/1311.0275v3.

Commentary.

The two marginal entropies are the diagonal entropy, and the joint entropy is the original entropy. The pinching entropy identity supplies the remaining relative entropy term.

References

• Truth anchor: D5/S3/Quantum/Information/CoherentCopyCorrelationTax.coherentCopyState_correlated_entry
• Truth anchor: D5/S3/Quantum/Information/CoherentCopyCorrelationTax.coherent_copy_correlation_tax
• Truth anchor: D5/S3/Quantum/Information/CoherentCopyCorrelationTax.marginalLeft_coherentCopyState
• Truth anchor: D5/S3/Quantum/Information/CoherentCopyCorrelationTax.marginalRight_coherentCopyState
• Truth anchor: D5/S3/Quantum/Information/CoherentCopyCorrelationTax.vonNeumannEntropy_coherentCopyState
• Dependency: D5/S3/Quantum/Dynamics/EntropyProductionCoherenceDeletionIdentity
• Dependency: D5/S3/Quantum/Information/PartialTraceMutualInformation