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Shadow Theory

Quantum Source Completions and Memory–Environment Tradeoffs

A four-dimensional source attaining the recorded-channel floor

Jeremy RodgersSection 7 of 14Original PDF ↗Publication DOI ↗

A four-dimensional source attaining the recorded-channel floor

Phase-enhanced pure-state stochastic encodings have established antecedents. The following explicit specialization realizes the delayed-trine law while attaining the smallest possible recorded-channel rank. It supplies the four-dimensional endpoint of the Pareto comparison. For r = 1 −λ, z = √ 1 −r2 and θj = 2πj/3, define
Technical passage, source page 13.
Searchable transcription
Phase-enhanced pure-state stochastic encodings have established antecedents. The following explicit
specialization realizes the delayed-trine law while attaining the smallest possible recorded-channel rank.
It supplies the four-dimensional endpoint of the Pareto comparison.
For r = 1 −λ, z =
√
1 −r2 and θj = 2πj/3, define
√︃1 + z 2 |0⟩+ eiθj √︃1 −z 2 |1⟩, |ej⟩= (|0⟩−eiθj|1⟩)/ √ |lj⟩= 2.
Source page 13.
Searchable transcription
√︃1 + z
2
|0⟩+ eiθj
√︃1 −z
2
|1⟩,
|ej⟩= (|0⟩−eiθj|1⟩)/
√
|lj⟩=
2.

The rank-one effects Ej = (2/3)|ej⟩⟨ej| sum to identity. With ρj = |lj⟩⟨lj| their response is

Tr(Ejρi) = 1 −r cos(θi −θj) 3 = Tij.
Source page 13.
Searchable transcription
Tr(Ejρi) = 1 −r cos(θi −θj)
3
= Tij.

On two qubits let S swap factors and put

Kj = S (︁√︂ 2/3|lj⟩⟨ej| ⊗I )︁, Φj(X) = KjXK† j.
Source page 14.
Searchable transcription
Kj = S
(︁√︂
2/3|lj⟩⟨ej| ⊗I
)︁,
Φj(X) = KjXK†
j.
Then ∑︁ j K† jKj = I4 and Φj(ρa ⊗ρb) = Tajρb ⊗ρj.
Source page 14.
Searchable transcription
Then ∑︁
j K†
jKj = I4 and
Φj(ρa ⊗ρb) = Tajρb ⊗ρj.
3 ∑︁ j ρj = (I + zσz)/2. The stationary preparation is τ ⊗τ. Since Tr(Ejτ) = 1/3, the first two emitted symbols are independent uniform trits; thereafter the displayed update proves every finite word of the delayed law by induction. This stationary preparation differs from that of the qutrit source and is explicitly budgeted. Each symbol has one nonzero Kraus operator, so the complete registered one-step channel has Choi rank exactly three. No registered three-symbol channel with three nonzero outputs can have smaller rank, since its orthogonal output blocks each contribute at least one. At 0 < λ < 1 all words are positive and every word Kraus operator is rank one after two steps. The full registered n-step channel therefore has Choi rank 3n for n ≥2, attaining the alphabet-imposed floor with retained final memory. It streams with one fresh three-level record port and one fresh three-level record-copy bath per step; old baths never return. These resources are not an uncounted fresh source or a correlated tape. The initial mixed preparation and its rank-four purifier are separate stock. For λ = 0 the exact-law support floor changes to 9 2n−2. At λ = 1 the process is iid and memory one suffices; this redundant four-dimensional implementation is not a memory optimum. All claimed two-point frontiers exclude that iid endpoint. A coherent archive or an initially mixed bath defines a different optimization task. Let τ = 1
Technical passage, source page 14.
Searchable transcription
3
∑︁
j ρj = (I + zσz)/2. The stationary preparation is τ ⊗τ. Since Tr(Ejτ) = 1/3, the first
two emitted symbols are independent uniform trits; thereafter the displayed update proves every finite
word of the delayed law by induction. This stationary preparation differs from that of the qutrit source
and is explicitly budgeted.
Each symbol has one nonzero Kraus operator, so the complete registered one-step channel has
Choi rank exactly three. No registered three-symbol channel with three nonzero outputs can have
smaller rank, since its orthogonal output blocks each contribute at least one. At 0 < λ < 1 all words
are positive and every word Kraus operator is rank one after two steps. The full registered n-step
channel therefore has Choi rank 3n for n ≥2, attaining the alphabet-imposed floor with retained final
memory. It streams with one fresh three-level record port and one fresh three-level record-copy bath
per step; old baths never return. These resources are not an uncounted fresh source or a correlated
tape. The initial mixed preparation and its rank-four purifier are separate stock.
For λ = 0 the exact-law support floor changes to 9 2n−2. At λ = 1 the process is iid and memory
one suffices; this redundant four-dimensional implementation is not a memory optimum. All claimed
two-point frontiers exclude that iid endpoint. A coherent archive or an initially mixed bath defines a
different optimization task.
Let τ = 1
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