Entropy isn’t just a feeling about uncertainty — it’s a hard, physical size. Schumacher proved the quantum version of Shannon’s great result: a stream of qubits can be squeezed down to exactly S qubits each, and not one bit further.
↩ before you start · keep these handy
·From Ch. 12: Shannon entropy H = bits of surprise per symbol — the classical message size.
·From Ch. 13: S(ρ) is Shannon’s formula run on a state’s eigenvalues — the quantum entropy.
·power-of-2 reminder: 2ₙ counts the strings of length n; 2ₙˢ counts the few that actually appear.
🔑 symbol decoder · every new mark, in plain words
nhow many qubits the source emits — the length of the stream.n·S(ρ)the compressed length — S qubits per symbol, the hard floor.typical setthe handful of outcomes that actually occur — about 2ₙˢ of them.pthe bias of the source — here, the probability each qubit is |0⟩.faithfulrecoverable with vanishing error — you get the original back as the stream grows.
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Squeeze out the redundancy
A source that almost always sends the same qubit is wasteful — most of what arrives, you could have guessed. Compression is the art of spending storage only on the surprising parts. The more predictable the source, the harder you can squeeze. But a perfectly fair, maximally uncertain source has nothing to throw away — every qubit is news, and the stream won’t shrink at all.
📦 everyday picture
Zip a file. A page of solid “AAAAA…” collapses to almost nothing — you just store “A × 10,000”. A page of random static won’t shrink one byte, because there’s no pattern to abbreviate. Every zip program hits the same wall: it can throw away predictability, but never genuine surprise. Schumacher’s theorem just names where that wall sits for qubits — exactly S qubits per symbol.
recapCompression discards predictability; what’s left — the surprise — cannot be squeezed away.
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Compress a biased source
A machine emits 100 qubits, each |0⟩ with probability p and |1⟩ otherwise. Bias the source and watch how far it compresses. The compressed block can never drop below the entropy floor — the dashed line at 100·S qubits.
▸ Schumacher compressionn qubits → ≈ n·S(ρ) qubits
raw source · 100 qubits{{ srcLabel }}
compressed length
{{ compQubits }}q
0floor = 100·S100
|0⟩ bias{{ pPct }}
entropy S{{ entropy }} bits
compressed{{ compQubits }} qubits
saved{{ savedPct }}
recapThe more biased the source, the lower its entropy floor 100·S — and the more the stream shrinks.
math
The typical subspace
Out of all 2n possible strings, only a vanishingly small typical set ever shows up with real probability — and there are about 2nS of them. Schumacher’s theorem says you only need enough qubits to label that set:
n qubits of source ρ → ≈ n·S(ρ) qubits
faithful: the original is recoverable with vanishing error as n → ∞
optimal: no scheme beats S qubits each — it is a hard floor
This is the precise sense in which von Neumann entropy is information: it is the number of qubits per symbol that genuinely must be sent. Classical Shannon compression (to H bits each) is just the special case where the source states are perfectly distinguishable.
✎ worked example · 100 qubits, 85% likely to be |0⟩
2.entropy S = 0.85×0.23 + 0.15×2.74 = 0.20 + 0.41 = 0.61 bits per qubit.
3.compressed length ≈ 100 × 0.61 = 61 qubits — the other 39 were redundant.
4.try to ship only 60 and the original can no longer be recovered: 61 is the floor. ✓
recapn qubits compress to ≈ n·S(ρ) — the size of the typical subspace, and no fewer.
⚠ common misconception
“With clever enough coding you could squeeze any source to almost nothing.” The entropy floor forbids it. Compress below S qubits per symbol and information is irretrievably lost — the original cannot be reconstructed. Von Neumann entropy is the incompressible core of the stream.
That answers “how much can a qubit store?” The mirror question — “how much classical information can you read back out of a qubit?” — has a different, even more surprising answer. That’s the Holevo bound, next.
✓ you can now
✓compute how far a biased qubit stream compresses — to about n·S(ρ) qubits
✓explain the entropy floor: below S qubits per symbol the original is lost for good
✓see why von Neumann entropy is the true, incompressible size of quantum information