Quantum Information Apps
Exercise PLAN-403
primary source nl only. Current content-bound mathematical second pass accepted. No new Lean verification is claimed for this NL view.
Assumptions and conventions
Question
It is a theorem that if rho_AB is k-extendible and N : B -> B' is any quantum channel, then (id_A tensor N)(rho_AB) is again k-extendible; applying a channel to one side can only preserve or improve extendibility. One might conjecture the converse: that if the output (id_A tensor N)(rho_AB) is k-extendible then the input rho_AB must already have been k-extendible. Disprove this converse. Build your counterexample from the isotropic family rho_AB(x) = (1 - x) Phi+ + (x/d^2) I, using that it is the image of the maximally entangled state under a depolarizing channel, rho_AB(x) = (id_A tensor D_x)(Phi+) with D_x(sigma) = (1 - x) sigma + x tr(sigma) I/d, together with the isotropic k-extendibility condition x >= (d/(d+1))(1 - 1/k). Exhibit an explicit input, channel, and k.
Hint: mathematical route
The converse of channel-preserved extendibility fails: the depolarizing channel sends the non-2-extendible Phi+ to a 2-extendible isotropic state, so a channel can create extendibility the input lacked.
- A pure entangled state such as Phi+ is not 2-extendible.
- The depolarizing channel maps Phi+ to the isotropic state rho(x).
- Choosing x in the 2-extendible window gives an extendible output from a non-extendible input.
Reference solution
The converse fails. Take the input to be the maximally entangled state itself, rho_AB = Phi+ = rho_AB(x=0). This is a pure entangled state, hence NOT 2-extendible: a pure entangled bipartite state has no 2-extension, because any state whose B'-marginal equals a pure psi_AB must factorize as psi_AB tensor omega_{B'} (Exercise 7.3), and then its A B'-marginal is the uncorrelated psi_A tensor omega_{B'}, which cannot equal the correlated psi_AB. Now let the channel be the depolarizing channel N = D_x. The output is (id_A tensor D_x)(Phi+) = rho_AB(x), an isotropic state, which by the isotropic condition is k-extendible precisely when x >= (d/(d+1))(1 - 1/k). Concretely for k = 2 the condition is x >= (d/(d+1))(1/2) = d/(2(d+1)); pick any such x -- for instance x = d/(d+1) >= d/(2(d+1)), which even makes the output separable and hence infinitely extendible. Then the output rho_AB(x) is 2-extendible, yet the input Phi+ is a pure entangled state that is not even 2-extendible. This is a k-extendible output arising from a non-k-extendible input, disproving the converse. The mechanism: the depolarizing channel degrades entanglement by mixing the state toward the identity, so a channel can create extendibility that the input lacked -- extendibility is monotone under channels in one direction only.
Assessment rubric
- Identifies that a pure entangled bipartite state (e.g. Phi+) has no 2-extension, with the Exercise-7.3 reason (1 weight)
- Uses rho_AB(x) = (id_A tensor D_x)(Phi+) to realize an isotropic output as a channel image of Phi+ (1 weight)
- Chooses x satisfying the isotropic 2-extendibility window x >= d/(2(d+1)) so the output IS 2-extendible (1 weight)
- Concludes a k-extendible output from a non-k-extendible input, refuting the converse, and explains the one-way monotonicity (1 weight)
Graph-grounded prerequisites
These are support/source anchors, not an automatically extracted theorem-dependency proof.
- Operator and semigroup theory (frozen curriculum anchor; see graph registry)
- Probability and sampling (frozen curriculum anchor; see graph registry)
- Tensor networks (frozen curriculum anchor; see graph registry)
Lean view and evidence boundary
No aligned Lean view has been admitted for this native core in this textbook checkpoint. This is not a claim that the underlying mathematics lacks a formal proof. Contributions must preserve this exact target contract and split.
Sources and provenance
- website/research/sources.json
leditzky-2025-repth-qit
Source SHA-256:9f85f45f20ed5a46931ad0dec3b94ee09748bc56e39fae29b0aefe910440b0ae· MIT metadata only - Primary source
Contract: 12791f17f0b5e85427e37c81fcb3cba157f767784b22f68806097a88107e171c
Item digest: e4a453dba730b81064c7eab905c12090da22e6fb67aa29a08ee0feccfc0ce3f0
Graph snapshot: qcb-snapshot-ffa644f0da3031a1ce754b9a
Authored support steps: 3. This is not measured graph distance or model difficulty.