Formalize Open Quantum Problem #31: Individual measurement strategies on geometrically uniform states

### What is the conjecture

This is problem [#31](https://oqp.iqoqi.oeaw.ac.at/individual-measurement-strategies-on-geometrically-uniform-states) on the [Open Quantum Problems list](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems).

> **Problem (Open Quantum Problem #31: “Individual measurement strategies on geometrically uniform states”).**
> We call a set of quantum states `{|ψ_0⟩, |ψ_1⟩, …, |ψ_{d-1}⟩}` geometrically uniform if there is a unitary operator `U` that transforms `|ψ_j⟩` into `|ψ_{j+1}⟩` for all `j`, with indices read mod `d`. Suppose now that `N` copies of those geometrically uniform states are given, i.e. `{|ψ_0⟩^{⊗ N}, |ψ_1⟩^{⊗ N}, …, |ψ_{d-1}⟩^{⊗ N}}`.
> If we have a quantum memory, and can do collective measurements on the `N` systems, the square-root collective measurement will be the optimal strategy and provide the minimum error. Otherwise, we must rely on measurements performed on individual copies. Does there always exist a suitably designed individual measurement strategy that asymptotically reaches the minimum error of the collective measurement?

A standard formalization is as a **minimum-error discrimination problem** for a cyclically symmetric ensemble:

* Fix a finite-dimensional Hilbert space `H` and pure states
`|ψ_0⟩, …, |ψ_{d-1}⟩ ∈ H`
such that there is a unitary `U` with
`U|ψ_j⟩ = |ψ_{j+1}⟩` for all `j` (indices mod `d`).
In the standard symmetric version, the prior is uniform: `Pr(j)=1/d`.
* For `N` copies, define the tensor-power states
`ρ_j^(N) := (|ψ_j⟩⟨ψ_j|)^{⊗ N}`.
* A **collective measurement** is a POVM `{M_j}_{j=0}^{d-1}` on `H^{⊗ N}`.
Its success probability is
`P_coll(N,M) = (1/d) ∑_{j=0}^{d-1} Tr(M_j ρ_j^(N))`,
and the optimal collective success probability is `P_coll^*(N)`.
* Let
`Σ_N := (1/d) ∑_{j=0}^{d-1} ρ_j^(N)`.
Because the tensor-power ensemble remains geometrically uniform under `U^{⊗ N}`, the **square-root measurement** / **pretty good measurement**
`M_j = Σ_N^{-1/2} (1/d) ρ_j^(N) Σ_N^{-1/2}`
(on the support of `Σ_N`) is optimal for the collective problem.

To formalize the “no quantum memory / individual measurements” side:

* An **individual measurement strategy** is a sequential scheme in which, at step `k`, one measures only the `k`-th copy. The measurement at step `k` may depend on the earlier classical outcomes (adaptive / feed-forward), but no entangling joint measurement across several copies is allowed.
* Equivalently, one specifies single-copy POVMs
`M^(1)`, `M^(2)_{x_1}`, `M^(3)_{x_1,x_2}`, …, `M^(N)_{x_1,…,x_{N-1}}`
together with a final classical decision rule
`g(x_1,…,x_N) ∈ {0,…,d-1}`.
* Let `P_ind^*(N)` be the optimal success probability over all such sequential/adaptive strategies.

The main question can then be stated as:

* **Main asymptotic attainability problem:** does every geometrically uniform ensemble satisfy that the best individual strategy asymptotically matches the optimal collective one?
Writing
`p_coll^*(N) := 1 - P_coll^*(N)`
and
`p_ind^*(N) := 1 - P_ind^*(N)`,
a natural strong formalization is
`lim_{N→∞} p_ind^*(N) / p_coll^*(N) = 1`.

Natural sharpenings / sub-problems are:

* Determine whether one already has exact equality
`p_ind^*(N) = p_coll^*(N)` for **every** `N` for some or all geometrically uniform ensembles.
* Determine whether adaptivity is essential by comparing outcome-dependent individual measurements with non-adaptive fixed single-copy measurements.
* Consider weaker asymptotic variants, e.g. equality of optimal error exponents
`lim_{N→∞} -(1/N) log p_ind^*(N) = lim_{N→∞} -(1/N) log p_coll^*(N)`.

The cited `d=2` partial result is especially striking: adaptive individual projective measurements can already match the collective optimum for every number of copies. The open difficulty is to understand whether analogous attainability persists for general geometrically uniform multi-state ensembles.

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #31)](https://oqp.iqoqi.oeaw.ac.at/individual-measurement-strategies-on-geometrically-uniform-states)
* [Open Quantum Problems master list (for numbering/metadata)](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)
* (Optional project background) O. Krüger & R. F. Werner, “Some Open Problems in Quantum Information Theory”:
[https://arxiv.org/abs/quant-ph/0504166](https://arxiv.org/abs/quant-ph/0504166)
[https://arxiv.org/pdf/quant-ph/0504166](https://arxiv.org/pdf/quant-ph/0504166)

Key related references (as listed on the OQP page):

* Y. C. Eldar and G. D. Forney Jr, “On quantum detection and the square-root measurement” (IEEE Trans. Inform. Theory 47, 858–872 (2001); arXiv: [quant-ph/0005132](https://arxiv.org/abs/quant-ph/0005132))
(shows that the square-root measurement is optimal for geometrically uniform state sets)
* N. Gisin and S. Wolf, “Quantum cryptography on noisy channels: quantum versus classical key-agreement protocols” (Phys. Rev. Lett. 83, 4200 (1999); arXiv: [quant-ph/9902048](https://arxiv.org/abs/quant-ph/9902048))
* A. Acín, Ll. Masanes, and N. Gisin, “Equivalence between two-qubit entanglement and secure key distribution” (Phys. Rev. Lett. 91, 167901 (2003); arXiv: [quant-ph/0303053](https://arxiv.org/abs/quant-ph/0303053))
* D. Bruß, M. Christandl, A. Ekert, B.-G. Englert, D. Kaszlikowski, and C. Macchiavello, “Tomographic quantum cryptography: equivalence of quantum and classical key distillation” (Phys. Rev. Lett. 91, 097901 (2003); arXiv: [quant-ph/0303184](https://arxiv.org/abs/quant-ph/0303184))
* A. Acín, J. Bae, E. Bagan, M. Baig, Ll. Masanes, and R. Muñoz-Tapia, “Secrecy content of two-qubit states” (Phys. Rev. A 73, 012327 (2006); arXiv: [quant-ph/0411092](https://arxiv.org/abs/quant-ph/0411092))
* B.-G. Englert, D. Kaszlikowski, L. C. Kwek, and J. Y. Lim, “Coherent eavesdropping attacks in tomographic quantum cryptography: nonequivalence of quantum and classical key distillation” (Phys. Rev. A 72, 042315 (2005); arXiv: [quant-ph/0312172](https://arxiv.org/abs/quant-ph/0312172))
* A. Acín, E. Bagan, M. Baig, Ll. Masanes, and R. Muñoz-Tapia, “Multiple copy 2-state discrimination with individual measurements” (Phys. Rev. A 71, 032338 (2005); arXiv: [quant-ph/0410097](https://arxiv.org/abs/quant-ph/0410097))
(key partial result: for `d=2`, fixed individual measurements asymptotically saturate the collective bound, while adaptive individual projective measurements can match the collective optimum for every `N`)
* B. L. Higgins, B. M. Booth, A. C. Doherty, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Mixed state discrimination using optimal control” (Phys. Rev. Lett. 103, 220503 (2009); arXiv: [0909.1572](https://arxiv.org/abs/0909.1572))
(experimental implementation of a related adaptive scheme for noisy two-state discrimination)

(Optional background review:)

* J. Bae and L.-C. Kwek, “Quantum state discrimination and its applications” (J. Phys. A 48, 083001 (2015); arXiv: [1707.02571](https://arxiv.org/abs/1707.02571))

### Prerequisites needed

* Finite-dimensional quantum mechanics: Hilbert spaces, pure states / density operators, tensor products
* POVMs and minimum-error quantum state discrimination
* Unitary symmetries and geometrically uniform (cyclically covariant) ensembles
* Linear algebra / matrix analysis: positive semidefinite operators, trace, supports, operator square roots
* Sequential / adaptive measurement schemes with classical feed-forward
* Basic asymptotic analysis of error probabilities and error exponents
* (Optional) quantum-cryptographic eavesdropping models, and the Bayesian-updating intuition from the two-state case

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-81 (Quantum theory)
* ams-94 (Information and communication theory)
* ams-15 (Linear and multilinear algebra; matrix theory)
* ams-47 (Operator theory)

### Choose either option

* [ ] I plan on adding this conjecture to the repository
* [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

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