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

[MarioKrenn6240]

What is the conjecture

This is problem #31 on the Open Quantum Problems list.

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)
• Open Quantum Problems master list (for numbering/metadata)
• (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/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)
  (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)
• 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)
• 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)
• 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)
• 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)
• 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)
  (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)
  (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)

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

• 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
• This issue is up for grabs: I would like to see this conjecture added by somebody else

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[HyunHoCha]

I think that "Realizable receivers for discriminating arbitrary coherent-state waveforms and multi-copy quantum states near the quantum limit" solved the weaker problem of error exponents.