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Quantum pure-state identification

A. Hayashi, M. Horibe, T. Hashimoto

arXiv:quant-ph/0507237v2quant-ph

TL;DR

The paper studies how to identify an input pure state known to equal one of two reference states when only finite copies, rather than classical reference descriptions, are available. It derives the optimal averaged identification strategy under independent unitary-invariant reference distributions for arbitrary copy number and dimension. The mean identification probability approaches the mean discrimination probability as the number of reference copies tends to infinity.

  • Problem

    The problem is identifying an input pure state with one of two reference pure states when the references are known only through a finite number of copies.

    Source: §I · p. 1

  • Method

    The paper derives optimal averaged identification strategies for arbitrary reference-copy number and dimension under independent unitary-invariant reference-state distributions.

    Source: §I · p. 1, §III · p. 3

  • Results

    The mean identification probability approaches the mean discrimination probability as the number of reference copies goes to infinity.

    Source: §I · p. 1, §V · p. 7

  • Takeaways & Limitations

    Quantum pure state identification is established as a natural generalization of quantum state discrimination with a complete solution for arbitrary N and d.

    Source: §V · p. 7

    The analysis assumes that the input state is supplied as a single copy.

    Source: §V · p. 7

Abstract

from arXiv · show

We address a problem of identifying a given pure state with one of two reference pure states, when no classical knowledge on the reference states is given, but a certain number of copies of them are available. We assume the input state is guaranteed to be either one of the two reference states. This problem, which we call quantum pure state identification, is a natural generalization of the standard state discrimination problem. The two reference states are assumed to be independently distributed in a unitary invariant way in the whole state space. We give a complete solution for the averaged maximal success probability of this problem for an arbitrary number of copies of the reference states in general dimension. It is explicitly shown that the obtained mean identification probability approaches the mean discrimination probability as the number of the reference copies goes to infinity.

I. INTRODUCTION

Quantum pure state identification asks how to identify an unknown input known to equal one of two reference pure states when only finite copies, not classical descriptions, of the references are available. The paper formulates this as a unitary-invariant generalization of discrimination and solves its averaged performance for arbitrary reference-copy number and dimension.

  • The task is to identify an unknown pure state with one of two reference pure states in a d-dimensional space.Source: §I · p. 1
  • With only N copies of each reference and no classical knowledge, the measurement must identify the input from the combined copies without increasing them through cloning.Source: §I · p. 1
  • Quantum state identification reduces to standard quantum state discrimination when N is infinite because infinitely many copies provide complete classical state knowledge.Source: §I · p. 1
  • The reference states are independently and uniformly distributed over the d-dimensional pure-state space, while the input is guaranteed to be one of them.Source: §I · p. 1
  • The paper gives a complete solution for optimal strategies and mean success probability as functions of N and d, and shows convergence to mean discrimination performance as N grows.Source: §I · p. 1

II. AVERAGED MAXIMUM PROBABILITY OF PURE STATE DISCRIMINATION

The paper averages optimal discrimination of two known pure states over independently uniform, unitary-invariant reference states. The resulting mean success probability approaches unity as the Hilbert-space dimension increases.

  • Equal occurrence probabilities are assumed, and the POVM may depend on the reference states because their classical descriptions are available.Source: §II · p. 2
  • For fixed pure references, the optimal success probability is expressed through their trace distance, whose pure-state form determines the averaging calculation.Source: §II · p. 2
  • The averaged maximum discrimination probability is defined by averaging the optimal known-state success probability over independently uniform pure reference states.Source: §II · p. 2
  • Unitary invariance permits fixing one reference state to a basis vector and averaging the other over coefficients uniformly distributed on the normalized hypersphere.Source: §II · p. 2
  • The integrations yield the averaged maximum discrimination probability, and the mean distance and success probability approach unity as d increases.Source: §II · p. 2

III. QUANTUM STATE IDENTIFICATION: CASE OF N = 1

For one copy of each reference state, identification becomes a measurement on three systems whose POVM cannot depend on unknown reference states. Averaging over the unitary-invariant distribution reduces the optimization to the spectrum of an operator built from pairwise symmetric projectors.

III. QUANTUM STATE IDENTIFICATION: CASE OF N = 1

  • The N = 1 task distinguishes two equally likely three-system states with a POVM independent of the unknown reference states.The input occupies system 0, while one copy of each reference state occupies systems 1 and 2.Source: §III · pp. 2–3
  • Unitary averaging replaces tensor products of identically prepared pure states by normalized projectors onto totally symmetric subspaces.This follows from unitary invariance, irreducibility on the symmetric subspace, and a trace normalization.Source: §III · p. 3
  • The optimal measurement is projective, with E1 = PD+ + P0 and E2 = 1 − E1, where P0 may act within the zero-eigenvalue subspace.The positive and zero eigenspaces of the operator D determine the measurement, and the freedom in P0 does not change the success probability.Source: §III · pp. 3–4
  • For general dimension, the N = 1 averaged identification probability is strictly below the averaged maximum discrimination probability.The calculation uses the nonzero eigenvalues and multiplicities of D, whose structure is related to symmetric and mixed-symmetric subspaces.Source: §III · p. 4

IV. QUANTUM STATE IDENTIFICATION: CASE OF ARBITRARY N

For arbitrary numbers of reference copies, the same averaged operator formulation applies on the full input-and-reference system. The optimization can again be solved through a projective measurement after restricting attention to the symmetric reference subspaces.

IV. QUANTUM STATE IDENTIFICATION: CASE OF ARBITRARY N

  • With N copies of each reference state, the POVM acts on (Cd)^(2N+1), and the averaged success probability is obtained analogously to the N = 1 case.Reference copies are grouped into systems 1 and 2, each containing N identical copies.Source: §IV · p. 4
  • Only the subspace where both reference registers are totally symmetric is needed to determine the nonzero eigenvalues and their multiplicities.This restriction captures the support generated by the identical-copy reference states.Source: §IV · p. 4
  • The optimal POVM remains projective for arbitrary N, with E1 = PD+ + P0 and E2 = 1 − E1.The same positive-, zero-, and negative-eigenspace construction used in the one-copy case applies generally.Source: §IV · p. 4

A. Case of qubits (d = 2)

For qubits, the symmetric reference registers are represented as angular momenta, reducing the spectral calculation to recoupling three angular momenta. The resulting mean identification probability converges to the mean discrimination probability for large N.

A. Case of qubits (d = 2)

  • The calculation reduces the problem to recoupling three angular momenta, with each relevant projector supported on the subspace where systems 0 and a couple to total angular momentum N.Source: A. Case of qubits (d = 2) · p. 5
  • The operator D has nonzero eigenvalues ±δ(J), each with multiplicity 2J + 1, obtained from the total angular momentum J.D^2 is evaluated using Pauli-matrix identities and angular-momentum commutation relations.Source: A. Case of qubits (d = 2) · p. 5
  • The resulting qubit mean identification probability is given explicitly as a function of N, with values listed for several small N.The paper also derives the large-N limit by replacing the finite sum with a continuous integral.Source: A. Case of qubits (d = 2) · p. 5
  • In the large-N limit, the qubit mean identification probability equals the mean discrimination probability.Thus increasing the number of reference copies recovers the known-state discrimination benchmark in this case.Source: A. Case of qubits (d = 2) · p. 5

B. Case of arbitrary dimension d

For arbitrary dimension, the paper uses permutation and unitary-group representations to derive the optimal state-identification measurement and averaged success probability. The resulting probability approaches averaged discrimination performance as N grows.

B. Case of arbitrary dimension d

  • The operator D is analyzed through irreducible representations of S2N+1 and U(d), with eigenvalues independent of the multiplicity label b.Only the subspace where systems 1 and 2 are totally symmetric contributes nonzero eigenvalues.Source: B. Case of arbitrary dimension d · p. 6
  • The representation basis includes the totally symmetric sector, two-row sectors appearing twice, and three-row sectors appearing once.The multiplicity m[λ1,λ2](d) determines the number of copies of each two-row representation in the total system.Source: B. Case of arbitrary dimension d · pp. 6–7
  • States with three-row Young diagrams have zero eigenvalue, while two-row representations split into opposite-eigenvalue sectors labeled a = ±.The exchange operator T (12) anticommutes with D and preserves the representation labels.Source: B. Case of arbitrary dimension d · p. 6
  • The averaged maximum identification probability is obtained for arbitrary N and d using the multiplicities of two-row representations.The result is achieved by the optimal projective POVM.Source: B. Case of arbitrary dimension d · p. 7
  • In the large-N limit, the averaged maximum identification probability approaches the averaged maximum discrimination probability.This establishes the expected reduction to discrimination when many reference copies are available.Source: B. Case of arbitrary dimension d · p. 7

V. CONCLUDING REMARKS

The paper formulates pure-state identification as discrimination with unknown reference states supplied through N copies, and solves its averaged success probability for arbitrary N and d. Its mean identification probability approaches mean discrimination probability as N becomes large, while the extension to multiple input copies remains unresolved.

V. CONCLUDING REMARKS

  • For arbitrary N and d, the paper determines the averaged maximal success probability and shows that a projective measurement achieves it.The reference states are independently and unitarily invariantly distributed in the d-dimensional pure-state space.Source: §V · p. 7
  • The mean identification probability approaches the mean discrimination probability in the large-N limit.This confirms the expected connection between identification with finite copies and discrimination with classical state knowledge.Source: §V · p. 7
  • The analysis assumes a single input-state copy, and evaluating the probability further for general M, N, and d is left for future work.The paper notes that the corresponding general evaluation does not seem straightforward.Source: §V · p. 7
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