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Reliable Quantum State Tomography
Matthias Christandl, Renato Renner
TL;DR
Finite-data tomography requires reliable error bars around state estimates, including when likelihood maxima lie on the boundary. The paper constructs confidence regions from induced state measures and shows their consistency with maximum-likelihood estimation while providing reliable error bars.
Problem
Finite measurement data can place likelihood maxima on the boundary of the state space, making the decay around maxima difficult to analyze and estimates operationally incomplete without error bars.
Method
The method constructs a probability measure on states by purifying them and taking a partial trace, then uses this induced measure in a data-analysis procedure for arbitrary POVM elements.
Results
The method provides reliable error bars, shows that likely states within them are generally not boundary states, and is theoretically consistent with maximum-likelihood estimation.
Takeaways & Limitations
Confidence regions can resolve the boundary-related unphysical-state problem associated with maximum-likelihood predictions while accommodating additional information such as rank or symmetry.
Takeaways & Limitations
The construction represents the prior state density through a density on purifications over an auxiliary space K isomorphic to H.
Abstract
from arXiv · showhide
Quantum state tomography is the task of inferring the state of a quantum system by appropriate measurements. Since the frequency distributions of the outcomes of any finite number of measurements will generally deviate from their asymptotic limits, the estimates computed by standard methods do not in general coincide with the true state, and therefore have no operational significance unless their accuracy is defined in terms of error bounds. Here we show that quantum state tomography, together with an appropriate data analysis procedure, yields reliable and tight error bounds, specified in terms of confidence regions - a concept originating from classical statistics. Confidence regions are subsets of the state space in which the true state lies with high probability, independently of any prior assumption on the distribution of the possible states. Our method for computing confidence regions can be applied to arbitrary measurements including fully coherent ones; it is practical and particularly well suited for tomography on systems consisting of a small number of qubits, which are currently in the focus of interest in experimental quantum information science.
Supplemental Information
The supplemental material contains technical proofs, a discussion of independent measurements, further remarks, state representations on symmetric subspaces, function representations, and examples.
- Supplemental Information: The document includes additional material beyond the journal version’s Supplemental Information.Appendices A–D are identified as additional material not included in the journal supplement.
- Supplemental Information: The first part presents precise technical statements and proofs, discusses independent measurements, and gives further remarks.The second part covers symmetric-subspace state representations, state-space functions, and examples.
1 Statements and Proofs
This section develops a general framework for reliable quantum-state inference, defines confidence regions from measurement-dependent distributions, and proves the associated reliability statements.
- 1 Statements and Proofs: The procedure maps measurement outcomes to a probability distribution µ_Bn over candidate states and derives confidence regions from it.The construction uses a high-probability set and its δ-extension to obtain the confidence region.
- 1 Statements and Proofs: Theorem 1 establishes reliable predictions from µ_Bn, while Corollary 1 converts the resulting criterion into confidence regions.The proof analyzes failure probabilities for tests associated with candidate states.
- 1 Statements and Proofs: The construction does not require a prior distribution and supports arbitrary quantum measurements, including fully coherent ones.The supplemental treatment also explains purification, partial tracing, and the relation between distributions on purified and original state spaces.
2 Independent Measurements
For independent measurements, the paper connects its distribution-based analysis to maximum likelihood estimation and characterizes the concentration of that distribution around its maxima.
- 2 Independent Measurements: For product-form measurements, the analysis uses POVM outcome frequencies to study the distribution µ_Bn over candidate density matrices.The probability weights are built from the observed relative frequencies and the POVM outcome probabilities.
- 2 Independent Measurements: The maximum of µ_Bn coincides with the density matrix inferred by maximum likelihood estimation.The maximizing expression is the standard log-likelihood function for the observed relative frequencies.
- 2 Independent Measurements: The method provides reliable error bars around the MLE maximum, including when the maximum lies on the boundary of the state space.Most likely states within the error bars need not lie on that boundary.
- 2 Independent Measurements: The extreme-point analysis is simplified by assuming a tomographically complete POVM with linearly independent effects.Under these assumptions, the relevant maximum is treated as unique.
- 2 Independent Measurements: The decay exponent of µ_Bn is asymptotically related to relative entropy, and Pinsker’s inequality bounds the resulting state-space errors.The paper uses this relation to characterize the error bars around interior maxima.
3 Remarks
The method can incorporate additional state information and retain measurement data in a representation suitable for later updates.
- 3 Remarks: Known symmetry can restrict the confidence region to states invariant under the specified local unitary actions.The resulting region is the intersection of the symmetry-compatible state set with Γδ.
- 3 Remarks: The output µ_Bn can be represented using generalized spherical harmonics whose moments of degree less than n are fixed by measurements on n systems.These moments contain the information needed to update µ_Bn with additional measurement data.
A Quasi-Probability Distributions
The section develops Q- and P-representations for operators on the symmetric subspace, showing when diagonal data uniquely determines an operator and how representation coefficients behave.
- Q-representation: The Q-representation uniquely determines an operator on Symn(Cd).It is based on the diagonal values ⟨x|⊗nB|x⟩⊗n.
- Q-representation: Diagonal matrix elements determine all off-diagonal matrix elements through polynomial dependence on x and x′.The relevant matrix elements form polynomials whose values on real parameters determine them uniquely.
- P-representation: Every operator on Symn(Cd) admits a P-representation.The proof uses the uniqueness of the Q-representation to show that the orthogonal complement of the representable space is trivial.
- P-representation: The coefficients pB(ℓ, m) are uniquely determined for ℓ≤n and arbitrary for ℓ>n.The same cutoff appears in the associated orthogonality and multiplicity statements.
- P-representation: The representable operator space is characterized by its relation to operators with vanishing Q-representation.The orthogonal complement of the P-representable space equals the space of operators with vanishing Q-representation.
B Spherical Harmonics for Higher Dimensions
The section constructs a Fourier-like basis on CP d−1 using unitary-group representation theory and shows that the resulting functions densely span L2(CP d−1).
- Geometric and representation-theoretic setup: Gelfand-Zetlin patterns provide an orthonormal basis for irreducible U(d) representations.The basis vectors are labeled by sequences of interlaced diagrams generated through the branching rule.
- Geometric and representation-theoretic setup: CP d−1 is identified with the homogeneous space U(d)/[U(d−1)×U(1)].This identifies projective space with the quotient by the stabilizer subgroup of a point.
- Basis construction: The functions yℓ,m are defined from representation matrix elements stabilized by U(d−1)×U(1).They are square integrable and orthonormal with respect to the induced measure on CP d−1.
- Basis construction: The functions yℓ,m densely span L2(CP d−1).The result follows by applying the Peter-Weyl theorem to U(d) and selecting the subgroup-invariant components.
- Product structure: Products of the basis functions are expressed using U(d) Clebsch-Gordan coefficients.The resulting product formula is used in the update rule, with relevant representations indexed by λ=(ℓ,0,…,0,−ℓ).
- Product structure: For the tensor-power representation ν=(n,0,…,0), the relevant representation coefficients are nonzero only for ℓ≤n.This cutoff follows from the corresponding Littlewood-Richardson coefficient behavior.
C Recovering the Spherical Harmonics on the Bloch Sphere
For d=2, the functions on CP 1 are analyzed on the Bloch sphere and related to ordinary spherical harmonics through SU(2) representation theory.
- Bloch-sphere representation: For d=2, CP 1 is represented as the sphere S2 with the measure 1/(4π) sin θ dθdφ.Points are parameterized by θ∈[0,π] and φ∈[0,2π).
- Recovery of spherical harmonics: The basis functions yℓ,m are related to ordinary spherical harmonics.The relation is established by converting U(2) Clebsch-Gordan coefficients into SU(2) coefficients.
- Recovery of spherical harmonics: The U(2) and SU(2) basis labels are connected through spin L and projection M.The mapping follows from the weights of Gelfand-Zetlin patterns and the z-direction spin projection.
- Coefficient formulas: For λ=(ℓ,−ℓ) and ν=(n,0), the relevant Clebsch-Gordan coefficient vanishes for odd parity conditions.In particular, when ℓ=ℓ′, it vanishes unless ℓ′′ is even; otherwise the coefficient vanishes in the stated case.
- Coefficient formulas: Explicit coefficient formulas and bounds are obtained for even n and ℓ.These formulas support the conversion between the representation-theoretic basis and spherical-harmonic expressions.
- Equatorial distribution: The equator-uniform distribution has Fourier components constrained by its U(1)×U(1) invariance.Only components with m=0 can be nonzero.
Holevo’s Covariant Measurement
The section analyzes Holevo’s fidelity-optimal covariant measurement using Fourier coefficients, showing asymptotic correctness and a compact treatment of convergence.
- Measurement analysis: Holevo’s fidelity-optimal state-estimation procedure is represented by the POVM {|y⟩⟨y|⊗n dy}.The analysis begins by examining outcomes associated with effects such as |d⟩⟨d|⊗n.
- Measurement analysis: The estimate density becomes asymptotically correct because dim(n,d)|⟨x|d⟩|2n converges to the δ-distribution.Replacing |d⟩⊗n with |z⟩⊗n yields the corresponding transformed analysis.
- Fourier analysis: Convergence can be tested by requiring the Fourier coefficients of the estimate density to converge to one.The coefficients are compared with the Fourier coefficients of δ(x).
- Fourier analysis: For the qubit case, explicit Fourier-coefficient formulas yield convergence bounds.The formulas include a cutoff at ℓ≤n.
- Fourier analysis: Fourier analysis compresses the convergence behavior and can be combined with basis rotations for protocols using several measurement bases.The update rule can incorporate the derived formula together with the basis-rotation lemma.
Basis Measurements
The analysis specializes product measurements in an orthonormal basis and examines qubit data with equal counts in two bases. In this setting, the estimate density converges to the uniform distribution on the Bloch-sphere equator, reflecting the measurement’s missing phase information.
- The analysis considers product measurements whose single-system measurement is given by an orthonormal basis.
- For qubits, the calculation examines measurements with equal numbers of outcomes 1s and 2s, corresponding to m = n/2.
- The estimate density converges to the uniform distribution on the equator of the Bloch sphere.
- The equatorial concentration occurs because this measurement provides no information about the state’s phase.
- For several bases, including BB84 and six-state protocols, the derived formula can be combined with a basis-rotation lemma in the update rule.