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Quantum Tomography Protocols with Positivity are Compressed Sensing Protocols
Amir Kalev, Robert L. Kosut, Ivan H. Deutsch
TL;DR
Quantum tomography can require substantial resources as quantum-system dimensions grow, motivating compressed-sensing methods that reduce measurement settings. The paper shows that positivity makes compressed-sensing measurements uniquely identify low-rank states among all positive matrices, with consequences for reconstruction methods and measurement design.
Problem
Quantum-state and process parameters grow poorly with Hilbert-space dimension, motivating methods that reduce the resources required for tomography.
Source: INTRODUCTION · p. 1Method
The paper analyzes how positivity of the recovered matrix changes the guarantees of compressed-sensing measurements in quantum tomography.
Source: INTRODUCTION · p. 1, RESULTS · p. 3Results
With positivity, compressed-sensing measurements uniquely identify a rank-at-most-r state among all positive matrices, making them strictly informationally complete rather than only rank-restricted informationally complete.
Source: RESULTS · p. 3, DISCUSSION · p. 6Takeaways & Limitations
Positivity lets diverse tomography optimizers, including maximum-likelihood methods, exhibit compressed-sensing recovery, while enabling efficient convex optimization and robust strictly informationally complete measurements.
Source: INTRODUCTION · p. 2, DISCUSSION · pp. 6, 8The noisy-record analysis assumes the noise contribution is bounded in norm by ϵ.
Source: RESULTS · p. 5
Abstract
from arXiv · showhide
Characterizing complex quantum systems is a vital task in quantum information science. Quantum tomography, the standard tool used for this purpose, uses a well-designed measurement record to reconstruct quantum states and processes. It is, however, notoriously inefficient. Recently, the classical signal reconstruction technique known as "compressed sensing" has been ported to quantum information science to overcome this challenge: accurate tomography can be achieved with substantially fewer measurement settings, thereby greatly enhancing the efficiency of quantum tomography. Here we show that compressed sensing tomography of quantum systems is essentially guaranteed by a special property of quantum mechanics itself---that the mathematical objects that describe the system in quantum mechanics are matrices with nonnegative eigenvalues. This result has an impact on the way quantum tomography is understood and implemented. In particular, it implies that the information obtained about a quantum system through compressed sensing methods exhibits a new sense of "informational completeness." This has important consequences on the efficiency of data taking for quantum tomography, and enables us to construct informationally complete measurements that are robust to noise and modeling errors. Moreover, our result shows that one can expand the numerical tool-box used in quantum tomography and employ highly efficient algorithms developed to handle large dimensional matrices on a large dimensional Hilbert space. While we mainly present our results in the context of quantum tomography, they apply to the general case of positive semidefinite matrix recovery.
INTRODUCTION
The introduction explains how positivity enables compressed-sensing recovery in quantum tomography, including unique identification and robustness to noise and model imperfection. It also extends the discussion beyond state tomography to positive semidefinite matrix recovery, outlining recovery conditions and numerical experiments.
INTRODUCTION
- Compressed sensing uses prior structure such as sparsity or low rank to reduce measurements, and quantum tomography applies it to states and processes whose parameter counts grow rapidly with system size.Successful recovery depends on using the associated compressed-sensing measurements and reconstruction heuristics, rather than merely using fewer measurements with an arbitrary optimization program.Source: INTRODUCTION · p. 1
RESULTS
- A rank-r positive semidefinite matrix measured by rank-r compressed-sensing measurements is uniquely identified among positive semidefinite matrices of any rank, making the record strictly-IC.Without positivity, the same measurements identify a matrix only within the rank-r set; with positivity, the result also applies beyond normalized density matrices, including process tomography and general positive semidefinite matrix recovery.Source: INTRODUCTION · p. 2, RESULTS · p. 3, Figure 1 · p. 4
- With positivity enforced, any convex objective or measurement-error norm recovers the unique state from noiseless compressed-sensing data, including maximum-likelihood estimation.The trace constraint is redundant in the noiseless case because the measurement data come from trace-preserving quantum measurements.Source: RESULTS · p. 4
- For noise bounded by ϵ, positive-cone convex minimization yields an estimate whose error is bounded by g(ϵ), with g(ϵ) → 0 as ϵ → 0; recovery also handles model imperfections by recovering the dominant rank-r part.Different optimization heuristics may have different quantitative performance depending on the noise model.Source: RESULTS · p. 5
- Compressed-sensing measurements can be robust rank-r strictly-IC: for many-qubit systems, O(rd poly(log d)) Pauli-product expectation values satisfy the required restricted isometry property with overwhelming probability.The passage also notes analogous high-probability restricted-isometry results for sparse process-matrix reconstruction using random input states and observables.Source: RESULTS · pp. 5–6
- In three-qubit simulations, positivity-constrained estimators all exhibit the compressed-sensing effect, while without positivity only nuclear-norm minimization succeeds with fewer than a full informationally complete record.Under the stated restricted-isometry condition, the estimators recover the exact state without noise and a close estimate with finite-sampling noise.Source: RESULTS · p. 6, Figure 2 · p. 7
- In a ten-qubit simulation with d = 1024, the positivity-constrained method captured the information in about 28 random Pauli bases, given sufficient statistics.The simulation used 30 random Pauli bases and 100d repetitions per observable.Source: RESULTS · p. 6
DISCUSSION
- Positivity unifies procedures such as maximum-likelihood and nuclear-norm minimization under compressed sensing, while enabling efficient convex algorithms for large matrices.The discussion specifically identifies ADMM algorithms as an example of methods suited to large-dimensional matrices.Source: INTRODUCTION · p. 2, DISCUSSION · p. 6
- With positivity constraints, nonnegative trace minimization, nonnegative least-squares, and maximum-(log)likelihood estimation are all compressed-sensing estimators; without positivity, least-squares and trace minimization require 27 Pauli bases for exact recovery.Source: DISCUSSION · p. 7
- A ten-qubit simulation using nonnegative least squares exhibits compressed sensing, with infidelity measured across varying numbers of settings and averaged over 10 Haar-random pure states.The numerical setup samples independent Pauli-basis measurements and generates outcome frequencies from Nrep repetitions.Source: Figure 3 · p. 7, METHODS · p. 8
- Compressed-sensing measurements that satisfy the restricted isometry property yield strictly informationally complete measurements robust to noise and model imperfection, even when the state is only close to rank r.The estimated matrix is guaranteed to be close to the unknown matrix under these conditions.Source: DISCUSSION · p. 8
- The results apply to positive sparse vectors and positive rank-deficient matrices, including quantum-process tomography, and extend to sensing maps with other robust-recovery guarantees.Examples include O(rd) generic rank-one projections and O(rd log(d)) projectors onto random elements of an approximate 4-design.Source: DISCUSSION · p. 8, METHODS · p. 8
METHODS
- For rank-r matrix recovery, compressed-sensing measurements use a nuclear-norm minimization program; when the matrix has rank r, sufficient samples scale as O(rd), with possible logarithmic corrections.The reconstruction error is O(ϵ) when the measurement noise satisfies ∥e∥2 ≤ ϵ.Source: METHODS · p. 8
Section A: Proof of Theorem 1
The proof reduces uniqueness among positive solutions to a fixed-trace auxiliary problem, then uses a restricted isometry condition to establish uniqueness. In quantum tomography, the auxiliary transformation is unnecessary for trace-preserving POVMs, so the result applies directly to the sensing map and density matrix.
Section A: Proof of Theorem 1
- Under the stated conditions, positivity makes the set of matrices consistent with the measurements contain a single element; the argument applies to general positive matrices and sensing maps.The proof does not require the matrix to be a quantum state or the sensing map to come from a POVM.Source: Section A: Proof of Theorem 1 · p. 10
- A change of representation maps the positive solutions one-to-one to an auxiliary problem whose solutions have fixed trace and preserve rank.The transformed problem uses D[⊙] = Tr(B−1EB⊺−1⊙) and Z = B⊺MB.Source: Section A: Proof of Theorem 1 · p. 10
- If the auxiliary sensing map satisfies the stated restricted isometry condition, the fixed-trace positive solution is unique, yielding uniqueness for the original positive-solution set.The proof obtains uniqueness of the fixed-trace set by combining the restricted isometry result with the trace-minimization argument.Source: Section A: Proof of Theorem 1 · p. 10
- For trace-preserving POVMs, the auxiliary map and matrix reduce directly to the original sensing map and density matrix, so the theorem follows without the auxiliary construction.In this case D = A and Z = ρ.Source: Section A: Proof of Theorem 1 · pp. 10–11
Section C: Proof of formal equivalence between equation (8) and equation (9) of the main text
The proof establishes equivalence between trace minimization with a residual constraint and least-squares minimization at a matching trace. It does so by comparing their dual problems, and the equivalence also holds with shared convex constraints or a different norm.
Section C: Proof of formal equivalence between equation (8) and equation (9) of the main text
- The proof constructs and solves each program’s dual; coincident dual solutions imply coincident primal solutions because both programs have zero duality gap.The programs’ objective functions and constraints are linear or quadratic.Source: Section C: Proof of formal equivalence… · p. 12
- For t = Tr(ρ_tr), the trace-minimization and least-squares programs return the same solution.The trace-minimization program minimizes trace subject to residual at most ε; the least-squares program minimizes residual at fixed trace.Source: Section C: Proof of formal equivalence… · pp. 12–13
- The equivalence remains when any convex constraint is added to both programs.The source states the shared-constraint extension as a continuation of the equivalence result.Source: Section C: Proof of formal equivalence… · p. 14
- The equivalence also holds when the two-norm in the programs is replaced by any other norm.Source: Section C: Proof of formal equivalence… · p. 14