Pretty-good simulation of all quantum measurements by projective measurements

Abstract

In quantum theory general measurements are described by so-called Positive Operator-Valued Measures (POVMs). We show that in $d$-dimensional quantum systems an application of depolarizing noise with constant (independent of $d$) visibility parameter makes any POVM simulable by a randomized implementation of projective measurements that do not require any auxiliary systems to be realized. This result significantly limits the asymptotic advantage that POVMs can offer over projective measurements in various information-processing tasks, including state discrimination, shadow tomography or quantum metrology. We also apply our findings to questions originating from quantum foundations by asymptotically improving the range of visibilities for which noisy pure states of two qudits admit a local model for generalized measurements. As a byproduct, we give asymptotically tight (in terms of dimension) bounds on critical visibility for which all POVMs are jointly measurable. On the technical side we use recent advances in POVM simulation, the solution to the celebrated Kadison-Singer problem, and a method of approximate implementation of nearly projective POVMs by a convex combination of projective measurements, which we call dimension-deficient Naimark theorem. Finally, some of our intermediate results show (on information-theoretic grounds) the existence of circuit-knitting strategies allowing to simulate general $2N$ qubit circuits by randomization of subcircuits operating on $N+1$ qubit systems, with a constant (independent of $N$) probabilistic overhead.

Figures (3)

• Figure 1: We tackle the problem of minimizing the size of the ancilla space necessary to implement a general POVM $\mathbf{M}$ on a $d$-dimensional system. (a) The standard method is based on the Naimark theorem (Theorem \ref{['th:Naimark']}) which, in conjunction with convex structure of POVMs (cf. Oszmaniec17), allows to implement arbitrary $\mathbf{M}$ by projective measurement $\mathbf{P}^\mathbf{M}$ on a system enlarged by a $d$-dimensional ancilla initialized in state $\left| \mathrm{anc}_d \right>$. (b) An alternative solution, formalized as Result \ref{['res:singleANC']}, allows to realize arbitrary qudit POVM via randomization over measurements $\mathbf{N}^{(\beta)}$ which require only a single auxiliary qubit (initialized in state $\left| \mathrm{anc}_2 \right>$) to be implemented. This simplification comes at a price -- the protocol works in every experimental shot with success probability $q=1/8$. (c) If no ancillas are permitted it is possible to realize a noisy version of $\mathbf{M}$ by a convex combination of projective measurements on $\mathbb{C}^d$. Specifically, in Result \ref{['res:projSIM']} we prove that for $c=0.02$ the noisy POVM $\mathrm{\Phi}_c(\mathbf{M})$ is projectively simulable, where $\mathrm{\Phi}_c$ is the depolarizing channel.
• Figure 2: A circuit knitting method originating from the POVM simulation protocol in Result \ref{['res:singleANC']}. The method realizes sampling from the output of a $2N$-qubit unitary $U$ on $\rho\otimes\left| 0_N \rangle\langle 0_N \right|$, where $\rho$ is an $N$-qubit state and $\left| 0_N \right>$ is a pure state of an $N$-qubit ancilla. It proceeds by by: (i) sampling $\beta$ according to the probability distribution $\{p_\beta\}$, (ii) implementing an $N+1$-qubit unitary $U_\beta$ on a state $\rho \otimes \left| 0 \rangle\langle 0 \right|$ (with $\left| 0 \right>$ being a pure state of a single qubit), (iii) performing a measurement in the computational basis on $N+1$ qubits and (iv) applying post-processing $\mathcal{Q}^{(\beta)}$ to the resulting outcome $\mathbf{y}$ to return $\mathbf{x}$ or $\emptyset$ (a flag indicating that the protocol was unsuccessful). Importantly, the method generates a sample $\mathbf{x}$ from the correct probability distribution with success probability $q=1/8$.
• Figure 3: Structure of the main results of the paper. Some of the arrows are labelled by auxiliary lemmas used in the proof of the respective result. We start with a POVM to be simulated $\mathbf{M} \in \mathrm{POVM}(\mathbb{C}^d)$. First, classical post-processing is used to replace $\mathbf{M}$ with a fine-grained POVM $\mathbf{M}'$ whose outcomes have nearly equal magnitude (Lemma \ref{['lem:flatPOVM']}). Our central intermediate result, Theorem \ref{['th:PostSimulation']}, states that such POVMs can be simulated with high success probability using a convex combination of POVMs with only $\Theta(d)$ outcomes. The simulation relies on an explicit probabilistic simulation protocol from SMO2022 (Theorem \ref{['th:OldProtocol']}) and the solution to the Kadison-Singer problem (Theorem \ref{['th:solutionKS']}). From this we derive our first main result, Result \ref{['res:singleANC']} -- the POVM $\mathbf{M}$ can be simulated with high probability by projective measurements using only a low-dimensional ancilla. Our second result, Result \ref{['res:projSIM']}, states that $\mathrm{\Phi}_c(\mathbf{M})$, a noisy version of $\mathbf{M}$ with constant noise parameter, can be simulated by projective measurements that do not require any ancilla. To prove this, we first show that the fine-grained POVM $\mathbf{M}'$ is simulable by nearly projective measurements, as stated in Lemma \ref{['lem:nearlyPROJpost']}. Noisy versions of such measurements are then shown to be easily simulable by projective measurements (Lemma \ref{['lem:simnearlyPROJ']}), which requires an additional result, the dimension-deficient Naimark theorem (Theorem \ref{['th:dimentionDefitient']}). An additional result (Theorem \ref{['th:PostSimulation-randomized']}) shows that one can obtain simulation success probability $\Theta(1/\log (d))$ with the use of random partitions (which can be efficiently generated).

Theorems & Definitions (41)

• Remark 1
• Theorem 1: Naimark extension theorem Peres2002
• Proposition 1: No unbounded advantage of general POVMs over projective measurements in minimal-error state discrimination
• proof
• Proposition 2: Limitations of single-shot classical shadows based on generalized measurements
• Proposition 3: Unitary compression via POVM simulation
• Proposition 4: New local hidden variable model for noisy pure qudit states
• Proposition 5: Improved compatibility region for noisy POVMs
• proof
• Lemma 1: Nearly flat fine-graining of POVMs