Estimation of multivariate traces of states given partial classical information

TL;DR

The paper addresses estimating high-order Bargmann invariants $\Delta_n(\pmb\varrho)=\text{Tr}[\rho_1\cdots\rho_n]$ with partial classical information by introducing a measurement-enhanced cycle test. By replacing the full unknown-input cycle test with a framework that uses classical descriptions of $m\le\lfloor n/2\rfloor$ states, the authors show how to estimate invariants of order $n=m+n'$ using a controlled cycle $\mathtt{cCYC}_{n'}$ and targeted POVMs, reducing quantum memory and gate requirements. They connect this approach to Chiribella et al.'s measurement-enhanced swap test and derive a general protocol that yields an unbiased estimator for quantities $\square_{n'+m}(\mathbf{j},\pmb\varrho)$, enabling efficient reconstruction of $\Delta_{n'+m}$ when the known states are used as projective measurements. The work also investigates destructive variants, providing a destructive 3-cycle test and an eigenbasis decomposition of $\mathtt{CYC}_n$ to access invariants, while noting practical limitations of destructive strategies. Overall, the framework broadens practical avenues for certifying quantum resources and analyzing high-order relational information with constrained quantum hardware, with implications for extended KD distributions and sequential weak-value protocols.

Abstract

Bargmann invariants of order $n$, defined as multivariate traces of quantum states $\text{Tr}[ρ_1ρ_2 \ldots ρ_n]$, are useful in applications ranging from quantum metrology to certification of nonclassicality. A standard quantum circuit used to estimate Bargmann invariants is the cycle test. In this work, we propose generalizations of the cycle test applicable to a situation where $n$ systems are given and unknown, and classical information on $m$ systems ($m\leq n)$ is available, allowing estimation of invariants of order $n+m$. Our main result is a generalization of results on 4th order invariants appearing in double weak values from Chiribella et al. [Phys. Rev. Research 6, 043043 (2024)]. The use of classical information on some of the states enables circuits on fewer qubits and with fewer gates, decreasing the experimental requirements for their estimation, and enabling multiple applications we briefly review.

Figures (8)

• Figure 1: Main result. Quantum circuits for estimating multivariate traces---expressed as $\Delta_n(\pmb\varrho) = \text{Tr}[\rho_1 \ldots \rho_n] = \text{Tr}[\mathtt{CYC}_n (\rho_1 \otimes \ldots \otimes \rho_n)]$ where $\mathtt{CYC}_n$ is a unitary---such as the cycle test oszmaniec2024measuring typically assume that all input states are either fully unknown (e.g., black-box preparations) or need to be physically prepared on demand. We show that if classical descriptions of $m \leq \lfloor n/2 \rfloor$ of these states are available, one can trade the cost of preparing those $m$ states for the cost of measurement and classical post-processing. This substitution relaxes the circuit requirements for estimating such invariants. The figure shows the case of $n=5$ and $m=2$.
• Figure 2: Circuit implementing a swap test. The inputs are two quantum states and an auxiliary qubit system (also known as a control system) that is put in a coherent state $\vert +\rangle \langle + \vert \otimes \rho_1 \otimes \rho_2$. A controlled swap (also known as a Fredkin gate) between the auxiliary qubit and the two states is performed. an Hadamard gate is applied to the auxiliary qubit that is then measured in the $Z$ basis $\{\vert 0\rangle, \vert 1\rangle\}$. The two-state overlap is recovered via the relation $p(0) = (1+\text{Tr}[\rho_1\rho_2])/2$.
• Figure 3: Circuit implementing a destructive swap test. The input state is $\rho_1 \otimes \rho_2$. One then performs a Bell measurement, which is implemented by a CNOT followed by an Hadamard on the first system, and local $Z$ measurements. Assuming each system is a single qubit, the two-state overlap is recovered via the relation $p(1,1) = {(1 - \text{Tr}[\rho_1\rho_2])}/{2}$. For generic multi-qubit systems see Ref. bandyopadhyay2023efficient.
• Figure 4: Circuit implementing a cycle test. We show an instance of an Hadamard test, where the unitary $\mathtt{CYC}_n$ is the unitary representation of a cyclic permutation $C_n$ of $n$ symbols. We initialize a quantum memory of $n+1$ systems in a product state. The first wire in the circuit represents a single qubit system. The remaining wires represent systems of dimension $d \geq 2$. After the controlled cycle operation a gate $P^s=\text{diag}(1,i^s)$ is applied to the auxiliary qubit, later measured with the computational basis. When $s = 0$, measuring the auxiliary qubit yields the real part of the Bargmann invariant, while when $s = 1$ the imaginary part.
• Figure 5: Quantum circuit considered by Chiribella et al. chiribella2024dimensionindependentweakvalueestimation. The input state is $\vert +\rangle \langle +\vert \otimes \rho_1 \otimes \rho_2$. The first wire represents a single qubit system, while the remaining wires represent generic single qudit systems. We apply a Fredkin gate between all three systems and perform local measurements. The auxiliary qubit is measured with the positive operator-valued measure (POVM) $R$ from Eq. \ref{['eq: povm R']}. The other systems are measured with generic POVMs $A$ and $B$. We dub this a measurement-enhanced swap test since the measurements of $A,\,B$ allow for the estimation of $\text{Tr}[\rho_1A\rho_2B]$, otherwise inaccessible to the standard swap test---hence the 'enhancement' in the order of the accessible multivariate trace.