The Signal
Kun Wang Sets Out Three Protocols to Compare Remote Bosonic States
Kun Wang’s paper compares three ways for separated quantum devices to estimate the overlap of bosonic states using local measurements. Its theoretical sample bounds favor coordinated Haar measurements, while homodyne uses more accessible op
Quantum.world Team · Written with AI · fact-checked ·
Kun Wang’s paper presents a trade-off: the protocol with the mildest sample-complexity scaling requires the hardest optical operations. Coordinated Haar measurements lead the theoretical comparison, but Wang rates their implementation as challenging; independent homodyne measurements use standard optical components, but their stated worst-case sample bound grows more steeply with mode count.
ArXiv lists the paper as submitted on October 29, 2026, and revised on October 30. The revised paper is titled “Cross-Platform Verification of Continuous-Variable Quantum Systems”; its original title was “Distributed Inner Product Estimation of Continuous-Variable Quantum States.” It addresses a basic problem for separated quantum devices: comparing states when the devices cannot send their quantum systems to a shared measurement station. Wang proposes three protocols that rely on local measurements and classical communication, with guarantees for energy-constrained states, including multimode states that are entangled or non-Gaussian.
Three routes to the same inner product
The protocols estimate tr(ρσ), the inner product of two states. It can quantify their overlap without reconstructing either state, and the paper notes that purity estimates can also be used to calculate measures such as max fidelity and squared Hilbert–Schmidt distance. The task is useful when a classical prediction of a device’s output is unavailable or difficult to compute.
“The leading terms of these bounds favor coordinated Haar measurements, whose global implementation is experimentally challenging.”
Kun Wang, paper author
Each method handles the infinite-dimensional bosonic state space by imposing a cutoff, then bounding the error introduced by excluding the remaining tail. That guarantee depends on a known energy constraint: without one, Wang shows that no fixed finite-volume region of phase space can keep the truncation error uniformly small across unrestricted states.
Haar measurements win the bound, not the hardware test
In the Haar protocol, each laboratory applies the same randomly selected measurement to its locally truncated state, then shares classical results. The paper’s sufficient copy bound has a leading mode-dependent scaling of roughly (eE/ε)^(m/2), where m is the number of modes, E the energy bound and ε the target error. Wang cautions that these are upper bounds, not matching lower bounds that establish the optimal resources needed.
The implementation demands more than the favorable scaling suggests. A generic Haar measurement must mix different total-photon-number sectors, which passive optical circuits on the original modes cannot do. Wang discusses encoding the truncated state into a single photon across optical rails, but that construction requires coherent encoding, a large interferometer and photon detection that resolves the rails; those hardware costs are not included in the sample bound.
Homodyne is practical; its bound grows faster
Independent homodyne shadows offer the most accessible measurement route in the comparison. Each laboratory measures quadratures using balanced homodyne detection and independently chosen local-oscillator phases; software applies the Fock cutoff when it builds the state snapshots, so the protocol does not require a physical projection onto a limited photon-number space. The leading mode-dependent term in its sufficient sample bound scales as roughly (eE/ε)^(3m).
Coordinated displaced parity sits between the two in the paper’s feasibility ratings. The method asks both laboratories to use shared random phase-space displacements and measure total photon-number parity, a readout that becomes more demanding as modes and occupied Fock levels increase. Its leading mode-dependent bound scales as roughly (eE/ε)^(2m).
Noise and optimality remain open tests
The three guarantees cover arbitrary states within the stated energy constraint, but they are theoretical worst-case bounds rather than reports of an optical demonstration. They also do not account for implementation costs in the Haar case, and the paper does not extend its guarantees to optical loss or detector noise. Wang identifies both the optimal copy complexity and the effect of shared randomness under fixed local measurement resources as open questions.
For experimental teams, the immediate choice is therefore not simply the protocol with the smallest exponent. Homodyne measurements use standard components; displaced parity adds demanding photon readout; and Haar measurements offer the mildest stated scaling alongside the most difficult optical implementation.