Imperfect Quantum Data Still Reveals Complete System States

A thorough investigation into uniquely determining a multiparticle quantum state from its local marginals has been undertaken by Wenjun Yu of the The University of Hong Kong and colleagues. The problem is addressed when experiments only have access to finite samples, introducing uncertainty. Their analysis reveals the conditions under which a quantum state remains uniquely determined despite noisy marginal information. They considered the impact of additive Gaussian noise with variance σ 2 on the marginal probabilities and established a bound on the noise level, below which unique state determination is still guaranteed. A key contribution is the demonstration that determinability is universally strong against such noise, holding true for a broad class of quantum states and system parameters. The study provides a quantitative measure of this robustness, expressed in terms of the noise variance σ 2, and advances understanding of the limits of quantum state reconstruction in realistic experimental scenarios.

Unique determinability of quantum states from marginal probabilities and its connection to

The relationship between local marginals and global quantum states is examined, originating from Schrödinger’s work on entanglement. Researchers examine the relationship between local marginals and global quantum states, originating from Schrödinger’s work on entanglement. The UDA property is crucial for reconstructing global properties from local measurements, including state tomography and entanglement detection.

Initial studies indicated that most few-qubit pure states are UDA, extending to multipartite systems with proportionally scaled marginals. Subsequent research identified state families retaining UDA from smaller marginals, alongside exceptions that do not exhibit this property. A link between the UDA property and quantum many-body physics has been demonstrated, revealing that unique ground states of local Hamiltonians are UDA. However, existing UDA theory assumes perfectly known marginals, a condition rarely met in experiments.

Consequently, a key question arises: is the UDA property robust under perturbations of the marginals. This work proves a universal durability of the UDA property under marginal perturbations, establishing a power-law relationship between deviations in marginals (ε) and global deviations (ε′), expressed as ε′ ∝ εα, where α ∈ (0, 1]. This result allows for a classification of UDA states based on their power-law exponents, with linear scaling (α = 1) representing the most favourable regime. Researchers derive a necessary and sufficient criterion for linear durability, alongside an executable semidefinite-programming certification method.

Applying this theory, stabilizer states are proven to be inherently square-root durable, and a complete durability classification for the Dicke family is provided. A scalable two-local genuine multipartite entanglement witness is also constructed, demonstrating the framework’s viability for practical applications. The team quantified deviations between the local marginals of two quantum states ρ and σ, considering the sum of the trace-norm distances of their marginals: P S’S ∥(σ − ρ)S∥1, where ∥A∥1:= Try √ A†A is the trace norm of a generic operator A. Let H denote the Hilbert space of a composite system with n subsystems labelled {1, , n} =: [n], and D(H) the set of density matrices on H. Given a density matrix ρ ∈ D(H) and a subsystem S ⊆ [n], the marginal is denoted by ρS:= TrS(ρ), where TrS represents the partial trace over S = [n] \ S. The compatibility set CS(ρ) comprises all global states sharing the same local marginals as ρ for the subsystems in S. If local marginals uniquely determine the state, namely if CS(ρ) = {ρ}, then ρ is called UDA with respect to S. The UDA condition can be expressed as D0(ρ) ∩ WS = {0}, where D0(ρ) is the set of state differences {σ − ρ | σ ∈ D(H)}, WS is the kernel of the marginal map MS, consisting of Hermitian operators with zero marginals on all systems in S, and MS(X) := (XS)S∈S for X ∈ V, where V is the space of traceless Hermitian operators.

To quantify deviations, the sum can be expressed as MS(X)S:= ∥XS∥1, for a generic Hermitian operator X ∈ V. Theorem 1 states that for a UDA state ρ ∈ D(H) with respect to a collection of subsystems S, there exist constants C > 0, 0 0 such that every σ ∈ D(H) with ε = ∥MS(σ − ρ)∥S ≤ ε0, satisfies ∥σ − ρ∥1 ≤ C εα. The proof relies on demonstrating that any traceless Hermitian X ∈ V with a small marginal norm ∥MS(X)∥S has a small distance from the subspace WS, and that the distance of any state difference δ ∈ D0(ρ) from WS is bounded by a finite polynomial rate: ∥δ∥1 ≤ K dist(δ, WS)α, where K is a constant and α ∈ (0, 1]. This establishes a universal power law for UDA durability and induces a classification of UDA states based on their power-law exponents. Linear durability, with α = 1, represents the optimal regime. The tangent cone KD0(ρ) is defined as X ∈ V such that X = lim k→∞ δk tk, tk → 0+, where {δk ∈ D0(ρ)}k is a sequence of differences approaching the origin.

Resilience of defined quantum states under measurement informs device development

Uniquely identifiable quantum states retain this property even with imperfect measurements, representing a key step towards practical quantum technologies. This durability is important because real-world experiments inevitably introduce errors, and understanding a system’s error tolerance is vital. While the team successfully classified the durability of well-known quantum states like the Dicke family, determining the criteria for initial unique determinability remains an open question.

This research establishes that uniquely identifiable quantum states maintain their determinability despite imperfections in measurements of their components. The team demonstrated a predictable relationship between errors in local marginals and resulting deviations in the overall global state, with these errors propagating following a power law. This finding classifies quantum states by how durable they are to such errors, with states exhibiting linear scaling proving most stable, and provides a method to certify this durability using semidefinite programming. The work concentrates on classifying the durability of states already confirmed as uniquely determined, rather than establishing criteria for initial unique determinability.

Uniquely identifiable quantum states were shown to remain distinguishable even when local measurements contain errors. This is significant because practical experiments always have some level of imperfection, and understanding how systems tolerate these errors is crucial for development. Researchers classified these states based on how quickly errors in individual components propagate to the overall system, finding that those with linear scaling are the most robust. The team also developed a method using semidefinite programming to verify this robustness for specific states, including those from the Dicke family.