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arXiv:2606.16549

What does measuring one qubit reveal about another? $K$-networks as a directed diagnostic for quantum circuits

Kostas Blekos, Paulo Vitor Itaboraí

quant-phSubmitted Jun 15, 2026• Announced Jun 16, 2026• 0 citations
arXivHTMLPDF

Abstract

Many-qubit circuit states are hard to inspect directly, so they are often summarized by pairwise graph weights. Common pairwise weights report symmetric correlations, while many circuit questions are directed and basis-specific: if qubit $i$ is measured in a given basis, how strongly does the outcome reshape the conditional state of qubit $j$? We define $K_{i\to j}$, a directed, basis-conditioned edge weight for this question. It is large when the two measurement outcomes occur with comparable probability and leave qubit $j$ in clearly different conditional states; it is zero when the source outcome is deterministic or the target states are indistinguishable. The scalar uses standard binary-ensemble distinguishability; the paper's contribution is to turn this conditional comparison into a directed network layer for circuit states. The resulting networks are computable from two-qubit reduced density matrices. They are diagnostic (not entanglement measures): for pure two-qubit states $K$ reduces to the tangle $C^2$ (squared concurrence)~\cite{WoottersConcurrence,CKWTangle}, while separable mixed states can reach $K=1$. Examples on teleportation, Grover, QAOA, and random circuit families show the intended use: $K$-networks map feed-forward, phase, and interaction-graph structure that symmetric or computational-basis summaries can leave weak or absent.

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What does measuring one qubit reveal about another? KKK-networks as a directed diagnostic for quantum circuits
Many-qubit circuit states are hard to inspect directly, so they are often summarized by pairwise graph weights. Common pairwise weights report symmetric correlations, while many circuit questions are directed and basis-specific: if qubit iii is measured in a given basis, how strongly does the outcome reshape the conditional state of qubit jjj? We define Ki→jK_{i\to j}Ki→j​, a directed, basis-conditioned edge weight for this question. It is large when the two measurement outcomes occur with comparable probability and leave qubit jjj in clearly different conditional states; it is zero when the source outcome is deterministic or the target states are indistinguishable. The scalar uses standard binary-ensemble distinguishability; the paper's contribution is to turn this conditional comparison into a directed network layer for circuit states. The resulting networks are computable from two-qubit reduced density matrices. They are diagnostic (not entanglement measures): for pure two-qubit states KKK reduces to the tangle C2C^2C2 (squared concurrence)~\cite{WoottersConcurrence,CKWTangle}, while separable mixed states can reach K=1K=1K=1. Examples on teleportation, Grover, QAOA, and random circuit families show the intended use: KKK-networks map feed-forward, phase, and interaction-graph structure that symmetric or computational-basis summaries can leave weak or absent.
Abstract Understanding how to decompose quantum computations in the language of the shortest possible sequence of quantum gates is of interest to many researchers due to the importance of the experimental implementation of the desired quantum computations. We contribute to this research by providing a quantum circuit to directly measure the three-tangle of three-qubit quantum states. Direct measurement of outcome probabilities in the computational basis quantifies the three-tangle of the three-qubit quantum states.
The WWW subspace is defined as the subspace of the nnn-qubit Hilbert space spanned by states containing at most one excitation. In this work, a separability criterion is established for pure and mixed states supported in this subspace: if all reduced two-qubit subsystems are separable, then the entire nnn-qubit state is necessarily separable. This result motivates the identification of the sum of two-tangles as an entanglement measure for pure states supported in the WWW subspace. Furthermore, it is shown that the commonly used πππ-tangle becomes ineffective for the nnn-qubit WWW state as the system size increases, and vanishes in the large-nnn limit. To address this limitation, the sum of πππ-tangles is introduced, which, like the sum of two-tangles, successfully quantifies the entanglement of the nnn-qubit WWW state in the large-nnn limit. In addition, a new condition for entanglement measures is introduced, which facilitates the formulation of a well-behaved and physically meaningful entanglement measure.
Abstract A central tenet of quantum theory is that ensembles sharing the same density matrix are indistinguishable under measurements on individual systems. We show that this indistinguishability can break down when additional structure is present across multiple copies. We investigate ensembles of qubits prepared in either the computational (ZZZ) or Hadamard (XXX) basis under a same-basis constraint. All qubits within a batch are prepared in the same basis, but with randomly distributed outcomes such that the reduced state of each individual qubit is maximally mixed. We introduce an experimentally realizable protocol that exploits multi-copy statistical structure to distinguish these ensembles. The graph-state protocol identifies the two basis through stabilizer measurements and higher-order statistical features. Our protocol has been implemented on the \texttt{ibm\_fez} superconducting quantum processor and demonstrates robust performance in the presence of realistic noise, achieving near-unity classification accuracy as the number of samples increases. While single-copy measurements are uninformative, aggregating outcomes across multiple instances produces a clear and scalable separation between basis families. Our result identifies a regime in which quantum information is not accessible at the level of individual systems but emerges through structured multi-copy measurements. This highlights the role of higher-order statistical correlations in quantum information processing and provides a practical route for exploiting such structure on near-term quantum devices.
We now have all the basic tools we need to describe systems of one and two qubits. In this final chapter of Part I we look at some of the consequences of the peculiar properties of qubits when they are used to encode information. Many of these results can be traced to the properties of quantum measurement. Measuring a single qubit A key result in quantum information theory is that it is impossible to accurately characterize a single qubit. In other words, there is no experiment, or sequence of experiments, which allows us to find out the state of a single quantum bit. The reason for this problem is twofold. Firstly, we have to make some sort of decision about the basis we will use for the measurement. For example, when measuring a single qubit the most popular choice is to make a measurement in the computational basis. This is equivalent to asking the qubit whether it is in state |0⧽ or state |1⧽. If the qubit is indeed in one of the basis states the measurement process is simple, and we will get the obvious answer of 0 if it is in |0⧽ and 1 if it is in |1⧽. If, however, the qubit is in a superposition, such as α|0⧽+β|1⧽, then the situation is more difficult. Characterizing the state now means determining the values of the two complex numbers α and β, but the measurement can only return the answer 0 or 1, and for a superposition state one of these two answers will be returned at random, with probabilities |α| 2 and |β| 2 , respectively.
A complete chapter on quantum entanglement measures and detection of entanglement is given because there are a wide variety of entanglement measures for pure and mixed states which are used very often. The interesting feature of these routines is the fact that, these operations can be done between any random set of qubits not necessarily contiguous qubits. Key recipes to perform operations like partial tracing and partial transpose are disscussed in detail. After that, it was decided to devote several routines to calculate the entanglement measures like concurrence and its variants, von Neumann entropy, block entropy, Renyi entropy, Q measure, negativity and log negativity, entanglement spectrum and residual entanglement (three tangle). Last but not least, certain entanglement detection procedures like Positive Partial Transpose (the Peres-Horodecki criteria) and the reduction criteria are also explained via examples.
The public 53-qubit, 20-cycle Google Sycamore random-circuit-sampling workload is used to test whether physical quantum width can be reduced when the frozen preparation and complete computational-basis sampling distribution, rather than the source wire diagram, define the external contract. The analysis is performed at the Quantum Transfer Level (QTL). Exact normalization maps the 3263-operation source to 430 dressed two-coordinate parents, while legal retiming gives an exact 53 → 52 live-qubit baseline. The active interaction graph has a 24-vertex cover, while its first interaction layer contains a matching of 24 disjoint edges; weak matching–cover dualitytherefore fixes 24 as the minimum interaction-separating frontier width. Compose-first temporal transfers on that frontier exhibit geometric suppression of their transverse defect with repetition depth, reaching the 10^−16 scale at the five repetitions of the frozen schedule. The final lowering changes parent class rather than forcing the transverse transfers to be unitary. One frontier output is emitted before the first instrument, freeing its physical carrier; the remaining 23 conditional-state carriers retain the frontier record and the released carrier is reused as a Stinespring meter. The peak width therefore remains 24 rather than increasing to 25. Recursive deterministic instruments emit the complementary outputs and leave the exact conditional state of the final 23 bits, proving equality of the complete 53-bit sampling distribution and hence DTV = 0 in exact arithmetic. Generic conditional-state preparation and instrument synthesis may remain exponentially expensive, so the result is a physical-width result rather than an efficiency claim.