Batch Entanglement Detection in Parameterized Qubit States using Classical Bandit Algorithms
Authors: K. Bharati
Vikesh Siddhu
Krishna Jagannathan
Venue: Accepted by TMLR ()

Abstract: Entanglement is a key property of quantum states that acts as a resource for a wide range of tasks in quantum computing. Entanglement detection is a key conceptual and practical challenge. Without adaptive or joint measurements,  entanglement detection is constrained by no-go theorems~\citep{tomography2016no-go}, necessitating full state tomography. Batch entanglement detection refers to the problem of identifying all entangled states from amongst a set of $K$ unknown states, which finds applications in quantum information processing. We devise a method for performing batch entanglement detection by measuring a single-parameter family of entanglement witnesses, as proposed by \citet{mintomography}, followed by a thresholding bandit algorithm on the measurement data. The proposed method can perform batch entanglement detection conclusively when the unknown states are drawn from a practically well-motivated class of two-qubit states $\mathcal{F}$, which includes Depolarised Bell states, Bell diagonal states, etc. Our key novelty lies in drawing a connection between batch entanglement detection and a Thresholding Bandit problem in classical Multi-Armed Bandits (MAB). The connection to the MAB problem also enables us to derive theoretical guarantees on the measurement/sample complexity of the proposed technique. We demonstrate the performance of the proposed method through numerical simulations and an experimental implementation. More broadly, this paper highlights the potential for employing classical machine learning techniques for quantum entanglement detection.

Published in Transactions on Machine Learning Research (01/2026)
Batch Entanglement Detection in Parameterized Qubit States using Classical Bandit Algorithms
K. Bharati
ee20d7000 sitm.oc.in
Department of Electrical Engineering, IIT Madros Chennai, India
Vikesh Siddhu
usiddhon.@prtonmmaaii com
IBM Quantum, IBM Reseorch India
Krishna Jagannathan
krishnaj@ee. iitt iitm, oc.
Department Electrical Engineering, III Madros Chenmai, India
Reviewed on OpenReview: https: // openreview. net/ forum? id= 0o27eBBBV
Abstract
Entanglement is key property of quantum states that acts resource for wide range of tasks in quantum computing. Entanglement detection is key conceptual and practical challenge. Without adaptive or joint measurements, entanglement detection is constrained by no-go theorems (La el all, 2016), necessitating full state tomography. Batch entangle- ment detection refers to the problem of identifying all entangled states from amongst a set of K unknown states, which finds applications in quantum information processing. We devise a method for performing batch entanglement detection by measuring a single-parameter family of entanglement witnesses, proposed by Zhu all (20m), followed by threshold- ing bandit algorithm the measurement data. The proposed method can perform batch entanglement detection conclusively when the unknown states are drawn from a practically well-molivatee class two-qubit states F, which includes Depolarised Bell states, Bell diag- onal states, etc. Our key novelty lies in drawing a connection between batch entanglement detection and a Thresholding Bandit problem in classical Multi-Armen Bandits (MAB). The connection to the MAB problem also enables us to derive theoretical guarantees on the mea- aurement/sampl complexity of the proposed technique. We demonstrate the performance of the proposed method through numerical simulations and experimental implementation. More broadly, this paper highlights the potential for employing classical machine learning techniques for quantum entanglement detection.
1 Introduction
Quantum information theory has redefined quantum entanglement from a descriptive property of quantum states a fundamental non-classical resource. the basis for applications such as quantum communica- tion, teleportation, and information processing (Bennett et all [ииК; [Bnheman all 2001; Horodecki all, 200g), entanglement detection and verification central problems in quantum information science. Tradi- tionally, this involves performing quantum measurements that yield probabilistic data, enabling techniques like full-state tomography (FST) for state reconstruction. However, this faces two challenges: theorctically, even after FST, determining entanglement remains computationally intractable, and this limitation even more pronounced in scenarios involving many qubits; practically, real-world noise and imperfections limit the accuracy of state reconstruction. Modern multi-qubit compute systems may generate a bunch of en- tangled states across different sets of qubits through quantum gate operations; however, gate noise (c.g.. phase flip errors, depolarization) compromise their entanglement, requiring precise verification ensure

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their reliability before use in applications such quantum computation and communication (Hong et all, 2010). In such contexts, FST can be employed for entanglement detection. However, it comes with a high computational burden, which may be unnecessary. We propose an alternative approach for simultaneously verifying or detecting entanglement among a given set of quantum states, dubbed batch entanglement de- lection. Instead of relying on FST, learning algorithms utilize statistical patterns to simultaneously analyze measurement data from a batch of quantum states and provide high probability guarantees on what they learn, i.e, whether or not the states are entangled.
Conventional techniques for learning quantum states include extensive research on FST (see Kueng et al. (2017);; Wang et all (2019);; O'Donnell & Wright (2 Banaszck cf all
(20T3); Flammia of all (20T2) and
references therein and also see Gula el all (2121); ; Torlai el (201)) Quek el al. (201S) Koutny el al. (2)12)
Schmale et all (2022); França et al. (2022] for machine learning based approaches). Measurements required
for FST scale exponentially with the number of qubits. While entangled measurements enable near-optimal
copy complexity for FST (O'Donnell & Wright, 2015; Haah ct all, 2017), practical implementations rely
single-copy measurements, utilizing reconstruction methods such as linear inversion, maximum likelihood
estimation, and maximum a posteriori estimation eLl, 2UT1; Siddmm MIB These reconstructed states can be tested for entanglement using well-known criteria (some are outlined in Sec
2. Alternatively,
entanglement can be detected by measuring entanglement witnesses (Horodecki et all, 1996a;
Terhal, 2000;
Lewenstein et all, 2001a Chrnciski & Sarbicki, MIEE),, observables that detect some entangled states. No single witness can detect all entangled states, but in the worst case, combining information obtained from measuring different witnesses aids in state reconstruction via FST. This is explored in Zmm et all (20M)) where measurement operators from family of six witnesses are used for bipartite qubit systems. The proposed approach for entanglement detection involves mcasuring witness and formulating separability crilerion based on the frequencies of measurement outcomes. A negative value of the criterion indicates entanglement; otherwise, the process is repeated with another witness. If the state remains undetected by all witnesses, a tomographic reconstruction is performed (see Section 2 for further details).
Recently, the authors of Lumbreras el al (2022) proposed using multi-armed bandit (MAB) frameworks for learning quantum states. The MAB algorithm repeatedly chooses from several options ('arms"), with the goal of finding the arm with the best outcome (the "best arm'). The algorithm balances between exploiting the known best options and exploring others to ensure no better option is missed (more details on MAB and policies can be found in Sec. 2 In Lumbreras el_al. all (2022), the inherent linearity in the quantum mechanical description of states is capitalized and a well-known classical learning algorithm that prescribes a sequential order of choosing measurements is employed. The MAB algorithms that are used provide guarantees on the quality of the estimate of the unknown quantui state. This MAB model Lumbreras ot all (2022) does not directly apply to batch entanglement detection. Instead, focuses on learning one entire quantum state, which may be unnecessary for entanglement detection.
Our first contribution builds on the based separability criterion in Zt (2010) and uses suitable MAB policies for learning the same. We establish a formal connection between batch entanglement detection and the thresholding bandit problem (TBP) (Kano el. all, 2015), enabling accurate and quick identification of m entangled states from a batch of K candidate states through adaptive measurement allocation. This formulation, which we refer to as the (m, K) -quuntuu MAB framework, differs structurally from the setting metrics without
in Lumbreras all (2122) (sce Remark M) and focuses on learning entanglement-speedie requiring full state reconstruction. Our second contribution uses classical MAB policies for adaptive mea- surement allocation and provides explicit measurement/ copp complexity guarantees for batch entanglement detection-guarrattes absent in FST and repeated witness testing Zhu all (2mm) Using statistically guided confidence bounds, these policies are sample-effecien since they prioritise measurement effort only on un- certain states. Finally, we demonstrate the complete MAB-based pipeline for batch entanglement detection across multiple IBM Quantum backends, validate the framework under realistic noise and benchmark our results with non-adaptive tomographic approaches.
The rest of this paper is organized as follows: In Sec. A we provide a brief recap of some preliminary concepts in entanglement theory and multi-armed bandits. Readers interested in our connection between the two can move directly to Sec. B. where we describe the (m, K) Ku)-qunnum Multi-Armed Bandit framework for entanglement detection, We define a class of parameterized two-qubit states F and identify measurement
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operators that conclusively detect entanglement in F, detailed in Sections I. 2 and A.R. In Section El we demonstrate two TBP policies for entanglement, detection. Section R analyzes the MAB policy performance on IBMQ backends and on an ibm--risbane device for a family of states in F and details the quantum circuits used for simulation. Section @ highlights measurement scheme limitations for entanglement detection in arbitrary states through numeric examples. In Section B, we contextualize the numerical performance gains and discuss the practical advantages of the proposed MAB approach comparison to existing state-of- the-art methods for entanglement detection like FST and fixed-witness testing (Zhn el all 201).. Finally, Section g conchudes the paper. Details on the non-adaptive tomography baseline and proofs of the results are presented in the paper and in Appendix []]
2 Preliminaries
Let H be a finite-dimensionall Hilbert space with dimension A pure quantum state is represented by a unit norm vector (6b) E H. Let L(H) be the space of linear operators on H. the Frobenius inner product for any A, B E L(H), (A, == TT)AtB) where f represents conjugate transpose. A Hermitian operator satisfies H H1. A density operator p L(H) is Hermitian, positive semi-definite, p 2 O, and has unit trace, Tr(p) 1; it can represent both pure and mixed states. A positive operator value measure (POVM) is a collection of positive operators {E, 2 0} that sum to the identity, E, E, = I. A POVM represents a mcasurement where E; corresponds to mcasurement outcome i, but sometimes we compress this and just say E, is a measurement outcome.
Let He and Hs be finite-dimensional Hilbert spaces with dimensions da and de, respectively, and Has Ha & Hs, where & represents tensor product, be bipartite Hilbert space with dimension d dg A densily operator Pab E L(Hor) is called separable if it can be wrilten convex combination of product states, that is,
Pub =
Σ Pi la xb) (0 xól.
(1)
where ps 2 such that Σ.p = and 10h xt) 1o)'s 6 1x) is a product of two pure states. We denote the convex set of all separable states by Sab. Conversely, Peb is entangled if it can not be written in the form equation M We discuss some preliminaries separability criteria for entanglement detection in Section 2. and background on stochastic multi-armed problems in Section 2N2.
2.1
Separability Criteria for Entanglement Detection
2.1.1 Standard Analytical Separability Tests
Using full state tomography (FST), one reconstruct any bipartite qubit state Pab and verify its entangle- ment through standard separability criteria. (Horodecki all 200g). Notably, the Peres-Horodeeks criterion establishes that Pab is separable if
(also called the PPT criterion) (Horodecki et all, 19966; Peres 1
and only if the eigenvalues of its partial transpose Pab T are non-negative. Here, To is the partial transpose with respect to Hs. This condition remains necessary and sufficient for (2 x 3) systems but fails in higher dimensions due to the existence of bound-entangled PPT states. Other separability criteria include the range criterion (Horadecki, 19977) the matrix realignment criterion (Rudolph) 2000). the covariance matrix (CM)
criterion (Gühne all 2007) and additional methods discussed in Gurvits (201) Doherty et al. (2004).
2.1.2 Entanglement Witness-baseel Separability Criterion
Fatanglement can be detected by measuring entanglement witnesses, which can be defined as follows:
Definition 1 (Entanglement Witness) An entanglement witness W L(Has) is Hermition operator salisfying,
(Pent: W) Tr(p W) 0, for some entangled Pent
(2)
(e, W) Tr(oW) 0, Vp Sab-
(3)
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Geometrically, witness W defines a hyperplane in the state space, delineating the set of detectable entangled states Dw = (p s.t. Tr(pW) < O) from all separable states. For two arbitrary witnesses W1 and W2, W2 is said to be finer than W1 if Dw, S Dw,, A witness is said to be optimal when no strictly finer one exists, implying that it lies tangent to the boundary of the convex sct Seb (Bengtsson & Zyczkowski, 2008). Further insights into this topology are detailed Lewenstein el all (2000), Lemma 1).
We brielly review the witness-based separability criterion from Zm st (201). The authors propose a parameer witness family,
witness family:
Pu(a) cos² al (1b) (b
(4)
where ( = cos a (00) + sin a [11) such that a E [0, π //4]. We denote & to be the set of projectors onto the cigenstates
p(a) = (1v) (+)) To 1 + cos 2a
100) (00) + 1 cos 2a
111) (11++ sin 2a
(1V+) -
1 (V)
2
The set E = {100) (00), [11) (11), + (1* v ") ) ( } forms a POVM and is referred to as Witness Basis Mensurement (WBM). For the remainder of the paper, we assume that the exact projective forms of the WBM are fixed and known.
The witness expectation value serves as detection statistic, that is, Tr(pW) < 0 certifies entanglement, while non-negativity renders the test inconchusive. If the test is incon- clusive for the base witness equation E, that is, Tr(pi(()))) 2 O, then subsequent witnesses are generated via local unitary transformations U1 and U2 as,
Quadratic WBM criterion:
Pa) - (U, @ U2) pm))((U & U2).
(5)
with (U1, U2) e (1, C), (Ct, (Ct, XC), (C, Ct), (C,) Here, the operator C cyclically per- mutes the Pauli operators X, Y and Z, satisfying that CX-YC, CY=ZC, CZ=XC. Instead of performing negativity test, the authors in Zhn et all (2ImII adopt more stringent criterion:
min Tr (psep (cos² at (oo(a))) 0, Apsop E Sab-
(6)
The criterion is violated by the set of entangled states that can be detected by this witness family. The above optimisation leads to the following quadratic WBM critcrion,
S = =
- - f4)² 2 0, Apso E Sab-
where ft == Trpp,p} are probabilities obtained from WBM E. The value of S in equation a depends on the underlying WBM. Thus, for a WBM & and state p. we denote equation M as Sz(p). We note that mcasuring the witness basis provides estimates for a distinct set of observables. For instance, the base witness in equation El yields estimates for three observables: 21 12, ZZ, and XX+YY The six-witness ensemble in total provides 15 independent expectation values, which provide sufficient information about the two-qubit state. Thus, if the state is undetected by the family of six witnesses, it can be reconstructed using these expectation values.
2.2 Sixed-Confidence Multi-Armed Bandit Policies
In this section, we briefly review some fixed-confidence policies for Best Arm Identification (BAI) and Good Arm Identification (GAT) in the stochastic Multi-Armed Bandit (MAB) setting, a canonical framework for sequential makisin problems under uncertainty. A bandit instance μ (problem instance) comprises K arms, cach described by reward distribution 24 supported on R with unknown mean p4. In cach round t, the learner chooses an arm X,, receives an independent reward Ze vx,, and chooses the subsequent action based on specified policy. We consider the class of d-correct policies, i.e., given fixed error 8 and
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problem instance μ, such policies terminate in finite time and return the correct solution with probability at least 1 - 6. Mathematically, a 6-correct policy π = (T, () satisfies:
(T <0) P Ptrue) 2 6,
(8)
where I is the stopping time, Φ is the learner's final recommendation, and Øtrue the actual correct answer for the specifie problem (BAT GAT). Below, we briefly review well-known MAB policies that operate under fized-confidence guarantees, aiming to make statistically reliable recommendations while minimising the number of samples.
2.2.1 Fixed-Confidence Best Arm Identification
In the BAI problem, the learner's objective is to identify the (best) arm i* = arg maX #i with the largest expected reward. Without loss of gencrality, arms are enumcrated based on their expected reward #1 M2 2 ... 2 MK and the sub-optimality gap of arm is given by A, = μ1 - p4. The performance of BAI policies is primarily characterised by the expected stopping time E,(r], which represents the expected number of samples required to recommend a best arm with confidence - 8. The sample complexity improves progressively across algorithms: O(A -2 logg -)) for Successive Elimination (Even- et all, 2012) and O(A log A ) for LUCB (Kalyanakrishnan et al. 21T2). Building upon this, Exponential Elimination (Karnin ct all 20е) and li (Jamicson of all 2014) utilize the law of the itcrated logarithm (LIL) bounds to reach the near-optimal complexity of (( -2 log log 2),, bridging the gap to the theoretical
lower bounds [arrell 19her
Mannor
Tsitsikis
2108
2.2.2
Good Arm Identification
The GAI problem generalises BAI by introducing threshold Ç and defining the set G {i E [K]: u4 2 } of "good" arms with unknown cardinality [9] = m, leading to the (m, K)-GAI formulation. Without loss of gencrality, assumc μ1 > H2 2 ... 2 pmm 2 2 jm++1... 2 Px and the learner is unawarc of this indexing. Fixed-confidence GAI policies adapt the notion of (X, ccoov-ccesss (Kano el all, 2MN)). GAI policy is said to be (A. cororeet if, with probability at least 1 5, it correctly identifies at least À true good arms and does not misclassify any bad arm. Here, À specifies the number of correctly identified good arms. Unlike BAI, the sub-optimality gaps denoted by A, = 1M4 Çl and A.jj 14 - Ml and the sample complexity is expressed in terms of A min(min,ggk] A,, minje[K11 12)+
The goal, as in BAI, is to minimise the expected stopping time Em However, a key difficulty in GAI is the exploration-explottatiio dilemma of confidence, where the learner explores arms other than the empirical best arm to identify potentially 'good' arms with fewer pulls, while simultaneously ex- ploiting the empirical best arm increase confidence its classification as a good arm. The Hybrid Dilemma-of-CCofideeee (HDoC) algorithu (Kano el. all, 2 combines UCB-based exploration (Auer et al, 2m12) with LUCB-basl based elimination (Kalyankkrishaan et al, 2012), achieving sample complexity J + K log K + K log 1)). The L.L-based refinement Tsai et all (2023) lil lil'HDoC, employs tighter confidence widths to achieve O (A (2-2 (Klog J + K log K + K log log 1)) samples, the best-known order for fxeed-confidecce GAT policies. The specific connections between BAT/GAT and entanglement de-
K log 1
tection are claborated in Section B and F.
3 The Quantum MAB Framework For Entanglement Detection
In this section, we introduce the quantum Multi-Armed Bandit (MAB) framework for batch entanglement detection. We formalise the structural similarity between the stochastic MAB model and its quantum analogue, where the learner interacts with a batch of quantum states by performing structured measurements.
3.1 Stochastic- Bocchattic-Quantm MAB Mapping
In the stochastic MAB setting, pulling an arm corresponds to sampling from a probability distribution pt(-) with known support and unknown mean p4- Each pull yields reward j with probability (w.p.) pt(s) and
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rewards across arm pulls are independent and identically distributed (i..d.).. Analogously, in the quantum setting, each arm represents unknown quantum state p. When p is measured, the outcome distribution is determined by the fixed WBM. Specifically, if a WBM & is chosen, measuring & on will result in outcome e (1,2,3, 4] with probability Tr(pE,). Once the mcasurement is fixed, repeated mcasurements of yield i.idd outcomes. The key distinction lies the source of the rewards: in the stochastic MAB model, rewards are sampled from classical distributions, whereas in the quantum MAB model, the rewards depend on i.i.d. outcomes obtained by measuring & on p.
3.2 Problem Setting and Objective
Given a batch of K unknown quantum states {pI....,Pk] of which an unknown subset m < K states are entangled, the learner's objective is to correctly identify all entangled states while minimizing the total number of measurements performed. Given fixed WBM E, the goal is to estimate the quadratic WBM criterion Se (p:) which indicates that P: is entangled if (e(p) < O and is inconclusive otherwisc. The learner applies the MAB routine to this (m, K) instance of quantum states under the chosen WBM & with the objective of accurately identifying Aent = (i E [K] such that Se(p:) < 0), using the fewest possible number of measurements. Since a single WBM may not detect all m entangled states, and the value of m itself is unknown, the MAB routine must be repeated for the six WBMs. Importantly, the measurement data collected under WBM is not used to decide the next WBM; each WBM configuration should be treated independent instance. We summarise the stochestic-quantum MAB correspondence concisely in Table m.
Table Stochastic-Quantun MAB
Attributes
Stochastic MAB
Quantum MAB
Arms Measurement
Probability distributions (Pt.P2.- Pt.P2....PK' .P))
Density operators {pi, {PI.Pa.... P2.-..,PKk PK]
WBM
Meosurement Data
w.p. p.(j), ViE [K]
j j.p. Th Pa). Vj [4) Vi [K
Parameters to estimate
(1), M2. (K)
S: (SE(p)). Se (p2), SE (p(KK) ]]
Objective
Identify go [K] such that 1y S C) Identify Aunt - { E [K] such that Se (p:): 0}
We now formalise this correspondence by defining the (m, K )Jqquantum MAB setting.
The (), -quantum Multi-Armen Bandit (MAB) setting for entanglement detection is fully characterized by the buple (A,E). Here, A denoles a sinile action set with |A] = K, consisting of (K -m) two-qubit separable states and m two-qubit entangled states. The term E corresponds to a suitable Witness Basis Measurement (WBM).
Definition 2
The objective of the (m, K)-quantum MAB problem for entanglement detection aligns with the classical (m, K)-Bad Arm identification where the goal is to find the set of "bad" arms g = {i E [K] such that μ4 S } whose mean rewards fall below a threshold Ç. Analogously, the (m, Juuquutuum MAB problem seeks to identify the set of entangled states Aeut whose quadratic WBM scores Se violate the separability threshold. In essence, the (m, K) -ad Arm identification setting and the (m, K))quantum MAB problem for entanglement detection share a unified statistical structure, differing only in the interpretation of the reward model. To the best of our knowledge, this work is the first to establish a direct connection between stochastic MAB and quantum entanglement detection. This correspondence enables existing MAB algorithms to be directly applied in quantum settings, where the reward is encoded in the outcomes of WBM.
Remark 1
The d- d-dimensional discrete malti-armed quantum bandit model (Lambrrerss et all ZIZE) is dif- ferent from our formulation. The authors consider arms to be a finite set of observables and the environ- mend, an wnknown quantum slate p. The objective is to learn the unknown quantum slate through an exploration-xxploitatinn tradeoff. Given sequential oracle access to copies of p. each round involves selecting an observable marimize its expectation value (reward). The information from previous rounds (history) aids in refining the action choice, thereby mindmizing the regret, which is the difference between the obtained and maximal rewards. The authors also exploit the inherent linear structure in measurement. outcomes and
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map it to the linear bandit setting. Specifically, let {or be set of orthogonal Hermition matrices. The unknown environment = Σ Tr(po;)o; = Ei=1 050; and arm O2 E Tr(O,o;)o; = E Ar,or Then, Tr(pO,) = At where f = (01, ...0....00................. 002) and A, = (Ac,1, Ar,,,... Ar, ). In round L, pulling arm On provides a reward X, = 0 A, + Tt, where Tt I-subgaussian.
4 Parameterized Qubit States F
To demonstrate the applicability of stochastic MAB policies for entanglement detection, we define class F of parameterized two-qubit states defined as a union of three sub-families, F U2 U F3. These correspond to Depolarized Bell states (Fi), Bell Diagonal states (F2); and amplitudeddamped Bell Diagonal states (Fs) whose explicit parametrisations are detailed in Sec I.
and Ews respectively. We identify suitable WBMs from the witness family in equation EI that can detect the same. We denote the first two witnesses in the witness family by & (base witness) and & (adapled using (U1, U2) = (I, respectively. Here, & = (100) [11), (111, [ (V+)), ( (V-I) and & = {101) (01), (10), (6+) ( 1,,,,,100 / ( (+-1).
4.1 Twoiqubit Depolarized Bell States
For w E R, 1/3 s s 1, two-qubit Depolarized Bell state p(w) is given by,
I
(9) Here, ()) represents any of the four Bell states |/+) (101) + (10)) /22 | (100) + (1)) /v2. When Y = " equation is called Werner state, and when r = 1 equation g. is called an Isotropic state, The Horodeckk criterion guarantees that p(w) is separable when -1/3 s w S 1/3 and is entangled when 1/3 < w < 1. Table & outlines the specific choices of WBM for the combination of the maximally mixed state with each of the four Bell states. When measured with these corresponding WBMs, the entangled depolarized Bell states are conclusively detected, determined by the value of S (w - 1)2 w² which is strictly positive for -1 < w S 1/3 and negative for w > 1/3.
p(w) (T) (T) + (1 w)
Table 2: WBM for Depolarized Bell States
Depolarized State
Pauli Basis
WBM
(++ (1 ww//4
1+a(XX - YY +Z2) /4
E
w
V (Vr+ (1 -/)
T+a(XX + -ZZ)
V (V
(1-w)//4
[+a(-XX- - -Y -ZZ) /4
El
④
( + -w)/4
1+a(-XX+YX +YY =Z)]
E2
4.2 Two-qubit Bell diagonal States
Bell diagonal states are a probabilistic mixture of the four Bell states. These states more general than the ones in equation g. Given parameters P1, P2, Ps and pa such that 2 0. E. = 1, the Bell diagonal state is defined,
Peus = P1 p (
P2 ( +
Ps V
(v | P1 4 <-> (
(10)
The eigenvalues of To are calculated to be 1/2 PI, 1/2 p2, 1/2 pa and 1/2- p4. Consequently, Bell diagonal state is entangled if any one of these probabilities exceeds 1/2, while the sum of the other three probabilities less than 1/2. Conversely, a Bell diagonal state is separable if all probabilities are less than or equal to 1/2. Expressing equation III in the Pauli basis yields,
Pais 4
[I axX + bYY + cZZ),
where a P1 + P2 P3 p4; b PP p2 P3 + P4 and Pl P2 ps P4-

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Table 3: WBM for Bell Diagonal States
Probabilistic mixture
a
b
c
WBM
pi > 0.5, p2 13 + p4 < 0.5
+
+
Eq
P2 > 0.5, P1 P P1 < 0.5
+
+
E1
Ps > 0.5, P1 +2 P4 < 0.5
El
P4 > 0.5, Pı + Ps < 0.5
E2
When Paus is entangled, the index for which P: > 1/2 determines the sign of a. and c, see Table B Tt is notable that the signs of a, b and e follow similar pattern to the Pauli basis expansion of various Depolarized Bell states listed in Table 2. We observe that, for suitable combinations of b, and с E {+1,- -1}, the Bell diagonal state reduces to one of the Depolarized Bell states, and states can be detected using the same WBMs, as in Table Specifically, the value of under the two WBMs in Table R equal to (1 (1-P: Pr -pa)² --pp- -p4)² and (1 (1-20 P2 - ps)² 4(p2 py)², respectively. Depending on the probabilistic mixture, one of the two WBMs will conclusively result. in S < O.
4.3 Two-qubit Amplitude Damping on Depolarized Bell States
A qubit amplitude damping channel is source of noise in superconducting circuit-based quantum computing and thus serves a realistic channel model for simulating lossy processes in these systems. Mathematically, it can be obtained from an isometry J,
J: H. + H4, & Hoi JJJ=
(11)
where Ha denotes the Hilbert space for the channel's input, and Hy and He represent the Hilbert spaces for the direct and complementary channel outputs, respectively. An isomelry of the form,
Ji 10), = 10), (1),,
J1 [1), = /1 - r1)) 11, (1), + 10)00)00)
(12)
where S r S 1 defines a pair of channels, B(A) (JAJt) and C(A) Tr,(JA)i). Here, is an amplitude damping channel with damping probability r for the state [1) to decay output state 10) The isometry J1 Ko @ 10) + K₃ & [1} where Kg and K2 (Kraus) damping operators such that Ko [0, 0, 0] and Ki [1, 0, -.. For single qubit represented by state p. the amplitude-damped output is given by,
B(p) = KopK! + Kipk!
(13)
We can extend equation IK for two-qubit states with damping probabilities and q for the first and second qubit, respectively. Assuming that = q, we consider Depolarized Bell states equation g with amplitude damping.
For any domping probubility > 0, Depolarized Bell state with amplitude damping con not be expressed as Bell diagonal state equation M.
Proposition 3
This fact can be readily demonstrated through straightforward calculation. Consider the Isotropic state p(w) w
w | (0+))((+++1+++0)++11))((1111 3 which can be represented by the Bell diagonal state formed with probability distribution (Pi.P2.P3) P3. P3.Pa. = ((3w + 1)/4, (1 w)/4, (1--o (1-w)/4, (1 - ww/4). In a Bell diagonal state, the diagonal elements corresponding to (00) (00) and 111} [11] are identical. In the case of an amplitude-damped Isotropic state, we obscrve that,
r
P2 = P3
(w wr 1).
4
However, obtaining closed-fore expressions for Pt and P4 when > is cumbersome. Specifically, the values on the diagonal corresponding (00) (00) and [11) (11) is given by w(r² + 1)/2 (w 10--1p1 1-)4(r + 1)2/4 and w(r - 1)2 /2 (wr - 1)(r 1)2 /4, respectively. These expressions are equal only when 0.
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Proposition 4 For every w E 13 1], there ezists i [0, 1] such that an amplitude-damped Depolarised Bell stale becomes separable.
The PPT criterion asserts that a two-qubit state entangled and only its partial transpose contains at least one negative eigenvalue. For Bell stales that are both amplitude damped and depolarized, we evaluate the eigenvalues and observe that one of them can exhibit either positive negative values contingent upon the range of Detailed findings presented in Table EI and depicted graphically in Fig. In and Fig. IRR Furthermore, the WBM for amplitude damped and Depolarized Bell states aligns with that of depolarized Bell states, as outlined in Table 2.
1
Positive eigenvalues
Positive eigenvalues
0.8
Negative eigenvalues
0.8
Negative eigenvalues
0.6
0.6
0.4
0.4
0.2
0.2
0.4
0.6
0.8
0.4
0.6
0.8
w
(a) Range of r for eigenvalue corresponding to (+
(b) Range of for eigenvalue corresponding to |M**)
Figure phase diagram representing the region of damping and depolarizing paramcters, r and w, respectively, where the dampeddepolarized Bell state has negative positive partial transpose.
Table The four eigenvalues amplitude-damped Depolarized Bell states
State with ( #
State with //)=
Sign of eigenvalue
(a++1((1-22) 1
(1rr((+++wwur))
Always positive
(w+ (
(1-r) --+
Always positive
-(r-1) (r+1))
P+1-w - +2 (1=(1-))2+r
Always positive
4
y - 1) ur 1- 3w]
+1 -n)² 22
Positive and Negative
5 Stochastic MAB policies for Entanglement Detection
We apply stochastic MAB algorithms for entanglement detection in the parameterized states F from Section a. The terminology aligns with classical counterparts, as outlined in Table M Consider set of K unknown states, denoted by A = {PI: P2...., ) E F. To perform measurements on the arms, the learner must know the underlying WBM. Thus, we assume knowledge of the specific forms of the arms in A. For instance, A could represent the set of isotropic states detectable under the WBM &2, where each state of the form A = w. (++ ( (1 w;) with ws being unknown for all i [K]. With this assumption, we describe the MAB routine as follows: In each round t N, the learner selects state Pa, E A, performs a measurement E and obtains i.id.. outcomes, and then updates the values Sg to identify the entangled arm(s) or continues. In the subsequent sections, we discuss two MAB policies: Successive Elimination, which is applicable when there is guarantee of one entangled arm among K arms, and the lil Hil'Doo policy, designed for scenarios where there are entangled arms among K, with being unknown. For proving 8-correctness of policies, we
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concentration inequalities applicable to o-subgaussian" random variablesspeciffellly, the law of iterated logarithm (Jamieson el all, 2014, Lemma 3) for finite sum of o-subgaussian random variables:
Lemma 5 Let X1, X2,...X, X be i.idd. sub-gaussian random variables with scale parameter a. For any S (0, de E (o, los(1te) ), one has with probability at least 1 - c- for all t 2 1.
Σ X, s U(L,6),
(14)
where Utt (1+ + vE)
20 (1+e) log lox(()+110) is the confidence width and c₂ 2te (1ox(145)
1tc
5.1 Successive Elimination Algorithm
Consider the set states A= {pi, P2.....PP. pk} detectable under WBM &, with the guarantee that exactly one arm in the set is entangled. Given a WBM E, for each state Pi we use the notation S and Sp(p:) interchangeably, and denote by S,,,,()) the estimate formed from Nt(t) ii.d samples. The underlying problem instance Sş satisfies SE(PI) Se(22) arm, the Successive Elimination algorithm Dar all 2TnZ) with modified elimination rule, as outlined in Algorithm M. The algorithm takes as input the set A, threshold = O, WBM & and the error probability 6, and it outputs the entangled state i* = arg min Se(p.). The algorithm maintains an active set Ω and measures every state in it. In order to identify i", the policy eliminates states whose Lower Confidence Bound (LCB) exceeds the threshold and halts when only one state remains in the active set.
SE(PK-1) > 0 > Se(pk). To identify the entangled
Algorithm SUCCESSIVE ELIMINATION ALGORITHM Input: < <= 0, 6, A, WBM
Output:
Initialize active set 1 A
Set: 0, (t)) +- 0, O,S.,, - 0, V
for / = 1,2,3,... do
for P: E f do
Perform measurement & on
Update N,(t), St,t), based on measurement outcomes
Update confidence width U (Ni()) 2 (see Lemma 5
Compute lower confidence bound: LCB,(t) + S.,N - U (Na(1,, 8) end for if LCB,(t) > for E f then
Update active set: Ω Ω {i} end
if [ = 1 then
Return ? end end for
Lemma 6 Algorithm M is 8-correct.
Proof: The proof is presented in Appendix A.2.
The correctness of Algorithm D| and the copy complexity of identifying the entangled arm are presented below.
A e-subgaussian random variable real, centered random variable X that satisfies Efe < 20/22 for any E R.
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Theorem With probability at least 1 1-6 6, the entangled state i* K arg min:e(k] Se(pi) remains in the active set Qt till lerminalion.
Proof:
The proof is presented in Appendix A.2.2
Theorem 8 With probability at least 1 6, Algorithm M identifies the entangled state i", requiring K (nnga))) copies. Here, A (p((;)) - <\ denotes the sub-optimality gap with re-
Eie[K] O (A,² log
spect to the threshold Ç.
Proof:
The proof is presented in Appendix A.2.3.
We observe that the sample complexity derived in Thcorem is within log(K) factor of the optimal bound, as demonstrated in Theorem of lammieoo all (20T4). This result follows from the concentration bound established in Lemma E. which forms the basis for the MAB policy described in the following section.
5.2 HDDH Algorithm
The HiCC algorithm (Q'sai ct all 2022 builds on the HDoC algorithm (Kano et all, 2018) by leveraging finite LIT, concentration bounds (Lemma E) instead of the LCB-based identification rule (Kalyanakrishnaa Et al, 2012). To explore among promising arms, HiCC adopts the sampling rule from Kano et all (201S), derived from the UCB algorithm for regret minimization (Auer 2002). improves sample complexity
over HDoC by utilizing the LIL bound, where the /log logt/t factor has higher growth rate than the
Vlogt/ t/t factor in the LCB bound. In other words, there exists a value T such that for all t > T, C1 R*
logt
log logt
The confidence bound for HDoC grows as (tt) V In( 1K(2) //t.. Through straightforward calculations, the smallest integer T such that the confidence bound U (T, 8/,8 K) is greater than a(T) is,
1 T log(K + 1)
max (t/2)) 2/2
(15)
Thus, if each stale is measured times initially, Hil'DDoC achieves comparable identification capabilities HDoC with O (Iog(K + 1) log (max (1/8,2)) copies of each state. We note that small values of e tighten the confidence radius and therefore incur more samples before climination, while large values reduce the number of samples with higher chance of premature arm elimination. The asymptotic growth of (l, with is sublinear, and the policy's correctness remains unaffected for > 0. The 'warm-start' phase parameter T controls the number of measurements collected before adaptive allocation begins. Once the threshold in equation IB is crossed, the 5-correct guarantees and copy complexity depend primarily on (A,, K, 5) and not on itself.
Consider K states such that Se(p1) Se(p2) Se(pK-m) 0 Se(pk-m41)... ... SE(PK), with being unknown. The algorithm takes as imput, the set of states A, threshold Ç 0, WBM & and the error probability å and outputs Aent {i E [K] such that Se(p,) < 0}. The algorithı maintains Ω and terminates when the set f2 = 0. To demonstrate the correctness of Algorithm & we first show that the algorithm is (A, o)oorrere for all X e [K] and then characterize the copy complexity of identifying m entangled states.
active set
Lemma 9 Algoritimm 8-correct.
Proof:
The proof is presented in Appendia
Theorem 10 With probability at least 5, the algorithm identifies all the states in Aeat-
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Algorithm 2 LIL H ALGORITIM
Input: Ç= 0, 6, A, WBM 3
Output: Aent
Initialize active set Ω <- A, Aent t
Set: f 0, (tt) + 0, N,() + 0, ViE
for i= 2,2, K do
Perform T measurements & Pj Update (t), St. based outcomes end for
while ft + 0 do
Find h4 = arg maxiaA (L) + logt (N.(C)
Perform measurement & on Ph. Update Nr, (t), St,, based outcomes Update confidence width U (N((1), o
SM,,N,, - (Nn, oe 2 25 then
Remove he from else if Sn,,N,, + (t), (t), oe
5
<
then
Add he to Aent. Remove he from end end while
Proof:
The proof is presented in Appendix A.2.2
With T in equation [t. can be seen from Theorem B that the number samples required to identify However, in practice, T chosen be larger than and the total sample complexity is expressed in terms of A minie[k] A..
an entangled state P. E is is O A log
K log (=100
A-- KlogA,""
Theorem 11 With probability 1 1-6 ô and an initialization T measurements, Algorithm B identifies the en- tangled states using O (A (K + K Klg K + K log log })) + +0 (K log(K K + + (max (f)))) copies.
Proof: The first term in the sample complexity derived in Appendix A.2. and the second term follows from equation
6 Implementation and Simulations on IBMQ Cloud
This section presents an experimental workflow for detecting entangled states from an ensemble of Bell Diagonal states. Sections mill and describe the procedures for generating Bell Diagonal states (BDS) and their corresponding WBMs, respectively. The performance of the MAB policies (see Section presented through numerical findings in Sections T
6.1 Generating Bell Diagonal States
Bell Diagonal States (BDS) are constructed as convex combinations of the four Bell states equation I.I forming geometric tetrahedron, T and are represented by:
Pate
Σ ps 1=1
IT)
(T)
=
I
+
to
& 0
(16)
4
1=1
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Here, oj are the Pauli operators and (2,))2, are the coordinates within the tetrahedron T. The mapping {p5} (1,, 12, (z) equation T is implemented through the quantum circuit proposed by Pozzobom X
Maztero (011);; Riedel Garding et al. [2021) and is shown in Fig. E.
VPi = cos( P2 = sin(v/ cos(
Ps sin (b)) (i cos(p), VP4 = sin(t) sin(f) sin(p) (17)
The sub-cireuit G encodes the probabilities {p,}}= into canonical coordinates (v), 0, ④) on the unit 3-sphere, and sub-circuit B entangles the states in the Bell basis. Finally, the Gell-Diaonnal state Paus = Ped is obtained by taking a partial trace over qubits and b.
(1b) ab
10),
G
unroad (env.)
10),
10)
H
Ped
10)a
B
G
10)
Ry
ab
10)
Ry (2
Ry (
(2)
Figure 2: Four-qubit circuit for generating BDS with canonical encoder G shown below.
6.2 Implementing Witness Basis Measurements
As outlined in Table B BDS are detectable under WBMs & and &g. To measure in the Pauli-Z (computa- tional) basis, we apply appropriate unitary transformations to & and &g. The corresponding transformations are realized through circuits CIRCa, and CIRCs, shown in Fig. B aadd applied to qubits e and (see Fig. 2 before measurement.
H
Ped
Pcd
Figure 3: Circuits CIRCs, (top) and CIRCè, (bottom) perform the unitary transformations required to map ε, and & into the Pauli-Z basis.
6.3 Workflow for entanglement detection
We propose a workflow for detecting entanglement in BDS without assuming prior knowledge of the specific WBM. Instead, WBMs are sequentially adapted using suitable unilary transformations, as detailed in Section {p1, ......,Kkk of BDS, we construct K sets of probabilistic mixtures for combining the four Bell states. Specifically, m states are generated with max, P3 > 1 while the remaining m states satisfy max, Py S J. These probabilities are encoded following the procedure outlined in Fig. where the BDS circuit for state A; is denoted as BDS; Subsequently, one of two WBM circuits, CIRCs,
To generate a set
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Algorithm 3 Workflow for Entanglement Detection in BDS
Input: Ç= 0, 6, (BDS,), CIRC,, WBM choice
Output: Acnt, Stopping time 7
Run Algorithm @ on }BDS;} with circuit CIRCs, on K states Return entangled states | = mn and stopping time Ti.
(m = K) then
Acut,2 f- 0, 72
else if m K then
Run Algorithm on {BDS,) with circuit CIRCs, K m states Return entangled states Aent. and stopping time 72.
end
Aent +- Aent.1 + Acnt,2 +
CIRCE,, is appended to the respective BDS circuit. Algorithm & outlines this workflow for BDS and takes the following inputs: threshold Ç O, error 6, BDS circuits (BDS,), WBM circuits CIRC,, and CIRCa, and the initial choice WBM. Notably, the initial WBM selection is arbitrary, as the sequence of WBM adaptations does not rely on state estimation.
The learner does not initially know under which WBM the BDS are detectable. Consequently, least one iteration of Algorithm D must be executed. In the first iteration, the algorithm processes circuits corresponding to K states with WBM & (or E2) and identifies a subset of entangled states, m, where 0 < f < K. In the second iteration, Algorithm Ø is applied to the K - m states that remain undetected by using circuits with WBM & (or E, as inputs.
Corollary 12 Let A min SS; A2 = min St,l and Amin min A1, A2), then, with probability and T = 1, Algorithm B identifies entangled BDS using O ( min copies.
K log 1 + K log K + K log log A )mmm)
6.4 Qiskit Experiment
The workflow presented in Algorithm & is simulated on IBM's Qiskit. The Python code implementation is available in Bharafi (2025). We present numerical results on the achievable copy complexity for entanglement detection in BDS. The experimental setup given as follows:
Simulation Environments: The workflow executed across three computational setups: (i) AerSimulator for idealized simulations, (tt) FakeBrisbane backend simulate noisy quantum environments, and (Mi) ibm-brisbane for real quantum processing unit (QPU) computations.
Problem Instance: We consider K = states of which m = 3 are entangled. The probabilities are suitably generated, and the true corresponding parameters under E, and Eg are,
Se. = [0.630 -.2668,, 0.5232, 0. 0.6695,.00695) Se, = 1-000744, 0.5963, - 0. 131735,000 0.2801, ,..20100.7766)
MAB routine: Each state was measured 10 times on backends (i, and 10a times on (iii). Algorithm & run 20 times on (i) and 5 times on (ii) for 5 (0, 1). We plot the average the x-axis, shown in
mumber of copies measured until stoppage the y-axis and log(1/5) Fig. Here, we note that the large standard deviation for the trend in backend (Mil) arises due to the limited number of experiment iterations, constrained by available compute resources.
Benchmarking: We benchmark our approach against non-adaptive BDS tomography, as described in Appendix I and compare the resulting copy complexity across the same range of 8. We present criterion in equation A that guarantees that the reconstructed states not misclassified (see
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10
Average number of copies
ibmn-brisbane
FakeBrisbane
Aer simulator
BDS Tomography 0.01
108
BDS Tomography 0.05
10
10
log(1/6)
Figure 4: Copy complexity for entanglement detection in BDS v/s log(1/8)
Appendix For the K 5 BDS instance considered in this experiment, the bottleneck is de- termined by State 5 (max, P; = 0.446), implying 0.054 to avoid misclassification theoretically. Choosing = 0.05 allows only a negligible margin for statistical fluctuations, resulting in low classi- fication accuracy. Thus, lo ensure a valid comparison that matches the high reliability of the MAB approach, we choose e 0.01 as the functional baseline.
From Corollary we observe that the factor log(1/8) has a multiplicative effect on the sample complexity, while the average copy complexity is primarily determined by Amin- The values of Sg governed by the four frequencies f1, fe, fs, and f1, defined in equation i While the true values of the fi's are calculated using Tr(P F, the values of fi obtained from register counts based on simulations performed on different backends differ from the true values upto O(10-2). Due to measurement noise and decoherence, the goalpost for Sg varies across different backends, and these differences influence Amin- One option to mitigate the measurement. noise (see details the appendix, Sec. A.)..
7 Entanglement Detection in Arbitrary Quantum States
This section outlines a routine for detecting entanglement in arbitrary two-qubit quantum states. Specifically, we consider K arbitrary states, one of which is entangled, and describe an MAB routine along with numerical results.
7.1 Numerical Experiment
The workflow outlined in Algorithm E is implemented in MATLAB. The algorithm takes the following inputs: a threshold Ç, an error parameter ö, a set K states A (with the promise that one state is entangled), and permutation of {1, 2, 4, 5,5,6) that specifies the order in which the WBMs should be adapted. As this is promise problem, the algorithm terminates
it identifies entangled state, without needing to measure with all six WBMs. The different modules in the software code are described below:
Generating arbitrary quantum states: To generate arbitrary density matrices, we follow the method described Zyezkowski Sommers (20по). Specifically, we start by generating complex matrix A C1x4 where the real and imaginary parts of each element independently sampled
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Algorithm 4 Entanglement detection for arbitrary stales
Input: ç=0, 6, A (P.. .K)) WBM Order P
Output: Acnt
flag t 1, t-1
while flag do
With + Er(), run Algorithm 2 for K arms Return entangled arm A
if | = then
flag +-
else
I+1+1
end
end while A
Aent
from a normal distribution. We then compute the density matrix by normalizing AAT, resulting p AAt /Tr(A))) This procedure ensures that p is a valid density matrix. We encountered pure states with Sx(-) O. For such cases, the algorithm took a significantly long time converge and, despite this, incorrectly estimated the value of Sp(p). Consequently, adjusted the threshold to = -1 x 10-3 and imposed cutoff on the sample complexity at 1 x 10 to better reflect the real-time performance of this policy.
Experiment Setup: In this experiment, we generate 1000 distinct instances of full rank arbitrary states, ensuring that each instance contains exactly one entangled state. These instances validated using the PPT criterion.
Detection ratio versus dz We test. the efficacy of using the single-paramede family of witnesses equation H to detect entanglement in arbitrary states. For å E (0, 1), we report the detection ratio, which is defined the fraction of times the entangled state is correctly identified by the MAB policy. This result is shown in Fig. E. We observe that the detection ratio diminishes with larger error margins.
0.95
0.9
Detection Ratio
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
8
Figure 5: Entanglement Detection ratio v/s 8 for arbitrary quantum states
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WBM usage under arbitrary ordering vs. d: For random WBM ordering, compute how many WBMs are required to detect single valid entangled state among set of K states. For d E (0, 1), we present the frequency distribution of the number of WBMs used, displayed as a cumulative histogram in Fig. B For significantly larger values of 6, the lower detection ratios indicate that the algorithm terminates
0.8
Number WBMs
0.6
0.01
0.11
0.21
0.31
0.41
0.51
0.61
0.71
0.81
0.91
Figure 6: The cunulative histograms compare the number of WBMs used to detect one valid enlangled state across different values of
upon identifying the wrong state, preventing further adaptation and primarily (around 85%) relying on up to three witnesses. This experiment can be extended to the scenario where there entangled states. However, since m is unknown and the states may be detectable under any of the WBMs, the routine would necessitate measuring under all WBMs to reliably identify the entangled states.
7.2 Numerical Outliers
We present an example of an entangled state certified by the PPT test that yields positive values for Se(p) under all six WBMs. Consider the pure entangled states and their corresponding Sg values, as shown in Table B. The state p Σ Ps (ı4) (1k1, where (1xt) are defined Table a and (p.) = (0.2936, 0. 0.0655, 0.6409), has negative eigenvalue of -0.029 after applying the partial transpose, thus confirming it an entangled state. However, the values of (Se) (0.0732, 0.1727, 0.1257, 0.1139, 0.0736, 0.0296) under the six WBMs are all non-negative. This indicates that the state cannot be detected by the witness family described in equation El
Table 5: Examples of arbitrary pure entangled states detected by the family of witnesses equation BI
Pure entangled states (v)) 1v2)) and ((a))
Values under (Se,)
(0.2687 0.0375i; 0.2406
4090;;
0.0502 00.6162;;0.220 0.2413 0.5107:]
(-0.1851, 0.3160, 0.988, -0.0058,
(0.0565 0.3355t; 0.0508 0. 06.66;; (). 08.8855 0.51911; 0.5689 0.2125i
(0.1562, -0.0280, -0.1 0. 0.1832, 0.079,0.0079 (-0.1851, 0.3160, O. 1595, -0.0058, 0.2177,
[0.1953 + 0.4438/; 0, 4958 0.4009i; 0. .0099 + 0.3495;; 0.0322 + 0.4848i]
We derive observation the nature of such states, focusing specifically the eigenstate [a] 10.3773 0.1445i, 0.4768 - 0.3244i, 0.4598 + 0.0809i, 0.5351], which corresponds to the largest cigenvalue of p. This eigenstate has Schmidd coefficient close to, but not exactly equal to 1, suggesting that it lies near the boundary of separable states while still remaining entangled. The pure state [A) (A) produces the following values for (5x) (0.0380, 0. 0269 0.0401, 0.1054, 0.0221, 0.0074). This highlights that both pure and mixed entangled states can yield inconclusive results when measured using this specific witness family, In these cases, it crucial to measure all six witnesses sufficient. number of times to accurately obtain
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the expected values of the corresponding observables. Additionally, performing FST can help determine the entanglement of these states using other separability criteria.
8 Discussions
8.1
The MAB Advantage
While traditional FST and fixed witness testing methods (Zhu et al, 201)) are direct and computationally feasible for bipartite qubit states, the proposed MAB approach provides practical and theoretical advantage for entanglement detection in parameterised two-qubit states considered in this work.
Fixed-confidence guarantees without state reconstruction: 6-correct MAB policies considered in this work output a statistically certified set of entangled parameterised two-qubit states, obtained directly from measurement data and without explicit state roconstruction. In contrast, FST necessitates state recon- struction up to specified trace accuracy and does not provide any explicit confidence guarantee on the entanglement aspect. Thus, any guarantee derived from FST is therefore indirect and depends on post-hoc analysis of the reconstructed state using analytical separability tests, whereas the MAB approach delivers decision-level guarantees by design.
Adaptive allocation of measurements: The key advantage of the MAB formulation is its adaptivity in measuring the witness basis on states based on their separability gaps (Ai). i.e., prioritising states whose gaps are smaller or whose entanglement status is uncertain. This yields destanncedepennentt copy complexity guarantces that scale polynomially in A; and have polylogarithmic dependence on K and (1/8). Adaptivity also allows early stopping when the criteria for entanglement certification are met. In contrast, non-adaplive methods like FST and repeated witness-testing approaches (Zhu et all, 20)0) tend to systematically over- sample with no principled stopping rule; that is, states far from the threshold (large A;) are measured extensively as the ones close to it (small A,) till the desired trace accuracy is achieved. This results in the copy complexity scaling linearly in K; sample-optimal FST with collective measurements requires (116K/e2)
copies Haah et all (2017).
Practical Advantage: As illustrated in Fig. g. the copy complexity achieved by the MAB policy is benchmarked against that of the non-adaptive tomography bascline for Bell-diagonal states described Appendix I.I. The MAB routine is run across IBMQ backends (Aer, FakeBrisbane, ibm-brisbane), achieving copy complexity of O(10t) O(1"). In contrast, the Bell Diagonal state tomography approach requires 0(10)) O(17)) copies to achieve a trace distance accuracy 0.01 for the same scale (K 5) and same range of 5. We emphasise that the adaptive sampling and early stopping.
reduction in copy complexity is cnabled via
Overall, the MAB- approach offers a quantitatively demonstrated reduction in copy complexity for Bell Diagonal states, explicit confidence guarantees, adaptivity in distributing measurement effort and scalability for batch detection tasks for the two-qubit, parameterised states.
8.2 Does the MAB routine optimize WBM ordering?
As outlined in Dai ct all (2014), there are WBM optimization strategies that prescribe an optimal WBM ordering for efficiently detecting whether a single arbitrary two-qubit quantum state is entangled. One such adaptive strategy uses the maximun-likellhood maxinum-entropy (MLME) estimate of the unknown state, based causal measurement data. Using this estimate, the subsequent WBM is identified to be the one that minimises the quadratic separability criterion. This leads to partial estimation of the quantum state.
In the context of batch entanglement detection, where an unknown number m of entangled stales out of a set of K states may be detectable under different witnesses, implementigg the WBM adaptive strategies from Dai et all (20r)) would be both timo-consuming and complex. This is bocause each of the K states may require unique permutation of the WBM ordering. Furthermore, the goal of the proposed MAB framework is to minimize the number of measurements needed for detecting entanglement in a given set of quantum
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states under specific WBM. Notably, this framework does not optimize the WBM ordering across multiple MAB runs.
The closest comparison is with Fig. E, which depicts the cumulative frequency of WBMs used. This aligns with the cumulative percentage of states identified under the WBM family, as seen in schemes 1A and 4A of the recently reported incomplete state estimation techniques (see (Dai et all, RUTE, Fig. 1)). However, this approach does not address the batch entanglement detection problem. The WBM adaptation scheme A in Dai et all (20E)) successfully detects 98% of random pure states but only 33% of full-rank mixed states. We specifically analyze the latter category, generating multiple instances of K states to quantify the number of WBMs required to detect a single entangled state, presenting results for varying 8. In this way, we provide numerical insight into the sample complexity and convergence rate of our proposed schema, in contrast to Dai
(012).
9 Future Work And Conclusion
Batch entanglement detection, as discussed in this paper, is particularly useful for verifying the integrity of a batch of practically relevant entangled stales, before use in applications like secure multi-channel quantum communication. We established a novel correspondence between the problem of batch entanglement detection and the Thresholding Bandit problem in stochastic Multi-Armed Bandits. We proposed the (m, K))quantum Multi- Armed Bandit framework for entanglement detection. The focus of this framework is on identifying m entangled states out of K states, where m is potentially unknown. We apply this framework to two-qubit states using two key ingredients: a specialized set of six measurements for two-qubit states called Witness Basis Measurements (WBM)& and a separability criterion Sg, which is based on the data obtained from these measurements and serves as the parameter that needs to be estimated. We present theoretical guarantees and numerical simulations to demonstrate how this parameter can be estimated quickly and accurately using MAB policies. First, we show that entangled states belonging a class of parameterised two-qubit states F can be detected by measuring subset of the six WBMs. With the knowledge of the WBM, show that we can directly apply some suitable MAB policies. Second, for the same parameterised states, we present a routine for entanglement detection when the WBM is not known by enabling arbitrary sequential adaptation of the WBMs. We exlend this to arbitrary two-qubit. quantum states and provide numerical results on the efficacy of using these measurements for detecting entanglement.
An exciting avenue for future rescarch lics in identifying WBMs for nigher-dimensional bipartite systems. The minimalistic tomographic scheme proposed in Zhu (2mm significantly reduces the number of required witnesses for two-qutrits from 81 to just 11, demonstrating the potential for more efficient entanglement detection. Mcanwhile, recent advancements in data-driven machine learning, particularly the use of SVMs to construct linear entanglement witnesses from local measurements (Girenwood et all, 202X), open new possibilities for tackling the (m, Jququauu MAB problem. By leveraging these techniques, one could optimize the number of witnesses needed to reliably detect all m states.
Entanglement detection can be reframed as a membership problem, where a state belongs to a set if it exhibits a specific propertysuch entanglement. This perspective aligns with the partition identification problem (Juneja & Krishnasamy. 20)), in which the objective is to determine the partition to which a data point belongs using a hyperplane structure. Extending this framework to the (m, Ku)quaatum MAB problem could pave the way for groundbreaking approaches to adaptive entanglement detection.
Acknowledgement
K.B. sincerely acknowledges the support from the Ministry of Education, Government of India, through the Prime Minister's Research Fellowship (PMRF) Scheme. V.S. is supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) contract (DE- SC0012704). K.J. gratefully acknowledges a grant from Mphasis to the Centre for Quantum Information, Communication, and Computing (CQuICC) at IIT Madras.
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A Appendix
A.1 Non-adaptive baseline:
Bell- Bell-Diagonal State Tomography
Copy Complexity Bound: For prescribed traco-distance accuracy C, we derive a copy-complexity guarantee for non-adaptive Bell-diagonal-statee tomography and compare it with adaptive bandit algorithms for entanglement detection. As seen in Table &, Bell-diagonal states are characterised by three two-qubit Pauli correlations XX, YY and ZZ. Thus, no other Pauli settings are required. Given Pau it is sufficient estimate
Caa (XX) PEUS: Cyp (YY) APDS: C.z (ZZ) Peos
(18)
Let = (Czz, Cyus Cza) The eigenvalues of Pas correspond to the statistical mixture {p.}: equation III and are given by the affine map
1 Ac),
p
4
(1
(19)
where
1
-1
1
1
-1
A
-1
1
Each measurement outcome for {xx, 22] is random variable X, E (-1,1) with mean cs Let é, := 1/t Xaj be the empirical mean obtained from t measurements. By Hoeffding's inequality,
P((é, - col 2 n) & 2exp (
tt² --
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Denoting è = (Cras Cyy Czx)" and taking union bound over the three correlators gives,
11p²
P(llè cllo > n) S Σ P(c, cal > n) s 6exp
(22
6.
(20)
From equation IH. p p= A_A(è c). hence
4 - pll1 HllA(e - c)]l 1 (A(e s 3/1 clloe.
(21)
Let Paus be the estimate of Peuss then the trace distance between Puoe and Paus given by,
(panss Punns Pins
Puns
1 pal b pll S 3 [| c|loo
(22)
where (a) holds because both Puns and Pars are diagonal in the Bell basis, so the eigenvalues of PBDS PRDS
are {pi-pi -p:p M To guarantee a trace distance between Pras and Press it suffices to enforce ||è -0c-cll xc S Ge.
Sctting 77 2 c, we solve for in equation 20.
9
6 ④
t
log
(23)
2e
shots per setting. Since three measurement settings are used, the total number of copies required is,
NBDS(E, 6) = 31 2 log
(24)
26
Choice of e for preserving the true status of reconstructed Bell-Diagonal States: From Section
122 we know that a Bell diagonal state entangled if p:- > &o where = arg max; pt and separable if Ps Since p and p both form valid probability distributions,
Σ
(p: p.) 0. 1=1
(25)
Preserving the status of Pus amounts to preserving the inequality defining the status, i.e., entangled means per > I and separable means P.r < +. Let x == pe- - -pel denote the magnitude of the estimation error on the largest eigenvalue. We consider the worst case in which p.. changes by æ in the most detrimental direction to status preservation. If f.- decreases by x, then
por Pr- = =
For equation A to hold, we must have
Σ 1
(p: -p) Pi) x.
By the triangle inequality,
Σ in
pb pil 2 Σ (p. - pr) =
Substituting in equation 2
D(pane Pence 2
1p:- - 1p-l 1 + E
10: - Bul)
2x
8
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Imposing the constraint Dfpuss Puus) therefore yields
pi- Pi- S. E.
(26)
We now present the condition for status preservation. If Pars is entangled, then pr > 2. The worst-case decrease according to equation 2B implics p.- Pa- Thus, entanglement is preserved if,
<< P.-
(27)
Similarly, if Pros is separable, then < 21 and the worst-case increase is p;. S Pi + te. Hence, separability is preserved
E << 2
(28)
Combining equation in and equation 28. the unified status prauusssccoovtaioon condition is,
6 ps- 2
(29)
A.2 Proofs for Section 5
The following lemma useful for some calculations.
Lemma 13 For t 2 1, 0,8 E
1,
log ((1 E)t) e)t)
1
2 2
(1+c)
log
cc ts log
(30)
w
w
A.2.1 Proof of Lemma
Proof:
Let B denote the "good" event that at any time 0 and for all arms [K], the true value Se(P:) is well concentrated around its estimate SL,.()) 1/N,(t) E.= Ji.s where i.id. samples J.,s 41 - (1 1()=4)) (1("-=3) - 1()= for i.id. outcomes Y, Y from measuring & on Pa.
K 00
8 } - S.1 SUSU (N(O). दτ
B
UU i=1==1
|S,,N.0)) 1
From Lemma and by applying the union bound, we get that
Lts
B 2 1 - c.K
21-8
(31)
c.K
where Eq. EIl holds because (0,1) and ce 1.
A.2.2 Proof of Theorem 7
Proof: Recall that the threshold 0 and problem instance Sy is such that Se(pi) > Se(p2) Se(ps)... > Se(pK-1) 0 Sp(px). Let us consider the case that the event B described in Lemma B holds. As outlined in Algorithm MI the arm i' will be dropped from the active set Ω if LCB, (t) > 0. That is,
6
S (.) - V
N (t),
> c.K
S,,,,- (1) -1, -15:-.;;00 - S5r >
= S > 0
This contradicts the assumption about the problem instance S because Sy. Se(pk) and so, the arm it will not be dropped from the active set Ω long event B holds.
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A.2.3 Proof of Theorem B
Proof: Let us consider the case where B holds. By the elimination rule of Algorithm D. an arm i is removed from the active set LtB,(t) > We have that,
(L,,10)) U (N(()) 8
25
ceK
8 SL,,(1) - -St A U (N(()), cok)
Ni(()), à ce
A, 2 20
(32)
Let us denote N, to be the number of samples of arm i, that is, N₂ = inf[t: U (Ni(t), eτ)) s ) The minimum value of N, can be obtained by solving,
= A
U No. c.K
2
2(1 + E)
log ((1 (1++)))))
A;
(1 (1+ (1+
log
N;
8/c.K
2
1
log ((1 + e)N.))
A
log
(33)
N
8/ccK
8(1 +1 + VE)2
From Lemma I get that,
Bc (1++))
8(1 +) + V
20g log
6
N
log
(34)
A
8
Thus, the total number of samples required to identify the arm it with probability of at least 1 8 is N < EK,,, N,.
A.3 Proofs for Section 5.2
A.3.1 Proof of Lemma
Firstly, we show that Algorithm 2 is (), -oorrett for arbitrary À E [K]. In the case where there are arms greater than or equal X, we show that F [Im UiEUAi {S; < C). S 8 where ni the number of good arms identified by the agent. Since implies that at least one good arm j [m] is identified bad arm by the agent. That is, for some E [m]
Proof:
considering the case when 2 X, the event {m < A}
5 and f E N, the upper confidence bound $3,N, + U (N)(t), E. Ç. Thus, we have that,
Plm < ~
Σ jE(m)
P
US +U NN,1), c <<>
LIEN
8 1tc
Σ
(By Lemma F
CeK
/E(m)
< mcs
(35)
K
The event U. tmr < <} considers all those outcomes where bad arm is identified be good Thus, for some bad arm 82, &m) such that j [K] \ [m)] we have,
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U
(s <)
E
P
U1$ - (N,(0),
_ > =c
CEK
LrEN
< (K - mm
(36)
c.K
Thus, putting Eq. 5 and Eq. together, we get that IP [m < λ} & &n les < {}} < 6. Next,
consider the case when the number of good arms is less than À and show that P(m 2 A < 6. Since
there are at most À good arms, the event { > ) implies that one of the output arms E {X X2) is such that there exists some index j such that Xj is a bad arm. Thus, we have that,
Σ
Flj (X, X2. ...1
jE jmm]
8 1te
$ (K micc
CeK
< K m
8
Ce
K
< 6
(37)
We see that the algorithm is (A, c)oorcet for all such x [K], thereby giving us that the algorithm is 8-correct.
A.3.2 Proof of Theorem
Proof: Recall that the threshold ( O and problem instance Se is such that SE(PI) 2 SE(P2)... > (p > > Se(Pk-m+1)... > Se (Px), with m being unknown. Let us consider the case that the event B described in Lemma R holds. As outlined in Algorithm 2, an arm i will be dropped if LCB, > 0. That is,
5 > S..N.00) - U (N(()),
c.K
S..M.10) (8,,N(1) St
Thus, as long as event holds, all the arms that have Sg < will not dropped. Thus the Hil'DDoC algorithm identifics all the arms correctly.
A.4 Integrating Error Mitigation in MAB Algorithms for Batch Entanglement Detection
In the MAAB-based workflow for entanglement detection described in Section E. one state is measured every time instant dictated by the sampling rule, and the statisticsnamely, the estimates of f2, fa, and faaee updated as new mcasurement outcomes are obtained. These estimates are susceptible to measurement errors, particularly readout errors, which induce inaccuracies in the measurement counts. To improve the accuracy the estimates, we characterize such errors and wish to mitigate them (Qiskit Community 2nzz)). To this end, we carry out a preliminary investigation by incorporating procodure for (a) error mitigation and (b) including error mitigation the MAB routine, and study the impact of error mitigation on the overall copy complexity of batch entanglement detection.
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A.4.1 Procedure for Error Mitigation
In Fig. B. we apply a unitary transformation to WBM E and & to measure the state of the system p in the computational (Pauli Z) basis. Consequently, we obtain expeclation values of the diagonal Pauli operators zz, 21, and IZ. The estimates of fI, f2, fa, and fa are linear combinations of these expectation values.
fi 0.25 (1 + (12), p + (Z1), + (ZZ),)
f2 0.25 (1 (TZ), + (TT), p- - (?7)p)
f3 = 0.25 (1 + (?Z), - (2I), - (77)p)
f1 = 0.25 (1- (1Z),- - (ZI)p + (ZZ)p).
(38)
Thus, it is essential to obtain precise expectation values for the diagonal Pauli operators to improve the accuracy of our estimates. To do this, we use LocalReadOut scheme from IBM's Qiskit Experiments library (Bravyi et all 202T). In this scheme characterize the readout errors of physical qubits the FakeBrisbane backend. These errors are assumed to be local in the sense that they are independent across qubits. Readout eгror mitigation uses a mitigator object (matrix) computed from an assignment matrix A, where each element Ar represents the probability of observing outcome when the true outcome is j. By applying this mitigator to unmitigated measurement counts, we refine our estimates by obtaining more accurate expectation values for ZZ, 21, and IZ.
A.4.2 How and where does it fit in the MAB Routine?
In each round of the MAB policy, based on an Upper Confidence Bound (UCB) score, the sampling rule selects a quantum state to measure. Since only a single-shot measurement is performed per round, the error mitigation procodure described in Section A.4. applied after a state has been measured several times. To illustrate this process, consider a specific round L = F, where state Pi has previously been measured T* times. The unmitigated measurement counts for the four possibrle outcomes are denoted as Fum. Fumm Py"m. and Ful. The empirical frequencies of these outcomes are given by,
(F) = Fum T i [4).
(39)
At this point, we invoke the error mitigation routine, supplying it with the unmitigated counts {Pum) } input. The mitigation routine corrects for readout errors and returns mitigated expectation values of the diagonal Pauli observables, yielding mitigated estimates (F)). With wost-processing adjustments to correct for decimal rounding errors, the corresponding mitigated measurement counts,
FT = (F) X e
(40)
We propose a nested mitigative process where the MAB algorithm invokes the error mitigation routine once every F measurement shots per state and uses the mitigated values in subsequent shots. For instance, at t= F, the routine produces mitigated estimates (F) from which obtain mitigated counts. Future measurement outcomes update on these mitigated counts. At t = 2F, the routine takes input these new counts and outputs a new set of mitigated estimates (21"). This creates mested-mitigation cycle, where each round of mitigation refines the previous one.
We conduct an empirical study to assess the impact of error mitigation on the average copy complexity of the MAB algorithm. Mitigation is invoked once every F' rounds, where F ranges from 50 to 10,000 in steps of 50. Here, smaller F values correspond to high-frequency mitigation and larger values indicate lower-frequeet frweuffryyyyccy mitigation. For the problem instance described in Section B. with 8 e (0, 1) and range of F, execute Algorithm E on FakeBrisbane, averaging the copy complexity at stoppage over 20 runs. The percentage of error mitigation is quantified as the relative reduction in copy complexity compared to the case without mitigation. To ensure the algorithm correctly identifies the entangled states, we employ an error indicator that verifies whether its error remains within the prescribed threshold 8. Using this framework, we generate the heatmap in Fig. W. which visualizes the percentage reduction in copy complexity due error mitigation. Notably, the white regions indicate cases where the algorithm converged in finite time but failed to identify
28

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Published in Transactions on Machine Learning Research (01/2026)
1e-06
1e-05
30
0.0001
0.001
20
0.01
0.1
10
0.2
0.3
600
2600
4600 6600
8600
Figure 7: Heatmap of percentage error mitigation FakeBrisbane backend for ō (0, 1) and various mitigation frequencies
the entangled states correctly. We observe and report the following inferences from Fig. @ First, the effect of miligation is d-dependent. For larger values of 6, the mitigation effect starts only as early as (F= 600) and stabilizes faster (F ~ 4000). In contrast, for smaller values of 5, the effect of mitigation is prominent only mid-range and stabilizes at F 7000. Second, for < 600 and smaller values of 6, the algorithm fails to detect the correct set of states under the prescribed which could potentially lead to random fluctuations in the estimates. Third, the observed stabilization zone (yellow) across values of d suggests critical threshold for F' beyond which reducing mitigation frequency (increasing the value of F) no longer reduces errors. It remains open question to fully understand and optimize for the use of error peror-mitigation and integrate them with MAB strategies.
This can be attributed to over-mitigation,
29


Reviews:
[{"review_id": "RecOnb3cut", "reviewer": "Reviewer_muTT", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper tackles the problem of efficiently detecting which quantum states in a given batch are entangled, without performing full quantum state tomography (FST) or adaptive measurements.\nIn quantum information processing, such as quantum communication, computation, or teleportation, many qubit pairs are produced, and it is crucial to verify which of them are genuinely entangled.\nHowever, existing methods for entanglement detection have significant limitations:\n1) Full-state tomography requires exponentially many measurements as the qubit number increases.\n2) Individual entanglement witnesses can only detect specific classes of entangled states, not all possible ones.\nTo address these challenges, the authors formulate the batch entangled state detection problem as a Good Arm Identification (GAI) problem in the multi-armed bandit framework.\nEach quantum state corresponds to an arm, and the goal is to identify which arms (states) are \"good\" (entangled) based on noisy measurement outcomes.\nThe authors propose two algorithms, Successive Elimination for exact one entangled state detection and lil'sHDoC for multiple entangled states without prior knowledge of their number.\nThe author provides theoretical guarantees on the sample complexity of these algorithms, showing they are $\\delta$-PAC (probably approximately correct) algorithms.\nAuthors also conduct numerical simulations to validate the performance of their algorithms, demonstrating significant reductions in the number of measurements required compared to naive approaches.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Strengths:\n\n1) The paper presents a novel intersection between quantum information theory and classical multi-armed bandit (MAB) frameworks. This interdisciplinary approach is innovative and has the potential to inspire further research at the intersection of these fields. Compared to the previous works, such as Lumbreras et al. (2022) which works on a global entanglement detection, this work focuses on batch entanglement detection, which is more practical in many quantum information processing scenarios.\n\n2) The author provides a rigorous theoretical analysis of the proposed algorithms, including sample complexity bounds and PAC guarantees. This adds significant value to the work, as it proposes new algorithms with strong theoretical guarantees of their performance.\n\n3) The authors demonstrate feasibility through IBM Qiskit simulations and hardware tests (ibm-brisbane backend). The inclusion of real-device results, noise considerations, and performance trends with respect to error probability $\\delta$ substantially strengthens the empirical credibility of the framework.\n\nWeaknesses:\n\n1) The writing could be condensed in places; Sections 2 and 3 span too long with textbook-level exposition before contributions emerge. It would be beneficial to streamline these sections to focus more on the novel contributions of the paper and how you formalize the entangled state detection problem as a GAI problem. This would help readers quickly grasp the significance of your work.\n\n2) The author mentions that the proposed algorithms outperform naive approaches in numerical simulations. It would be good to provide a fair comparison with existing state-of-the-art methods for entanglement detection, if applicable. This would help to contextualize the performance improvements and demonstrate the practical advantages of your algorithms.", "audience_yn": "Yes", "audience_explanation": "Formalizing Qubit States as a multi-armed bandit is an interesting topics. Researchers have developed many MAB algorithms on abstract environments for a long time. Finding a good application is good for the community of MAB.", "requested_changes": "1: Condense the writing:\nThe background on entanglement witnesses, separability criteria, and stochastic bandits is lengthy and partly textbook-like. Consider moving derivations and detailed examples to an appendix and summarizing the essentials in 1-2 paragraphs within the main text.\nMerge or clearly separate theoretical versus experimental parts. Sections 3 and 4 could start with short “Problem-Goal-Contribution” paragraphs to orient the reader. Include a short subsection or figure summarizing how the “quantum reward model” corresponds to classical MAB components.\n\n2: Clarify assumptions on witness basis measurements:\nExplicitly state that the witness basis measurements (WBMs) are known and fixed. Discuss how the method might extend if the WBM were partially unknown or noisy.", "broader_impact_concerns": "This work formalizes a new application of Multi-armed Bandit (MAB) problem in the Qubit States detection. I believe this will broader the application of MAB algorithms in the long turn.", "additional_comments": "I am more familiar with the multi-armed bandit literature than quantum information theory. From the MAB perspective, I am glad to see the authors provide theoretical guarantees and empirical validation of their algorithms.\nFor the quantum information side, while I am not an expert, the problem of batch entanglement detection seems relevant and important.\nI hope the authors can address the clarity and presentation issues mentioned above to make the paper more accessible to a broader audience.\nThe interdisciplinary nature of the work is a strength, and I believe it will inspire further research at the intersection of quantum information and classical learning theory.", "extra_scores": {}, "extra_text": {}}, {"review_id": "HajdsQCAPk", "reviewer": "Reviewer_AdQr", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper addresses the batch entanglement detection problem, which aims to identify all entangled states from a batch of K unknown quantum states without performing full state tomography efficiently. The main contribution is establishing a connection between this quantum problem and the classical Thresholding Bandit Problem (TBP) in Multi-Armed Bandits (MAB). The authors propose a so-called (m, K)-quantum MAB framework where each quantum state is an \"arm\" and a separability criterion derived from a family of six WBMs serves as the parameter to be estimated. They apply classical MAB policies like Successive Elimination and lil'HDoC to this framework, demonstrating that they can efficiently identify entangled states with high probability and with theoretical guarantees on sample complexity. The approach is verified through numerical simulations on parameterized two-qubit states and real quantum hardware experiments, as well as on arbitrary quantum states.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The claims made in the submission are supported by evidence, which is presented clearly through a combination of theoretical analysis, numerical simulations, and real-world quantum hardware experiments.", "audience_yn": "Yes", "audience_explanation": "The paper's novelty lies in drawing a connection between batch entanglement detection and a Thresholding Bandit problem in classical MAB. This demonstrates the potential for employing classical machine learning techniques to solve complex problems in quantum mechanics. This direct application of classical ML theory to quantum physics is a rapidly growing field of interest.", "requested_changes": "Here are some suggestions to improve the conceptual clarity and rigor.\n\n1. Can the authors summarize in the introduction how the MAB framework overcomes the exponential complexity of FST?\n    \n2. Summarize the exponential measurement complexity of FST into a lemma to compare with Theorems 10 and 13.\n    \n3. Add the ablation to discuss the algorithm's sensitivity to hyperparameters, such as $\\epsilon$ and T.", "broader_impact_concerns": "The paper focuses on an efficient technical method for detecting quantum entanglement. No Broader Impact Statement section is required.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "G2kGTsm2Tv", "reviewer": "Reviewer_Vty7", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces a framework for batch entanglement detection in quantum systems by leveraging techniques from classical multi-armed bandit (MAB) problem. Instead of the full state tomography, the proposed method focuses on detecting entanglement through a targeted set of measurements. Specifically, the authors employ a family of \"Witness Basis Measurements\" (WBMs) derived from entanglement witnesses, which allow them to compute a separability parameter indicating whether a quantum state is entangled. By mapping each quantum state in a batch to an arm in a bandit model, the detection problem is reformulated as identifying the subset of states for which the separability parameter falls below a certain threshold. \n\nStrength: \n- The paper is generally well written, even though too much background material in the beginning of the paper compromises the readability.\n- Experimental results using numerical simulations on IBM’s Qiskit platform are provided. \n\nWeakness: \n- The main weakness is that this work combines existing literature heavily. In particular, the witness operators measurements and the separability criterion are mostly from Zhu et al. (2010) (in fact, the considered form of witnesses are more confined than that of Zhu et al. (2010)), and the algorithms are from multi-armed bandit community. \n- Above is not a critical weakness per se, especially for journals like TMLR. However, there is no evidence regarding if the proposed method is superior to existing methods, nor the scale of the experiments provided confirm the utility of the proposed method.", "claims_and_evidence_yn": "No", "claims_explanation": "This work is particularly hard to evaluate, as it utilizes many existing works, both in the multi-armed-bandit (MAB) community as well as entanglement detection community. Here is an exemplary sentence from pg 6: \"In HIDoC, the sampling rule is derived from the UCB algorithm for cumulative regret minimization, while the identification rule is based on the LUCB algorithm for BAI, and the APT algorithm for the thresholding bandits problem. The proposed HDoC algorithm (LUCB-G) requires $O( \\Delta^{-2} ( K \\log 1/\\delta + K \\log K + K \\log 1/\\Delta ) )$ samples.\" The reviewer is unfortunately not an expert in MAB, and thus it is hard to evaluate the novelty of such construction.\n\nApart from the MAB complication, the provided \"Qiskit experiments\" consider identifying 3 entangled states out of the 5 candidate states. \nUnfortunately, the proposed method is not compared against any existing work, which again hinders evaluating the efficacy of the proposed method.", "audience_yn": "No", "audience_explanation": "The reviewer generally finds this work to not fit well with TMLR. While some researchers from multi-armed-bandit (MAB) community might find the entanglement detection as an interesting application, this work hardly provides any novelty in terms of MAB. Rather, the claimed novelty lies in applying existing MAB algorithms for batch entanglement detection. In my humble opinion, other venues more tailored for quantum computing / quantum information science would be a much better fit.", "requested_changes": "Major changes \n- Too much background information: the core argument of this paper, MAB for entanglement detection, starts in pg 7. Most of the previous texts are devoted to explaining preliminary concepts. As a result, the readability is quite compromised.\n- Comparison against existing methods should be provided. The main experiment seems to concern with identifying 3 entangled states, out of 5; it is hard to gauge the usefulness of the proposed method given the scale of such experiments (full state tomography seems easy in such scale), especially in the absence of comparison to any other algorithm.  \n\n\nQuestions\n- End of Sec 2.2: \"We note that most two-qubit entangled states can be detected under the six witnesses listed in Table 1.\" What do you mean by \"most\"? Also is the content of Sec 2.2 is entirely from Zhu et al (2010)?\n- What reward is being used for the MAB formulation? What problem is being solved, for a \"problem instance denoted by ${\\mu}$\"? \n- Assuming that a unique arm exists -- doesn't this impose some sort of convexity? Is it justified in this framework? \n- Below Definition 2: Farrell (1964) already achieves $O( \\Delta^{-2} \\log \\log \\Delta^{-2} )$? Then what do the more recent cited works (e.g., Karnin et al. 2013) do?\n- Why is the \"primary objective\" of BAI setting to characterize the expected stopping time $\\mathbb{E}_\\mu [\\tau]$? \n- Where is $\\mathcal{F}$ defined? (parameterized two-qubit states) \n- How does this method compare with simply running the algorithm of Zhu et al. (2010) multiple times? \n- How does the theoretical guarantees on the sample complexity compare with existing methods? \n\nTypos\n- Top of pg 3: it can represents both pure and mixed states…\n- Top of pg 5: The value of S equation 7 depends on …\n- Bottom of pg 5: citation styles / notations for lograthims are inconsistent. \n- Above Definition 2: $\\delta$-PAC? Or $\\delta$-PC? \n- Pg 15: a sentence stops in the middle after \"…, then\" before corollary 14", "broader_impact_concerns": "This work provides a framework for entanglement detection in quantum computing, and there is not much ethical implication to be discussed.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)