In quantum systems, purification can map mixed states into pure states and a nonunitary evolution into a unitary one by enlarging the Hilbert space. We establish a connection between the complexities of mixed quantum states and their purification, proposing new inequalities among these complexities. By examining single qubits, two-qubit Werner states, eight-dimensional Gaussian random unitary ensembles, and infinite-dimensional systems, we demonstrate how these relationships manifest across a broad class of systems. We find that the spread complexity of purification of a vacuum state evolving into a thermal state equals the average number of Rindler particles. This complexity is also shown to adhere to the Lloyd-like bound, indicating a further relation to the quantum speed limit. Finally, using mutual Krylov complexity, we observe subadditivity of the Krylov complexities, which contrasts with known results from holographic volume complexity. We put forward Krylov mutual complexity as a diagnosis of a potential gravity dual of Krylov complexities.

Germany

Rathindra Nath Das

Canada

Japan

Takato Mori

Krylov Complexity of Purification

We numerically construct asymptotically global <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>AdS</mi><mn>3</mn></msub><mo>×</mo><msup><mi mathvariant="normal">S</mi><mn>3</mn></msup><mo>×</mo><msup><mi mathvariant="normal">T</mi><mn>4</mn></msup></math> black holes in type IIB supergravity, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">R</mi><mi>t</mi></msub><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>4</mn></msup></math> symmetry, localized on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">S</mi><mn>3</mn></msup></math> and translationally invariant along the torus. These solutions, with horizons whose spatial cross sections have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">S</mi><mn>4</mn></msup><mo>×</mo><msup><mi mathvariant="normal">T</mi><mn>4</mn></msup></math> topology, dominate the microcanonical ensemble at low energies. At higher energies, a first-order phase transition occurs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>BTZ</mi><mo>×</mo><msup><mi mathvariant="normal">S</mi><mn>3</mn></msup><mo>×</mo><msup><mi mathvariant="normal">T</mi><mn>4</mn></msup></math> black holes—possessing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">R</mi><mi>t</mi></msub><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mrow><mi>SO</mi></mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>×</mo><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>4</mn></msup></math> symmetry and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">S</mi><mn>1</mn></msup><mo>×</mo><msup><mi mathvariant="normal">S</mi><mn>3</mn></msup><mo>×</mo><msup><mi mathvariant="normal">T</mi><mn>4</mn></msup></math> horizon topology. By the AdS/CFT correspondence, this transition reflects the spontaneous breaking of the SO(4) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>R</mi></math> symmetry of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>D</mi><mn>1</mn><mtext>−</mtext><mi>D</mi><mn>5</mn></mrow></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>CFT</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math> to SO(3). We also compute the expectation value of the scalar operator with lowest conformal dimension in the low-energy phase. Our SO(3)-localized black holes—together with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math>-localized solutions of Bena et al. [<a href="http://dx.doi.org/10.1007/JHEP10(2025)165">J. High Energy Phys. 10 (2025) 165</a>.]—point to a rich landscape of novel black holes that may approach the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>CFT</mi><mn>2</mn></msub></math> “sparseness bootstrap condition,” and shed light on how macroscopic entanglement in thermal phases encodes microscopic structure via internal directions.

United Kingdom

Óscar J.C. Dias

Jorge E. Santos

Localized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>×</mo></mrow><msup><mrow><mi mathvariant="normal">S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><mtext> </mtext><msup><mrow><mi mathvariant="normal">T</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></math> Black Holes

We provide evidence that three-party entanglement signals in holography obey a relation that is not satisfied by generalized Greenberger-Horne-Zeilinger (GHZ) states. Using proposed holographic duals for these entanglement signals, we provide a geometric argument establishing this relation. This is the first known inequality on the structure of pure three-party holographic states, and shows that time-symmetric holographic states can never have purely GHZ-like entanglement. We also discuss similar relations for four parties.

United States

Belgium

Vijay Balasubramanian

Monica Jinwoo Kang

Charlie Cummings

Chitraang Murdia

Simon F. Ross

Purely Greenberger-Horne-Zeilinger–like Entanglement is Forbidden in Holography

The LEGEND Collaboration is searching for neutrinoless double-beta (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>0</mn><mi>ν</mi><mi>β</mi><mi>β</mi></mrow></math>) decay by operating high-purity germanium detectors enriched in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Ge</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>76</mn></mrow></mmultiscripts></mrow></math> in a low-background liquid argon environment. Building on key technological innovations from the GERmanium Detector Array (GERDA) experiment and the MAJORANA DEMONSTRATOR experiment, LEGEND-200 has performed a first <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>0</mn><mi>ν</mi><mi>β</mi><mi>β</mi></mrow></math> decay search based on 61.0 kg yr of data. Over half of this exposure comes from our highest performing detectors, including newly developed inverted-coaxial detectors, and is characterized by an estimated background level of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><msubsup><mrow><mn>0.5</mn></mrow><mrow><mo>−</mo><mn>0.2</mn></mrow><mrow><mo>+</mo><mn>0.3</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext></mrow><mi>cts</mi><mo>/</mo><mo stretchy="false">(</mo><mi>keV</mi><mtext> </mtext><mi>ton</mi><mtext> </mtext><mi>yr</mi><mo stretchy="false">)</mo></mrow></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>0</mn><mi>ν</mi><mi>β</mi><mi>β</mi></mrow></math> decay signal region. A combined analysis of data from GERDA, the MAJORANA DEMONSTRATOR, and LEGEND-200, characterized by a 90% confidence level exclusion sensitivity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2.8</mn><mo>×</mo><mrow><msup><mn>10</mn><mn>26</mn></msup><mtext> </mtext><mtext> </mtext><mi>yr</mi></mrow></math> on the half-life of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>0</mn><mi>ν</mi><mi>β</mi><mi>β</mi></mrow></math> decay, reveals no evidence for a signal and sets a new observed lower limit at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>T</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mrow><mn>0</mn><mi>ν</mi></mrow></msubsup><mo>&gt;</mo><mn>1.9</mn><mo>×</mo><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>26</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mi>yr</mi></mrow></mrow></math> (90% confidence level). Assuming the decay is mediated by Majorana neutrinos, this corresponds to an upper limit on the effective Majorana mass in the range <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>β</mi><mi>β</mi></mrow></msub><mo>&lt;</mo><mn>75</mn><mi>–</mi><mn>200</mn><mtext> </mtext><mtext> </mtext><mi>meV</mi></mrow></math>, depending on the adopted nuclear matrix element.

H. Acharya

N. Ackermann

M. Agostini

A. Alexander

C. Andreoiu

Switzerland

G.R. Araujo

F.T. Avignone

M. Babicz

W. Bae

Russia

A.M. Bakalyarov

Italy

M. Balata

A.S. Barabash

P.S. Barbeau

C.J. Barton

L. Baudis

C. Bauer

E. Bernieri

L. Bezrukov

K.H. Bhimani

V. Biancacci

E. Blalock

S.J. Borden

G. Borghi

F. Borra

B. Bos

A. Boston

V. Bothe

R. Bouabid

R. Brugnera

N. Burlac

M. Busch

S. Calgaro

L. Canonica

S. Capra

M. Carminati

R.M.D. Carney

C. Cattadori

R. Cesarano

Y.D. Chan

J.R. Chapman

A. Chernogorov

P.J. Chiu

C.D. Christofferson

M.L. Clark

A.I. Colon-Rivera

T. Comellato

V. D’Andrea

R. Deckert

J.A. Detwiler

A. Di Giacinto

N. Di Marco

T. Dixon

K.M. Dong

A. Drobizhev

G. Duran

Yu. Efremenko

S.R. Elliott

C.H.J. Emmanuel

E. Engelhardt

E. Esch

M.T. Febbraro

F. Ferella

D.E. Fields

C. Fiorini

JINR

M. Fomina

N. Fuad

R. Gala

A. Galindo-Uribarri

A. Gangapshev

A. Garfagnini

S. Gazzana

A. Geraci

L. Gessler

C. Ghiano

A. Gieb

S. Giri

M. Gold

C. Gooch

G. Grünauer

M.P. Green

J. Gruszko

I. Guinn

V.E. Guiseppe

V. Gurentsov

Y. Gurov

K. Gusev

B. Hackett

F. Hagemann

Poland

M. Haranczyk

F. Henkes

R. Henning

J. Herrera

D. Hervas Aguilar

J. Hinton

Czechia

R. Hodák

H.F.R. Hoffmann

M.A. Howe

M. Huber

M. Hult

A. Ianni

First Results on the Search for Lepton Number Violating Neutrinoless Double-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>β</mi></math> Decay with the LEGEND-200 Experiment

Searches are presented for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><mi>D</mi><msup><mi>h</mi><mo>+</mo></msup><msup><mi>h</mi><mo>−</mo></msup></math> decays, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi></math> is a charmed meson and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>h</mi><mo>±</mo></msup></math> is a charged pion or kaon, using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>p</mi></math> collision data collected by the LHCb experiment corresponding to an integrated luminosity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>9</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>. The decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msup><mi>D</mi><mo>+</mo></msup><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msup><mi>D</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>−</mo></msup></math> are observed for the first time. Their branching fractions, expressed as ratios relative to that of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msubsup><mi>B</mi><mi>s</mi><mn>0</mn></msubsup><msup><mi>π</mi><mo>+</mo></msup></math> decay, are determined to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">R</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1.96</mn><mo>±</mo><mn>0.23</mn><mo>±</mo><mn>0.08</mn><mo>±</mo><mn>0.10</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">R</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mspace linebreak="goodbreak"></mspace><mo stretchy="false">(</mo><mn>3.67</mn><mo>±</mo><mn>0.55</mn><mo>±</mo><mn>0.24</mn><mo>±</mo><mn>0.20</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">R</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1.61</mn><mo>±</mo><mn>0.35</mn><mo>±</mo><mn>0.13</mn><mo>±</mo><mn>0.07</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math>, where the first uncertainty is statistical, the second is systematic, and the third is due to the limited precision on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi></math>-meson branching fractions. The decay channels proceed primarily through excited <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mn>0</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mn>0</mn></msup></math> resonances or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi></math> mesons, and open a new avenue for studies of charge-parity violation in beauty mesons.

Netherlands

R. Aaij

A.S.W. Abdelmotteleb

C. Abellan Beteta

F. Abudinén

T. Ackernley

A.A. Adefisoye

Spain

B. Adeva

M. Adinolfi

P. Adlarson

France

C. Agapopoulou

C.A. Aidala

Z. Ajaltouni

S. Akar

K. Akiba

P. Albicocco

J. Albrecht

R. Aleksiejunas

CERN

F. Alessio

P. Alvarez Cartelle

R. Amalric

Brazil

S. Amato

J.L. Amey

Y. Amhis

China

L. An

L. Anderlini

M. Andersson

P. Andreola

M. Andreotti

S. Andres Estrada

A. Anelli

D. Ao

C. Arata

F. Archilli

Z. Areg

M. Argenton

Costa Rica

S. Arguedas Cuendis

L. Arnone

A. Artamonov

M. Artuso

E. Aslanides

R. Ataíde Da Silva

M. Atzeni

B. Audurier

J.A. Authier

D. Bacher

I. Bachiller Perea

S. Bachmann

M. Bachmayer

J.J. Back

P. Baladron Rodriguez

V. Balagura

A. Balboni

W. Baldini

Z. Baldwin

L. Balzani

H. Bao

J. Baptista de Souza Leite

C. Barbero Pretel

M. Barbetti

I.R. Barbosa

R.J. Barlow

M. Barnyakov

S. Barsuk

W. Barter

Observation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><mi>D</mi><msup><mi>h</mi><mo>+</mo></msup><msup><mi>h</mi><mo>−</mo></msup></math> Decays