Entanglement of Formation of an Arbitrary State of Two Qubits
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The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state $\ensuremath{\rho}$ is the minimum average entanglement of an ensemble of pure states that represents \ensuremath{\rho}. An earlier paper conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula for special states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.
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Concurrence, entanglement of formation, and tangle are measures of the quantum-coherence properties of a mixed quantum state.
“The concurrence, entanglement of formation and tangle are measures of the quantum-coherence properties of a mixed quantum state [29].”
From Measurement of qubits (2001) · cites this paper as [29]
The Abstract (pp. 1, 4) defines entanglement of formation for mixed states of two qubits and says concurrence can serve as a measure of entanglement. The paper does not address tangle as a measure, and it describes these quantities as measures of entanglement rather than of quantum-coherence properties generally. (LLM, conf=0.88)
Experimental — accuracy not yet measuredWootters' original definition of concurrence is invalid for two-qudit states with d > 2.
“As for the concurrence, rigorously, its original Wootters' definition [20] is invalid for two-qudit states with d > 2.”
From Entanglement of biphoton states: qutrits and ququarts (2011) · cites this paper as [20]
The discussion on pp. 2–4 defines concurrence and proves its use for two-qubit states. It does not address two-qudit states with d > 2 or establish that the definition is invalid for them. (LLM, conf=0.82)
Experimental — accuracy not yet measuredFor bipartite states with one-particle Hilbert space dimension d=2, there is a simple algebraic relation between Wootters' concurrence C and the Schmidt entanglement parameter K, giving equation (5.3).
“It is also known [19] that in the case of bipartite states with the dimensionality of the oneparticle Hilbert space d = 2, there is a simple algebraic relation between Wootters' concurrence C [20] and the Schmidt entanglement parameter K , owing to which”
From Entanglement of biphoton states: qutrits and ququarts (2011) · cites this paper as [20]
Expressions for concurrence C (5.3 and last formula of 5.4) can also be derived directly from Wootters' original definition [20].
“Note that expressions for the concurrence C (equation (5.3) and the last formula of equation (5.4)) can also be derived directly from the original Wootters' definition [20].”
From Entanglement of biphoton states: qutrits and ququarts (2011) · cites this paper as [20]
The entanglement shared through the channel can be distilled into pure singlets and used for teleportation.
“This entanglement, being that of a 2 × 2 system, can also be distilled [7] into pure singlets and used for teleportation.”
From Quantum Communication through an Unmodulated Spin Chain (2003) · cites this paper as [7]
Concurrence (CQ) measures the degree of entanglement in a system, where CQ > 0.5 indicates a partially entangled system.
“Concurrence (CQ) measures the degree of entanglement in a system, where CQ > 0.5 indicates a partially entangled system [73].”
From Mathematical proof of the Fisher-Escolà <i>Q</i> statistical distribution in quantum consciousness modeling (2025) · cites this paper as [73]
Only for bipartite two-level systems do a few entanglement measures exist that can be evaluated algebraically.
“Only for states of smallest possible dimensions, i.e., for bipartite two-level systems, do a few measures exist that can be evaluated algebraically [27,28,29]”
From Measures and dynamics of entangled states (2005) · cites this paper as [27]
For 2×2 systems, the entanglement of a pure state can be expressed as a function of its concurrence, E(Ψ)=E(c(Ψ)).
“For these low dimensional systems, the entanglement of a pure state can be expressed as a function of its concurrence, E (Ψ) = E ( c (Ψ)), with [27]”
From Measures and dynamics of entangled states (2005) · cites this paper as [27]
The infimum of the convex-roof expression for concurrence of mixed states is given by Eq. (39), where the S_i are the singular values of τ in decreasing order.
“The infimum of this quantity is known [27] to be given by”
From Measures and dynamics of entangled states (2005) · cites this paper as [27]
Concurrence can be considered as an independent entanglement measure.
“However, concurrence can also be considered as an independent entanglement measure [27].”
From Measures and dynamics of entangled states (2005) · cites this paper as [27]
The matrix τ defined here coincides with that introduced in Eq. (38) in the 2×2 case.
“The relation between the present and the original approach to concurrence [27,57] becomes apparent with the observation that the matrix τ defined here coincides with that introduced in Eq. (38)”
From Measures and dynamics of entangled states (2005) · cites this paper as [27]
The amount of entanglement can be quantified by concurrence, which is directly related to eij by C(eij) = max{0, -1/2 + 3/2 eij}.
“The amount of entanglement can also be quantified by concurrence[45], which is directly related to e ij with C ( e ij ) = max { 0 , -1 2 + 3 2 e ij } .”
From Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems (2011) · cites this paper as [45]
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Cites
Horn, R. A., & Johnson, C. R. (1985) · Cited by 22,203 · see citing works
On the Einstein Podolsky Rosen paradox
Bell, J. S. (1964) · Physics Physique Fizika · doi:10.1103/physicsphysiquefizika.1.195 · Cited by 12,252 · see citing works
Horn, R. A., & Johnson, C. R. (1985) · Cambridge University Press eBooks · doi:10.1017/cbo9780511810817 · Cited by 10,298 · see citing works
Mixed-state entanglement and quantum error correction
Bennett, C. H., DiVincenzo, D. P., Smolin, J. A., & Wootters, W. K. (1996) · Physical Review A · doi:10.1103/physreva.54.3824 · Cited by 5,272 · see citing works
Self-citationSakurai, J. J., & Napolitano, J. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108587280 · Cited by 3,672 · see citing works
Discussion of Probability Relations between Separated Systems
Schrödinger, E. (1935) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100013554 · Cited by 3,124 · see citing works
Concentrating partial entanglement by local operations
Bennett, C. H., Bernstein, H. J., Popescu, S., & Schumacher, B. W. (1996) · Physical Review A · doi:10.1103/physreva.53.2046 · Cited by 3,051 · see citing works
Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels
Bennett, C. H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J. A., & Wootters, W. K. (1996) · Physical Review Letters · doi:10.1103/physrevlett.76.722 · Cited by 2,961 · see citing works
Self-citationEntanglement of a pair of quantum bits
Hill, S., & Wootters, W. K. (1997) · Physical Review Letters · doi:10.1103/physrevlett.78.5022 · Cited by 2,823 · see citing works
Self-citationVedral, V., Plenio, M. B., Rippin, M. A., & Knight, P. L. (1997) · Physical Review Letters · doi:10.1103/physrevlett.78.2275 · Cited by 1,967 · see citing works
Sakurai, J. J., & Napolitano, J. (2017) · Cambridge University Press eBooks · doi:10.1017/9781108499996 · Cited by 1,653 · see citing works
DiVincenzo, D. P. (1995) · Science · doi:10.1126/science.270.5234.255 · Cited by 1,651 · see citing works
Probability relations between separated systems
Schrödinger, E. (1936) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100019137 · Cited by 991 · see citing works
A complete classification of quantum ensembles having a given density matrix
Hughston, L. P., Jozsa, R., & Wootters, W. K. (1993) · Physics Letters A · doi:10.1016/0375-9601(93)90880-9 · Cited by 663 · see citing works
Self-citationOptimal eavesdropping in quantum cryptography. I. Information bound and optimal strategy
Fuchs, C. A., Gisin, N., Griffiths, R. B., Niu, C.-S., & Peres, A. (1997) · Physical Review A · doi:10.1103/physreva.56.1163 · Cited by 475 · see citing works
Information theory of quantum entanglement and measurement
Cerf, N. J., & Adami, C. (1998) · Physica D Nonlinear Phenomena · doi:10.1016/s0167-2789(98)00045-1 · Cited by 185 · see citing works
Fundamental Problems in Quantum Theory
Popescu, S., Aharonov, Y., Coleman, S., et al. (1995) · Cited by 124 · see citing works
Fundamental Problems in Quantum Theory
Greenberger, D. M., & Zelinger, A. (1995) · Cited by 119 · see citing works
Properties of mixtures on non-orthogonal states
Hadjisavvas, N. (1981) · Letters in Mathematical Physics · doi:10.1007/bf00401481 · Cited by 39 · see citing works
Optimal decompositions of quantum states with respect to entropy
Benatti, F., Narnhofer, H., & Uhlmann, A. (1996) · Reports on Mathematical Physics · doi:10.1016/0034-4877(96)87681-6 · Cited by 31 · see citing works
Optimizing entropy relative to a channel or a subalgebraPreprint
Uhlmann, A. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/9701014 · Cited by 16 · see citing works
On the measure of entanglement for pure statesPreprint
Popescu, S., & Rohrlich, D. (1996) · arXiv (Cornell University) · Cited by 10 · see citing works
Flagged first, then by citations
Cited by
Sorted by citations
Horodecki, R., Horodecki, P., Horodecki, M., & Horodecki, K. (2009) · Reviews of Modern Physics · doi:10.1103/revmodphys.81.865 · Cited by 10,118 · see citing works
Kimble, H. J. (2008) · Nature · doi:10.1038/nature07127 · Cited by 6,023 · see citing works
Computable measure of entanglement
Vidal, G., & Werner, R. F. (2002) · Physical Review A · doi:10.1103/physreva.65.032314 · Cited by 4,901 · see citing works
Quantum information with continuous variables
Braunstein, S. L., & van Loock, P. (2005) · Reviews of Modern Physics · doi:10.1103/revmodphys.77.513 · Cited by 3,858 · see citing works
Entanglement in many-body systems
Amico, L., Fazio, R., Osterloh, A., & Vedral, V. (2008) · Reviews of Modern Physics · doi:10.1103/revmodphys.80.517 · Cited by 3,745 · see citing works
Coffman, V., Kundu, J., & Wootters, W. K. (2000) · Physical Review A · doi:10.1103/physreva.61.052306 · Cited by 2,619 · see citing works
Self-citationGühne, O., & Tóth, G. (2009) · Physics Reports · doi:10.1016/j.physrep.2009.02.004 · Cited by 2,401 · see citing works
James, D. F. V., Kwiat, P. G., Munro, W. J., & White, A. G. (2001) · Physical Review A · doi:10.1103/physreva.64.052312 · Cited by 2,176 · see citing works
Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond
Lewenstein, M., Sanpera, A., Ahufinger, V., Damski, B., Sen(De), A., & Sen, U. (2007) · Advances in Physics · doi:10.1080/00018730701223200 · Cited by 2,104 · see citing works
Quantum repeaters based on atomic ensembles and linear optics
Sangouard, N., Simon, C., de Riedmatten, H., & Gisin, N. (2011) · Reviews of Modern Physics · doi:10.1103/revmodphys.83.33 · Cited by 2,004 · see citing works
Scaling of entanglement close to a quantum phase transition
Osterloh, A., Amico, L., Falci, G., & Fazio, R. (2002) · Nature · doi:10.1038/416608a · Cited by 1,860 · see citing works
Colloquium : Quantum coherence as a resource
Streltsov, A., Adesso, G., & Plenio, M. B. (2017) · Reviews of Modern Physics · doi:10.1103/revmodphys.89.041003 · Cited by 1,787 · see citing works
Quantum metrology with nonclassical states of atomic ensembles
Pezzè, L., Smerzi, A., Oberthaler, M. K., Schmied, R., & Treutlein, P. (2018) · Reviews of Modern Physics · doi:10.1103/revmodphys.90.035005 · Cited by 1,705 · see citing works
Finite-Time Disentanglement Via Spontaneous Emission
Yu, T., & Eberly, J. H. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.140404 · Cited by 1,643 · see citing works
The classical-quantum boundary for correlations: Discord and related measures
Modi, K., Brodutch, A., Cable, H., Paterek, T., & Vedral, V. (2012) · Reviews of Modern Physics · doi:10.1103/revmodphys.84.1655 · Cited by 1,600 · see citing works
Logarithmic Negativity: A Full Entanglement Monotone That is not Convex
Plenio, M. B. (2005) · Physical Review Letters · doi:10.1103/physrevlett.95.090503 · Cited by 1,530 · see citing works
Entanglement in a simple quantum phase transition
Osborne, T. J., & Nielsen, M. A. (2002) · Physical Review A · doi:10.1103/physreva.66.032110 · Cited by 1,472 · see citing works
Experimental entanglement of four particles
Sackett, C. A., Kielpinski, D., King, B. E., Langer, C., Meyer, V., Myatt, C. J., Rowe, M., Turchette, Q. A., Itano, W. M., Wineland, D. J., et al. (2000) · Nature · doi:10.1038/35005011 · Cited by 1,436 · see citing works
Quantum Communication through an Unmodulated Spin Chain
Bose, S. (2003) · Physical Review Letters · doi:10.1103/physrevlett.91.207901 · Cited by 1,407 · see citing works
Volume of the set of separable states
Życzkowski, K., Horodecki, P., Sanpera, A., & Lewenstein, M. (1998) · Physical Review A · doi:10.1103/physreva.58.883 · Cited by 1,411 · see citing works
Multiphoton entanglement and interferometry
Pan, J.-W., Chen, Z.-B., Lu, C.-Y., Weinfurter, H., Zeilinger, A., & Żukowski, M. (2012) · Reviews of Modern Physics · doi:10.1103/revmodphys.84.777 · Cited by 1,410 · see citing works
Quantum interface between light and atomic ensembles
Hammerer, K., Sørensen, A. S., & Polzik, E. S. (2010) · Reviews of Modern Physics · doi:10.1103/revmodphys.82.1041 · Cited by 1,290 · see citing works
Scalable multiparticle entanglement of trapped ions
Häffner, H., Hänsel, W., Roos, C. F., Benhelm, J., Chek-al-kar, D., Chwalla, M., Körber, T., Rapol, U. D., Riebe, M., Schmidt, P. O., et al. (2005) · Nature · doi:10.1038/nature04279 · Cited by 1,253 · see citing works
Demonstration of two-qubit algorithms with a superconducting quantum processor
DiCarlo, L., Chow, J. M., Gambetta, J., Bishop, L. S., Johnson, B., Schuster, D., Majer, J., Blais, A., Frunzio, L., Girvin, S. M., et al. (2009) · Nature · doi:10.1038/nature08121 · Cited by 1,158 · see citing works
The role of relative entropy in quantum information theory
Vedral, V. (2002) · Reviews of Modern Physics · doi:10.1103/revmodphys.74.197 · Cited by 1,140 · see citing works
Yu, T., & Eberly, J. H. (2009) · Science · doi:10.1126/science.1167343 · Cited by 1,097 · see citing works
Conditions for a class of entanglement transformations
Nielsen, M. A. (1999) · Physical Review Letters · doi:10.1103/physrevlett.83.436 · Cited by 1,036 · see citing works
Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
Gogolin, C., & Eisert, J. (2016) · Reports on Progress in Physics · doi:10.1088/0034-4885/79/5/056001 · Cited by 1,024 · see citing works
Quantum discord for two-qubit systems
Luo, S. (2008) · Physical Review A · doi:10.1103/physreva.77.042303 · Cited by 1,060 · see citing works
Operational Resource Theory of Coherence
Winter, A., & Yang, D. (2016) · Physical Review Letters · doi:10.1103/physrevlett.116.120404 · Cited by 916 · see citing works
Environment-Induced Sudden Death of Entanglement
de Almeida, M. P., de Melo, F., Hor-Meyll, M. O., Salles, A., Walborn, S. P., Ribeiro, P. H. S., & Davidovich, L. (2007) · Science · doi:10.1126/science.1139892 · Cited by 913 · see citing works
Measuring Quantum Coherence with Entanglement
Streltsov, A., Singh, U., Dhar, H. S., Bera, M. N., & Adesso, G. (2015) · Physical Review Letters · doi:10.1103/physrevlett.115.020403 · Cited by 900 · see citing works
Quasiparticle engineering and entanglement propagation in a quantum many-body system
Jurcevic, P., Lanyon, B. P., Hauke, P., Hempel, C., Zoller, P., Blatt, R., & Roos, C. F. (2014) · Nature · doi:10.1038/nature13461 · Cited by 863 · see citing works
Universal state inversion and concurrence in arbitrary dimensions
Rungta, P., Bužek, V., Caves, C. M., Hillery, M., & Milburn, G. J. (2001) · Physical Review A · doi:10.1103/physreva.64.042315 · Cited by 848 · see citing works
Quantum metrology from a quantum information science perspective
Tóth, G., & Apellaniz, I. (2014) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/47/42/424006 · Cited by 862 · see citing works
Ma, J., Wang, X., Sun, C. P., & Nori, F. (2011) · Physics Reports · doi:10.1016/j.physrep.2011.08.003 · Cited by 843 · see citing works
An integrated diamond nanophotonics platform for quantum-optical networks
Sipahigil, A., Evans, R. E., Sukachev, D. D., Burek, M. J., Borregaard, J., Bhaskar, M. K., Nguyen, C. T., Pacheco, J. L., Atikian, H. A., Meuwly, C., et al. (2016) · Science · doi:10.1126/science.aah6875 · Cited by 844 · see citing works
Quantum discord for two-qubit X states
Ali, M., Rau, A. R. P., & Alber, G. (2010) · Physical Review A · doi:10.1103/physreva.81.042105 · Cited by 837 · see citing works
Non-Markovian Effects on the Dynamics of Entanglement
Bellomo, B., Franco, R. L., & Compagno, G. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.160502 · Cited by 818 · see citing works
Quantum entanglement in condensed matter systems
Laflorencie, N. (2016) · Physics Reports · doi:10.1016/j.physrep.2016.06.008 · Cited by 790 · see citing works
Natural Thermal and Magnetic Entanglement in the 1D Heisenberg Model
Arnesen, M. C., Bose, S., & Vedral, V. (2001) · Physical Review Letters · doi:10.1103/physrevlett.87.017901 · Cited by 787 · see citing works
Entanglement in continuous-variable systems: recent advances and current perspectives
Adesso, G., & Illuminati, F. (2007) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/40/28/s01 · Cited by 728 · see citing works
Geometric measure of entanglement and applications to bipartite and multipartite quantum states
Wei, T.-C., & Goldbart, P. M. (2003) · Physical Review A · doi:10.1103/physreva.68.042307 · Cited by 744 · see citing works
General Monogamy Inequality for Bipartite Qubit Entanglement
Osborne, T. J., & Verstraete, F. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.220503 · Cited by 733 · see citing works
Quantum information and relativity theory
Peres, A., & Terno, D. R. (2004) · Reviews of Modern Physics · doi:10.1103/revmodphys.76.93 · Cited by 696 · see citing works
Generation of Hyperentangled Photon Pairs
Barreiro, J. T., Langford, N. K., Peters, N. A., & Kwiat, P. G. (2005) · Physical Review Letters · doi:10.1103/physrevlett.95.260501 · Cited by 750 · see citing works
Quantum metrology from a quantum information science perspective
Tóth, G., & Apellaniz, I. (2014) · Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) · Cited by 672 · see citing works
Four qubits can be entangled in nine different ways
Verstraete, F., Dehaene, J., Moor, B. D., & Verschelde, H. (2002) · Physical Review A · doi:10.1103/physreva.65.052112 · Cited by 661 · see citing works
Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications
Reid, M. D., Drummond, P. D., Bowen, W. P., Cavalcanti, E. G., Lam, P. K., Bachor, H., Andersen, U. L., & Leuchs, G. (2009) · Reviews of Modern Physics · doi:10.1103/revmodphys.81.1727 · Cited by 650 · see citing works
Quantum Open System Theory: Bipartite Aspects
Yu, T., & Eberly, J. H. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.140403 · Cited by 632 · see citing works
Quantum magnonics: When magnon spintronics meets quantum information science
Yuan, H. Y., Cao, Y., Kamra, A., Duine, R. A., & Yan, P. (2022) · Physics Reports · doi:10.1016/j.physrep.2022.03.002 · Cited by 623 · see citing works
Vidal, G., & Tarrach, R. (1999) · Physical Review A · doi:10.1103/physreva.59.141 · Cited by 601 · see citing works
FIDELITY APPROACH TO QUANTUM PHASE TRANSITIONS
GU, S.-J. (2010) · International Journal of Modern Physics B · doi:10.1142/s0217979210056335 · Cited by 663 · see citing works
Ultrabright source of entangled photon pairs
Dousse, A., Suffczyński, J., Beveratos, A., Krebs, O., Lemaître, A., Sagnes, I., Bloch, J., Voisin, P., & Senellart, P. (2010) · Nature · doi:10.1038/nature09148 · Cited by 624 · see citing works
Measurement-induced entanglement for excitation stored in remote atomic ensembles
Chou, C. W., de Riedmatten, H., Felinto, D., Polyakov, S. V., van Enk, S. J., & Kimble, H. J. (2005) · Nature · doi:10.1038/nature04353 · Cited by 590 · see citing works
Liu, B.-H., Li, L., Huang, Y.-F., Li, C.-F., Guo, G.-C., Laine, E.-M., Breuer, H.-P., & Piilo, J. (2011) · Nature Physics · doi:10.1038/nphys2085 · Cited by 574 · see citing works
Entanglement in the quantum Heisenberg XY model
Wang, X. (2001) · Physical Review A · doi:10.1103/physreva.64.012313 · Cited by 576 · see citing works
Quantum repeaters: From quantum networks to the quantum internet
Azuma, K., Economou, S. E., Elkouss, D., Hilaire, P., Jiang, L., Lo, H., & Tzitrin, I. (2023) · Reviews of Modern Physics · doi:10.1103/revmodphys.95.045006 · Cited by 578 · see citing works
Classical correlation and quantum discord in critical systems
Sarandy, M. S. (2009) · Physical Review A · doi:10.1103/physreva.80.022108 · Cited by 545 · see citing works
Simple Test for Hidden Variables in Spin-1 Systems
Klyachko, A. A., Can, M. A., Binicioǧlu, S., & Shumovsky, A. S. (2008) · Physical Review Letters · doi:10.1103/physrevlett.101.020403 · Cited by 537 · see citing works
Robustness of quantum discord to sudden death
Werlang, T., Souza, S., Fanchini, F. F., & Villas-Bôas, C. J. (2009) · Physical Review A · doi:10.1103/physreva.80.024103 · Cited by 544 · see citing works
Entanglement of Two Qubits Mediated by One-Dimensional Plasmonic Waveguides
González-Tudela, A., Martín-Cano, D., Moreno, E., et al. (2011) · Physical Review Letters · doi:10.1103/physrevlett.106.020501 · Cited by 544 · see citing works
Fujii, K. (2009) · Institute of Electrical and Electronics Engineers eBooks · doi:10.1109/9780470544907.ch24 · Cited by 642 · see citing works
<i>Colloquium</i>: Quantum networks with trapped ions
Duan, L.-M., & Monroe, C. (2010) · Reviews of Modern Physics · doi:10.1103/revmodphys.82.1209 · Cited by 530 · see citing works
Mapping photonic entanglement into and out of a quantum memory
Choi, K. S., Deng, H., Laurat, J., & Kimble, H. J. (2008) · Nature · doi:10.1038/nature06670 · Cited by 530 · see citing works
Cavity-loss-induced generation of entangled atoms
Plenio, M. B., Huelga, S. F., Beige, A., & Knight, P. L. (1999) · Physical Review A · doi:10.1103/physreva.59.2468 · Cited by 526 · see citing works
Quantum Process Tomography of a Controlled-NOT Gate
O'Brien, J. L., Pryde, G. J., Gilchrist, A., James, D. F. V., Langford, N. K., Ralph, T. C., & White, A. G. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.080502 · Cited by 499 · see citing works
Protecting Entanglement via the Quantum Zeno Effect
Maniscalco, S., Francica, F., Zaffino, R. L., Gullo, N. L., & Plastina, F. (2008) · Physical Review Letters · doi:10.1103/physrevlett.100.090503 · Cited by 495 · see citing works
Protecting entanglement from decoherence using weak measurement and quantum measurement reversal
Kim, Y., Lee, J.-C., 권오성, & Kim, Y.-H. (2011) · Nature Physics · doi:10.1038/nphys2178 · Cited by 488 · see citing works
Entanglement measures under symmetry
Vollbrecht, K. G. H., & Werner, R. F. (2001) · Physical Review A · doi:10.1103/physreva.64.062307 · Cited by 485 · see citing works
Quantum Phase Transitions and Bipartite Entanglement
Wu, L.-A., Sarandy, M. S., & Lidar, D. A. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.250404 · Cited by 481 · see citing works
Quantum communication through spin chain dynamics: an introductory overview
Bose, S. (2007) · Contemporary Physics · doi:10.1080/00107510701342313 · Cited by 484 · see citing works
Entanglement and the Phase Transition in Single-Mode Superradiance
Lambert, N., Emary, C., & Brandes, T. (2004) · Physical Review Letters · doi:10.1103/physrevlett.92.073602 · Cited by 479 · see citing works
Entanglement of Dirac fields in noninertial frames
Alsing, P. M., Fuentes, I., Mann, R. B., & Tessier, T. E. (2006) · Physical Review A · doi:10.1103/physreva.74.032326 · Cited by 483 · see citing works
Research on hidden variable theories: A review of recent progresses
Genovese, M. (2005) · Physics Reports · doi:10.1016/j.physrep.2005.03.003 · Cited by 462 · see citing works
Non-Markovian dynamics of quantum discord
Fanchini, F. F., Werlang, T., Brasil, C. A., Arruda, L. G. E., & Caldeira, A. O. (2010) · Physical Review A · doi:10.1103/physreva.81.052107 · Cited by 464 · see citing works
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Dakić, B., Lipp, Y. O., Ma, X., Ringbauer, M., Kropatschek, S., Barz, S., Paterek, T., Vedral, V., Zeilinger, A., Brukner, Č., et al. (2012) · Nature Physics · doi:10.1038/nphys2377 · Cited by 463 · see citing works
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Yeo, Y., & Chua, W. K. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.060502 · Cited by 453 · see citing works
Bengtsson, I., & Życzkowski, K. (2006) · doi:10.2277/0521814510 · Cited by 502 · see citing works
Quantum communication through spin chain dynamics: an introductory overview
Bose, S. (2007) · UCL Discovery (University College London) · Cited by 443 · see citing works
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Werlang, T., Trippe, C., Ribeiro, G. A. P., & Rigolin, G. (2010) · Physical Review Letters · doi:10.1103/physrevlett.105.095702 · Cited by 443 · see citing works
Optimal creation of entanglement using a two-qubit gate
Kraus, B., & Cirac, J. I. (2001) · Physical Review A · doi:10.1103/physreva.63.062309 · Cited by 432 · see citing works
Experimental investigation of classical and quantum correlations under decoherence
Xu, J., Xu, X., Li, C., Zhang, C., Zou, X., & Guo, G. (2010) · Nature Communications · doi:10.1038/ncomms1005 · Cited by 434 · see citing works
Coherent and collective quantum optical effects in mesoscopic systems
Brandes, T. (2005) · Physics Reports · doi:10.1016/j.physrep.2004.12.002 · Cited by 430 · see citing works
Induced measures in the space of mixed quantum states
Życzkowski, K., & Sommers, H.-J. (2001) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/34/35/335 · Cited by 427 · see citing works
Fidelity and concurrence of conjugated states
Uhlmann, A. (2000) · Physical Review A · doi:10.1103/physreva.62.032307 · Cited by 421 · see citing works
Quantum discord and quantum phase transition in spin chains
Dillenschneider, R. (2008) · Physical Review B · doi:10.1103/physrevb.78.224413 · Cited by 420 · see citing works
Measures and dynamics of entangled states
Mintert, F., Carvalho, A. R. R., Kuś, M., & Buchleitner, A. (2005) · Physics Reports · doi:10.1016/j.physrep.2005.04.006 · Cited by 420 · see citing works
Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)
Chitambar, E., Leung, D., Mančinska, L., Ozols, M., & Winter, A. (2014) · Communications in Mathematical Physics · doi:10.1007/s00220-014-1953-9 · Cited by 409 · see citing works
Quantum coherence and geometric quantum discord
Hu, M.-L., Hu, X., Wang, J., Peng, Y., Zhang, Y.-R., & Fan, H. (2018) · Physics Reports · doi:10.1016/j.physrep.2018.07.004 · Cited by 434 · see citing works
Intrinsic randomness as a measure of quantum coherence
Yuan, X., Zhou, H., Cao, Z., & Ma, X. (2015) · Physical Review A · doi:10.1103/physreva.92.022124 · Cited by 400 · see citing works
The Dicke model in quantum optics: Dicke model revisited
Garraway, B. M. (2011) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2010.0333 · Cited by 397 · see citing works
Spatial correlations in parametric down-conversion
Walborn, S. P., Monken, C. H., Pádua, S., & Ribeiro, P. H. S. (2010) · Physics Reports · doi:10.1016/j.physrep.2010.06.003 · Cited by 393 · see citing works
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Carvalho, A. R. R., Mintert, F., & Buchleitner, A. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.230501 · Cited by 400 · see citing works
Violation of local uncertainty relations as a signature of entanglement
Hofmann, H. F., & Takeuchi, S. (2003) · Physical Review A · doi:10.1103/physreva.68.032103 · Cited by 399 · see citing works
Quantum Correlations in Systems of Indistinguishable Particles
Eckert, K., Schliemann, J., Bruß, D., & Lewenstein, M. (2002) · Annals of Physics · doi:10.1006/aphy.2002.6268 · Cited by 388 · see citing works
Quantum correlations in two-fermion systems
Schliemann, J., Cirac, J. I., Kuś, M., Lewenstein, M., & Loss, D. (2001) · Physical Review A · doi:10.1103/physreva.64.022303 · Cited by 393 · see citing works
Entangling Macroscopic Diamonds at Room Temperature
Lee, K. C., Sprague, M., Sussman, B., Nunn, J., Langford, N. K., Jin, X., Champion, T. F. M., Michelberger, P., Reim, K. F., England, D. G., et al. (2011) · Science · doi:10.1126/science.1211914 · Cited by 393 · see citing works
Entanglement certification from theory to experiment
Friis, N., Vitagliano, G., Malik, M., & Huber, M. (2018) · Nature Reviews Physics · doi:10.1038/s42254-018-0003-5 · Cited by 378 · see citing works
- Authors
- Wootters, W. K.
- Affiliations
- Williams College (US)
- Corresponding author
- Wootters, William K.
- ORCID
- Wootters, William K.
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1998-03-09
- Volume
- 80
- Issue
- 10
- Pages
- 2245-2248
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2011208902
- Cited by
- 8,290
- References
- 21
- Citations by year
- 2026: 320 · 2025: 387 · 2024: 339 · 2023: 340 · 2022: 349 · 2021: 371 · 2020: 328 · 2019: 336 · 2018: 325 · 2017: 301 · 2016: 406 · 2015: 339 · 2014: 381 · 2013: 368 · 2012: 381
- FWCI
- 62.08
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- entanglement of formation, two-qubit states, mixed states, density matrix, quantum entanglement
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