Virtual Cloning of Quantum States

Jing-Tao Qiu; Xiao-Dong Yu; Zhi-Hao Bi

Virtual Cloning of Quantum States

T0 review · 0 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A virtual cloning operation exists exactly for linearly independent sets of quantum states, and for two pure states the optimal 1-to-2 cost is $\sqrt{1+|\langle\psi_1|\psi_2\rangle|^2}$.

desk verdict A clean, correct result: linear independence characterizes virtual cloning, with an exact pure-state cost formula; the paper deserves serious refereeing. read the letter →

arxiv 2507.17279 v1 pith:5DPGLNI5 submitted 2025-07-23 quant-ph

Zhi-Hao Bi , Jing-Tao Qiu , Xiao-Dong Yu This is my paper · ORCID

classification quant-ph MSC 81P6881P1590C22 PACS 03.67.-a

keywords virtualquantumoperationsHPTPmapsno-cloningtheoremstatecloningsemidefiniteprogrammingdiscriminationsimulationcostlinearindependence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to show that the quantum no-cloning barrier disappears once operations are allowed to be virtual (Hermitian-preserving, trace-preserving maps rather than completely positive maps). Its central result is a criterion: a virtual cloning operation that sends each state $\rho_i$ to $\rho_i \otimes \rho_i$ exists exactly when the states are linearly independent, so any two distinct states — orthogonal or not — are virtually clonable. For two pure states the paper derives the optimal simulation cost, the sampling overhead needed to implement the virtual map by random physical operations and post-processing, as $\eta = \sqrt{1+|\langle\psi_1|\psi_2\rangle|^2}$ for 1-to-2 cloning, with generalizations to $k$-to-$n$ cloning and general pure-state conversions. Because a virtual operation is used only to reproduce measurement statistics, not to produce physical copies, this bypasses no-cloning while preserving the linearity restriction that makes universal cloning impossible. If the results are right, cloning becomes a statistical simulation task with a precisely quantified overhead rather than a physically impossible transformation.

What carries the argument

The central object is an HPTP (Hermitian-preserving, trace-preserving) map, written in its Choi-matrix form and decomposed as $\tilde\Lambda = \lambda_+\Lambda_+ - \lambda_-\Lambda_-$ into two ordinary quantum operations $\Lambda_\pm$. The simulation cost $\eta = \lambda_+ + \lambda_-$ is the factor by which measurement variances and required sample counts inflate when the virtual map is realized by randomly sampling $\Lambda_\pm$ and post-processing outcomes by $\pm\eta$. The proof machinery is the Choi–Jamiołkowski isomorphism, which converts the optimization over clone maps into a semidefinite program (Eq. 13) whose dual, via Helstrom measurements, supplies the trace-norm lower bounds; a Vandermonde-matrix argument is the technical engine for the linear-independence criterion.

What would settle it

Solve the SDP in Eq. (13) for the two qubit states $|0\rangle$ and $|+\rangle$ to high precision; the paper predicts $\eta_{\rm opt}=\sqrt{1+|\langle0|+\rangle|^2}=\sqrt{3/2}\approx 1.2247$. A feasible virtual map with cost below that, or a dual feasible solution with objective above it, would disprove Theorem 2. For a linearly dependent set such as $|0\rangle\langle0|$, $|1\rangle\langle1|$, and $\mathbb{1}/2$, exhibiting any HPTP map satisfying Eq. (2) for all three would refute Theorem 1, which asserts that no such map exists.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that virtual quantum operations—linear maps that preserve Hermiticity and trace but need not be positive—are exactly capable of cloning any linearly independent set of quantum states. Theorem 1 states that a virtual operation $\tilde\Lambda$ with $\tilde\Lambda(\rho_i)=\rho_i\otimes\rho_i$ exists if and only if $\{\rho_i\}$ is linearly independent; the proof builds a basis from the states when possible and uses a Vandermonde-matrix argument to show that a linear dependence among inputs would force a linear dependence among tensor powers, contradicting distinctness. Theorem 2 solves the optimal 1-to-2 cost for pure states $|\psi_1\rangle, |\psi_2\rangle$ as $\eta = \sqrt{1+|\langle\psi_1|\psi_2\rangle|^2}$, with the optimal map implementable by randomized unitary operations, and Lemma B1 generalizes this to arbitrary pure-state-pair conversion, where the cost is $\sqrt{(1-F')/(1-F)}$ when the output fidelity is lower. The paper also proves universal bounds relating the cloning cost to trace-norm distances, hence to optimal state-discrimination probabilities (Theorem 3).

Load-bearing premise

All quantitative claims depend on the assumption that the true implementation cost of a virtual operation is the minimum of $\lambda_+ + \lambda_-$ over two-term decompositions into ordinary quantum operations, as computed by the SDP from Ref. [27]; if more general decompositions or nonlinear post-processing could realize the same statistics more cheaply, the reported optimal costs would be upper bounds, while the existence criterion in Theorem 1 would stand.

Editorial extensions

If this is right

Any finite linearly independent set of pure or mixed states can have its two-copy (and hence $n$-copy) measurement statistics simulated exactly, with no disturbance to the inputs.
Two arbitrary distinct pure states can be cloned virtually with cost $\sqrt{1+|\langle\psi_1|\psi_2\rangle|^2}$ for 1-to-2 and $\sqrt{(1-|\langle\psi_1|\psi_2\rangle|^{2n})/(1-|\langle\psi_1|\psi_2\rangle|^2)}$ for 1-to-$n$, which stays finite as $n\to\infty$.
For any clonable set the optimal cost is obtained by solving an SDP, so in principle the protocol is computable and the overhead is certified by the SDP dual.
A finite number of input copies (at most $m-1$) always makes a set of $m$ states linearly independent, so $k\to n$ virtual cloning is possible for arbitrary finite sets, with $n$ arbitrarily large.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An unstated corollary is that virtual cloning supplies a generic way to estimate nonlinear functions of a state, such as $\mathrm{Tr}(\rho^2)$, by simulating the statistics of $\rho\otimes\rho$; this connects the framework to entanglement certification and error mitigation.
The exact pure-state formula suggests a resource-theoretic reading: pairwise distinguishability (through the trace-norm ratio) quantifies the simulation overhead, so for more than two states the optimal cost might be governed by the hardest pair in the set.
A natural testable extension is virtual cloning of quantum gates: the same linear-algebra criterion should carry over to transfer matrices of processes, with the SDP formulation replacing states by process matrices.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version: this paper is solid and deserves a real referee. The main result is a clean necessary and sufficient condition—a set of states can be virtually cloned iff they are linearly independent—and for a pair of pure states the optimal 1-to-2 cloning cost is sqrt(1+|⟨ψ1|ψ2⟩|^2). That's new relative to the virtual-broadcasting papers, which did not isolate this cloning characterization or give the exact cost formula. The paper also formulates the optimization as an SDP and derives universal bounds for mixed states via a state-discrimination connection. The proofs are generally rigorous; I checked the Vandermonde argument for Theorem 1 and the Choi-matrix construction in Appendix B, and they hold. The lower-bound construction in Theorem 3 is a bit terse—'one can easily verify'—but the verification is straightforward and correct.

The main caveat is that the quantitative cost claims are tied to the specific cost model from Jiang, Wang, and Wang: decompositions of the virtual map into two CPTP maps, with cost λ+ + λ−. Within that framework, everything works. If someone later finds a cheaper multi-term decomposition or nonlinear strategy, the numbers would change, but that is a framework-level caveat, not a flaw in this paper. The existence criterion in Theorem 1 is framework-independent, and that is the headline result.

I would send this to a serious referee. It is a meaningful advance in the theory of virtual quantum operations, with clean statements and a useful connection to state discrimination. The paper is also honest about its limitations, e.g., universal cloning remains impossible even virtually. I would cite this in my own work if I were working on virtual operations. Bring it to the reading group if you want a good example of HPTP maps doing something useful.

Referee Report

0 major / 5 minor

Summary. The paper introduces the notion of virtual cloning, in which a Hermitian-preserving trace-preserving (HPTP) map, implemented via quasiprobabilistic mixtures of CPTP maps, is used to clone a set of quantum states. The main results are: (i) a necessary and sufficient criterion for the existence of a virtual-cloning operation, namely that the states be linearly independent (Theorem 1); (ii) a semidefinite programming formulation of the optimal simulation cost (Observation 1); (iii) an exact formula for the optimal 1→n cloning cost of any two pure states, η = sqrt((1−|⟨ψ₁|ψ₂⟩|^{2n})/(1−|⟨ψ₁|ψ₂⟩|²)) (Theorem 2 and Eq. (15)); and (iv) universal lower and upper bounds for the optimal cost of cloning two mixed states in terms of their trace-norm distance, connecting the problem to state discrimination (Theorem 3). The proofs use standard tools: a Vandermonde argument for linear independence, the Choi isomorphism for the SDP, duality for lower bounds, and explicit constructions for upper bounds.

Significance. The results are a valuable contribution to the theory of virtual quantum operations. They give a clean, complete characterization of when perfect deterministic cloning is possible once complete positivity is relaxed to Hermitian preservation, and they provide exact optimal overheads for pure states. The connection to state discrimination is natural and yields interpretable bounds. The paper is rigorous: proofs are detailed, the SDP formulation is standard, and the lower-bound constructions are backed by explicit feasible points. The work is likely to stimulate further research on virtual protocols and their experimental implementation.

minor comments (5)

[Section II, proof of Theorem 1] The step 'Without loss of generality, we can always assume m = d²' needs a brief justification: one should explain that a linearly independent set of m < d² density operators can always be extended to a basis of the Hermitian operator space consisting of density operators, because the linear span of all density operators is the full Hermitian space. This is true, but it is not immediate from the text.
[Section IV, lower bound in Theorem 3] In the dual-feasible construction, the matrices M± are defined as ±[(P₊−P₋)(ρ₁−ρ₂)]ᵀ/‖ρ₁−ρ₂‖. For M± to be Hermitian, the measurement operators P± must be chosen as the spectral projectors of ρ₁−ρ₂ (or of p₁ρ₁−p₂ρ₂ in the prior-weighted case of Eq. (21)). Please state this explicitly, since the operator inequalities in Eq. (17) require Hermitian M±.
[Appendix B, Lemma B1] In the parametrization of J in Eq. (B6), the symmetry reductions are described only briefly. In particular, the derivation of the relations x = sqrt((1−F')/(1−F)) and z = sqrt(F'/F) − r/sqrt(F) from the first constraint in Eq. (B5) is omitted. Adding a few lines showing these calculations would make the proof easier to follow.
[Section IV, upper bound in Theorem 3] The verification that D̃ = η₊D₊ − η₋D₋ satisfies D̃(ρᵢ) = |i⟩⟨i| for i = 1, 2 is left as 'a direct calculation'. Including the short algebra (using the Helstrom relations) would improve the transparency of this construction.
[Section II and other typographical issues] There are a few typographical issues: in the proof of Theorem 1, the union of subspaces is written as 'S = S i̸=j{...}', which should be 'S = ⋃_{i≠j} {...}'. Please also check the notation around Eq. (B6) for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained; the only self-citation appears in the outlook and is not load-bearing.

full rationale

The paper's central results are derived from its stated definitions and standard external tools, not from the conclusions. Theorem 1 is a linear-algebra argument: linear independence gives a Hermitian basis, so an HPTP map is uniquely defined; the converse uses a Vandermonde matrix after choosing Y outside a finite union of hyperplanes. Observation 1 follows from the Choi isomorphism and the decomposition Lambda = lambda+ Lambda+ - lambda- Lambda-, with the SDP equivalence taken from an external reference [27] (Jiang, Wang, Wang), not from the authors' own prior work. Theorem 2 and Lemma B1 supply an explicit Choi matrix and explicit decomposition with cost xi = sqrt((1-F')/(1-F)), together with a matching lower bound via the trace-norm inequality; no parameter is fitted to the formula. Theorem 3's lower bound uses a dual-feasible solution built from Helstrom measurements, and its upper bound is an explicit construction D = eta+ D+ - eta- D-. The only self-citation, Ref. [53] by Xiao-Dong Yu et al., occurs in the concluding outlook as a possible application and carries no load in any proof. Thus there are no circular steps; the results are self-contained given the paper's explicitly framed HPTP cost model.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest only on standard quantum information tools and the chosen cost metric. There are no fitted parameters or ad hoc entities. The most load-bearing external input is the HPTP decomposition theorem from Ref. [27], which the paper cites but does not prove.

assumptions (5)

domain assumption Any HPTP map can be decomposed as Λ = λ+Λ+ - λ-Λ- with Λ± CPTP and λ± ≥ 0, with λ+ - λ- = 1.
Invoked in Eq. (6) and central to the definition of simulation cost and the SDP in Observation 1; cited to Ref. [27].
domain assumption The simulation cost η = λ+ + λ- is the correct measure of the sampling overhead of a virtual operation.
Defined in Section III based on the variance growth of the estimator and Hoeffding's inequality; this is the paper's chosen cost metric.
domain assumption Strong duality holds for the SDP in Eq. (13), so the dual program (17) attains the optimum.
Stated in the proof of Theorem 3, justified by Slater's condition.
standard math An invertible Vandermonde matrix with distinct nodes exists for any set of distinct density operators.
Used in the necessity proof of Theorem 1 and in Appendix A; standard linear algebra.
standard math The trace norm is monotone under positive trace-preserving maps.
Used in the introduction to rule out cloning under positive maps; cited to Ref. [41].

how reviews work

0 comments

Cite this review

Pith. "Pith review of Virtual Cloning of Quantum States." pith.science (2026). https://pith.science/paper/5DPGLNI5

@misc{pith2026250717279,
  author       = {Pith},
  title        = {Pith review of: Virtual Cloning of Quantum States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5DPGLNI5}},
  note         = {Machine review of arXiv:2507.17279}
}

read the original abstract

The inherent limitations of physical processes prevent the copying of arbitrary quantum states. Furthermore, even if we only aim to clone two distinct quantum states, it remains impossible unless they are mutually orthogonal. To overcome this limitation, we propose a virtual-cloning protocol that bypasses the restrictions imposed by the quantum no-cloning theorem. Specifically, we begin by outlining the general framework for virtual cloning and deriving a necessary and sufficient criterion for the existence of a virtual operation capable of simultaneously cloning a set of states. Subsequently, through an analysis of the simulation cost of the virtual-cloning process, we demonstrate that the problem of identifying an optimal virtual-cloning protocol can be cast as a semidefinite programming problem. Finally, we establish a connection between virtual cloning and state discrimination, from which universal bounds on the optimal cloning cost are derived.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 53 canonical work pages

[1]

W. K. Wootters and W. H. Zurek, A single quantum can- not be cloned, Nature 299, 802 (1982)

work page 1982
[2]

Dieks, Communication by EPR devices, Phys

D. Dieks, Communication by EPR devices, Phys. Lett. A 92, 271 (1982)

work page 1982
[3]

Scarani, S

V . Scarani, S. Iblisdir, N. Gisin, and A. Acín, Quantum cloning, Rev. Mod. Phys. 77, 1225 (2005)

work page 2005
[4]

Fan, Y.-N

H. Fan, Y.-N. Wang, L. Jing, J.-D. Yue, H.-D. Shi, Y.-L. Zhang, and L.-Z. Mu, Quantum cloning machines and the applications, Phys. Rep. 544, 241 (2014)

work page 2014
[5]

Bužek and M

V . Bužek and M. Hillery, Quantum copying: Beyond the no-cloning theorem, Phys. Rev. A 54, 1844 (1996)

work page 1996
[6]

Gisin and S

N. Gisin and S. Massar, Optimal quantum cloning ma- chines, Phys. Rev. Lett. 79, 2153 (1997)

work page 1997
[7]

Bužek and M

V . Bužek and M. Hillery, Universal optimal cloning of arbitrary quantum states: From qubits to quantum regis- ters, Phys. Rev. Lett. 81, 5003 (1998)

work page 1998
[8]

Niu and R

C.-S. Niu and R. B. Griffiths, Optimal copying of one quantum bit, Phys. Rev. A 58, 4377 (1998)

work page 1998

Show all 56 references

[9]

N. J. Cerf, Pauli cloning of a quantum bit, Phys. Rev. Lett. 84, 4497 (2000)

2000
[10]

Duan and G.-C

L.-M. Duan and G.-C. Guo, Probabilistic cloning and identification of linearly independent quantum states, Phys. Rev. Lett. 80, 4999 (1998)

1998
[11]

A. K. Pati, Quantum superposition of multiple clones and the novel cloning machine, Phys. Rev. Lett. 83, 2849 (1999)

1999
[12]

D. Bruß, M. Cinchetti, G. Mauro D’Ariano, and C. Mac- chiavello, Phase-covariant quantum cloning, Phys. Rev. A 62, 012302 (2000)

2000
[13]

H. Fan, K. Matsumoto, X.-B. Wang, and M. Wadati, Quantum cloning machines for equatorial qubits, Phys. Rev. A 65, 012304 (2001)

2001
[14]

Niu and R

C.-S. Niu and R. B. Griffiths, Two-qubit copying machine for economical quantum eavesdropping, Phys. Rev. A60, 2764 (1999)

1999
[15]

Bruß and C

D. Bruß and C. Macchiavello, Optimal eavesdropping in cryptography with three-dimensional quantum states, Phys. Rev. Lett. 88, 127901 (2002)

2002
[16]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, Se- curity of quantum key distribution using d-level systems, Phys. Rev. Lett. 88, 127902 (2002)

2002
[17]

D. Bruß, A. Ekert, and C. Macchiavello, Optimal uni- versal quantum cloning and state estimation, Phys. Rev. Lett. 81, 2598 (1998)

1998
[18]

Brougham, E

T. Brougham, E. Andersson, and S. M. Barnett, Cloning and joint measurements of incompatible components of spin, Phys. Rev. A 73, 062319 (2006)

2006
[19]

Sagawa and M

T. Sagawa and M. Ueda, Accuracy matrix in a general- ized simultaneous measurement of a qubit system, Phys. Rev. A 77, 012313 (2008)

2008
[20]

G. S. Thekkadath, R. Y. Saaltink, L. Giner, and J. S. Lundeen, Determining complementary properties with quantum clones, Phys. Rev. Lett. 119, 050405 (2017)

2017
[21]

Bruß and C

D. Bruß and C. Macchiavello, On the entanglement struc- ture in quantum cloning, Found. Phys. 33, 1617 (2003)

2003
[22]

Lamoureux, P

L.-P . Lamoureux, P . Navez, J. Fiurášek, and N. J. Cerf, Cloning the entanglement of a pair of quantum bits, Phys. Rev. A 69, 040301 (2004)

2004
[23]

Weedbrook, N

C. Weedbrook, N. B. Grosse, T. Symul, P . K. Lam, and T. C. Ralph, Quantum cloning of continuous-variable en- tangled states, Phys. Rev. A 77, 052313 (2008)

2008
[24]

H. P . Yuen, Amplification of quantum states and noise- less photon amplifiers, Phys. Lett. A 113, 405 (1986)

1986
[25]

Barnum, C

H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Noncommuting mixed states cannot be broadcast, Phys. Rev. Lett. 76, 2818 (1996)

1996
[26]

Temme, S

K. Temme, S. Bravyi, and J. M. Gambetta, Error mitiga- tion for short-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017)

2017
[27]

Jiang, K

J. Jiang, K. Wang, and X. Wang, Physical implementabil- ity of linear maps and its application in error mitigation, Quantum 5, 600 (2021)

2021
[28]

Takagi, S

R. Takagi, S. Endo, S. Minagawa, and M. Gu, Fundamen- tal limits of quantum error mitigation, npj Quantum Inf. 8, 114 (2022)

2022
[29]

A. J. Parzygnat, J. Fullwood, F. Buscemi, and G. Chiri- bella, Virtual quantum broadcasting, Phys. Rev. Lett.132, 110203 (2024)

2024
[30]

H. Yao, X. Liu, C. Zhu, and X. Wang, Optimal unilocal virtual quantum broadcasting, Phys. Rev. A 110, 012458 (2024)

2024
[31]

Y. Xiao, X. Liu, and Z. Liu, No practical quantum broad- casting: Even virtually, arXiv:2503.16380
[32]

X. Yuan, B. Regula, R. Takagi, and M. Gu, Virtual quan- tum resource distillation, Phys. Rev. Lett. 132, 050203 (2024)

2024
[33]

Takagi, X

R. Takagi, X. Yuan, B. Regula, and M. Gu, Virtual quan- tum resource distillation: General framework and appli- cations, Phys. Rev. A 109, 022403 (2024)

2024
[34]

Buscemi, M

F. Buscemi, M. Dall’Arno, M. Ozawa, and V . Vedral, Di- rect observation of any two-point quantum correlation function, arXiv:1312.4240

arXiv
[35]

Buscemi, M

F. Buscemi, M. Dall’Arno, M. Ozawa, and V . Vedral, Uni- versal optimal quantum correlator, Int. J. Quantum Inf. 12, 1560002 (2014)

2014
[36]

C. Zhu, Y. Mo, Y.-A. Chen, and X. Wang, Reversing un- known quantum processes via virtual combs for channels with limited information, Phys. Rev. Lett. 133, 030801 (2024)

2024
[37]

Rossini, D

M. Rossini, D. Maile, J. Ankerhold, and B. I. C. Donvil, Single-qubit error mitigation by simulating non- Markovian dynamics, Phys. Rev. Lett. 131, 110603 (2023)

2023
[38]

Zhang, Y

T. Zhang, Y. Zhang, L. Liu, X.-X. Fang, Q.-X. Zhang, X. Yuan, and H. Lu, Experimental virtual distillation of entanglement and coherence, Phys. Rev. Lett. 132, 180201 (2024)

2024
[39]

Zheng, X

Y. Zheng, X. Nie, H. Liu, Y. Luo, D. Lu, and X. Liu, Exper- imental virtual quantum broadcasting, arXiv: 2501.11390

arXiv
[40]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed. (Cambridge University Press, Cambridge, 2010)

2010
[41]

M. B. Ruskai, Beyond strong subadditivity? Improved bounds on the contraction of generalized relative en- tropy, Rev. Math. Phys. 06, 1147 (1994)

1994
[42]

The case n = 0 is obtained by taking the trace of Eq. ( 3)
[43]

R. A. Horn and C. R. Johnson, Matrix Analysis (Cam- bridge University Press, Cambridge, 1985)

1985
[44]

For the general observable X, the results are similar, but the influence of the factor η may be state dependent
[45]

Hoeffding, Probability inequalities for sums of bounded random variables, J

W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Am. Stat. Assoc. 58, 13 (1963). 9

1963
[46]

Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep

A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 275 (1972)

1972
[47]

Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl 10, 285 (1975)

M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl 10, 285 (1975)

1975
[48]

C. W. Helstrom, Quantum detection and estimation the- ory, J. Stat. Phys. 1, 231 (1969)

1969
[49]

Watrous, The Theory of Quantum Information (Cam- bridge University Press, Cambridge, 2018)

J. Watrous, The Theory of Quantum Information (Cam- bridge University Press, Cambridge, 2018)

2018
[50]

Boyd and L

S. Boyd and L. Vandenberghe, Convex Optimization (Cam- bridge University Press, Cambridge, 2004)

2004
[51]

Eisert, D

J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Quantum certifi- cation and benchmarking, Nat. Rev. Phys. 2, 382 (2020)

2020
[52]

Elben, R

A. Elben, R. Kueng, H.-Y. R. Huang, R. van Bijnen, C. Kokail, M. Dalmonte, P . Calabrese, B. Kraus, J. Preskill, P . Zoller, and B. Vermersch, Mixed-state entanglement from local randomized measurements, Phys. Rev. Lett. 125, 200501 (2020)

2020
[53]

X.-D. Yu, S. Imai, and O. Gühne, Optimal entanglement certification from moments of the partial transpose, Phys. Rev. Lett. 127, 060504 (2021)

2021
[54]

Chiribella, G

G. Chiribella, G. M. D’Ariano, and P . Perinotti, Optimal cloning of unitary transformation, Phys. Rev. Lett. 101, 180504 (2008)

2008
[55]

W. Dür, P . Sekatski, and M. Skotiniotis, Deterministic su- perreplication of one-parameter unitary transformations, Phys. Rev. Lett. 114, 120503 (2015)

2015
[56]

Chiribella, Y

G. Chiribella, Y. Yang, and C. Huang, Universal super- replication of unitary gates, Phys. Rev. Lett. 114, 120504 (2015)

2015

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

W. K. Wootters and W. H. Zurek, A single quantum can- not be cloned, Nature 299, 802 (1982)

work page 1982

[2] [2]

Dieks, Communication by EPR devices, Phys

D. Dieks, Communication by EPR devices, Phys. Lett. A 92, 271 (1982)

work page 1982

[3] [3]

Scarani, S

V . Scarani, S. Iblisdir, N. Gisin, and A. Acín, Quantum cloning, Rev. Mod. Phys. 77, 1225 (2005)

work page 2005

[4] [4]

Fan, Y.-N

H. Fan, Y.-N. Wang, L. Jing, J.-D. Yue, H.-D. Shi, Y.-L. Zhang, and L.-Z. Mu, Quantum cloning machines and the applications, Phys. Rep. 544, 241 (2014)

work page 2014

[5] [5]

Bužek and M

V . Bužek and M. Hillery, Quantum copying: Beyond the no-cloning theorem, Phys. Rev. A 54, 1844 (1996)

work page 1996

[6] [6]

Gisin and S

N. Gisin and S. Massar, Optimal quantum cloning ma- chines, Phys. Rev. Lett. 79, 2153 (1997)

work page 1997

[7] [7]

Bužek and M

V . Bužek and M. Hillery, Universal optimal cloning of arbitrary quantum states: From qubits to quantum regis- ters, Phys. Rev. Lett. 81, 5003 (1998)

work page 1998

[8] [8]

Niu and R

C.-S. Niu and R. B. Griffiths, Optimal copying of one quantum bit, Phys. Rev. A 58, 4377 (1998)

work page 1998