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Quantum Cloning Machines and the Applications

Heng Fan, Yi-Nan Wang, Li Jing, Jie-Dong Yue, Han-Duo Shi, Yong-Liang Zhang, Liang-Zhu Mu

arXiv:1301.2956v4quant-ph

TL;DR

Because quantum states cannot be perfectly cloned deterministically, the review examines how approximate or probabilistic cloning can achieve useful copying fidelity. It synthesizes cloning machines and their applications, including links between optimal machines and QKD state sets, while identifying unresolved minimal-input questions in higher dimensions.

  • Problem

    Given that arbitrary states cannot be cloned perfectly, the review asks what fidelity approximate cloning can achieve across qubit and higher-dimensional states.

    Source: §III · p. 18

  • Method

    The review surveys quantum cloning machines, related topics, and detailed formulations of major results and applications.

    Source: §I · p. 3

  • Results

    Phase-covariant cloning is suited to BB84-like states, while the universal machine optimally clones the six-state set and arbitrary qubits.

    Source: §I · p. 3, H. Minimal input set, six-state… · p. 30

  • Takeaways & Limitations

    Cloning machines connect QKD protocol state sets with optimal cloning and also have applications in quantum-state or phase estimation.

    Source: §I · p. 3

    The minimal input sets for higher-dimensional phase-covariant cloning and universal cloning remain unclear.

    Source: H. Symmetry condition and minimal sets… · p. 48

Abstract

from arXiv · show

No-cloning theorem is fundamental for quantum mechanics and for quantum information science that states an unknown quantum state cannot be cloned perfectly. However, we can try to clone a quantum state approximately with the optimal fidelity, or instead, we can try to clone it perfectly with the largest probability. Thus various quantum cloning machines have been designed for different quantum information protocols. Specifically, quantum cloning machines can be designed to analyze the security of quantum key distribution protocols such as BB84 protocol, six-state protocol, B92 protocol and their generalizations. Some well-known quantum cloning machines include universal quantum cloning machine, phase-covariant cloning machine, the asymmetric quantum cloning machine and the probabilistic quantum cloning machine etc. In the past years, much progress has been made in studying quantum cloning machines and their applications and implementations, both theoretically and experimentally. In this review, we will give a complete description of those important developments about quantum cloning and some related topics. On the other hand, this review is self-consistent, and in particular, we try to present some detailed formulations so that further study can be taken based on those results.

I. INTRODUCTION

The introduction motivates quantum cloning through the impossibility of perfect quantum copying and its roles in quantum information and QKD. It presents the review’s scope, foundational background, and two proof approaches to the no-cloning theorem.

A. Quantum information, qubit and quantum entanglement

  • Quantum information encodes information in qubits, but the no-cloning theorem forbids perfect copying of arbitrary quantum states.Approximate or probabilistic cloning remains possible, and cloning machines are applied to QKD protocols including BB84, six-state, and B92.Source: §I · pp. 2–3

I. INTRODUCTION

  • The review surveys quantum cloning developments and related topics, with detailed formulations intended to support further study.It also notes that some important references may be missed.Source: §I · p. 3

B. Quantum gates

  • Single-qubit rotations and CNOT gates form a complete set of fundamental gates for universal quantum computation.The CNOT flips its target when the controlled qubit is |1⟩ and leaves it unchanged when the control is |0⟩.Source: B. Quantum gates · p. 7

A. A simple proof of no-cloning theorem

  • Linearity rules out a machine that perfectly and deterministically duplicates every pure state.The proof assumes such a machine and shows that its outputs for superposed inputs cannot agree.Source: A. A simple proof of no-cloning theorem · p. 8
  • A unitary cloning operation must preserve inner products, so it can copy only identical or orthogonal states.The inner-product argument is attributed to Yuen (1986), with a reference to Peres (1995).Source: A. A simple proof of no-cloning theorem · p. 8

B. No-broadcasting theorem

The review connects state broadcasting, correlation broadcasting, and local correlation broadcasting through conditions that characterize when each is possible. It presents their equivalence as a unified information-theoretic picture of no-broadcasting.

B. No-broadcasting theorem

  • A set of quantum states is broadcastable if and only if its states commute with each other.The review proves the two-state case using relative entropy and extends it to more complex cases.Source: B. No-broadcasting theorem · pp. 9–10
  • Under a weak assumption, some information about a quantum state can be broadcast.This qualifies the no-broadcasting discussion without claiming that the whole state can be broadcast.Source: B. No-broadcasting theorem · p. 10

C. No-broadcasting for correlations

  • Bipartite-state correlations are locally broadcastable by both parties if and only if the state is classical, and by party a if and only if it is classical-quantum.Local broadcasting by party a means an operation on that party produces two reduced states whose correlations with party b match the original amount.Source: C. No-broadcasting for correlations · p. 11

D. A unified no-cloning theorem from information theoretical point of view

  • The review establishes the equivalence of state broadcastability, local broadcastability of correlations, and local broadcastability by one party.This equivalence provides a unified information-theoretic picture of no-broadcasting in quantum systems.Source: D. A unified no-cloning theorem from… · pp. 11, 14

E. No-cloning and no-signaling

The review uses no-signaling to constrain quantum cloning, since perfect cloning of an entangled qubit could let Bob infer Alice’s measurement choice. It derives a tight fidelity bound for universal cloning and notes that the bound is attainable.

E. No-cloning and no-signaling

  • Perfect cloning of Bob’s entangled qubit would let him identify Alice’s measurement basis, enabling superluminal signaling; no-cloning rules out that scheme.Without Alice’s measurement result, Bob’s qubit density matrix does not depend on her measurement.Source: E. No-cloning and no-signaling · pp. 14–15
  • A linear, trace-preserving operation on Bob’s system leaves its density matrix unchanged by Alice’s measurement choice.The review uses this no-signaling condition to determine a cloning-fidelity limit.Source: E. No-cloning and no-signaling · p. 15
  • F = 5/6 is an attainable upper bound for the fidelity of 1→2 qubit universal cloning.The review states that the bound follows from no-signaling and is reached by a practical universal quantum cloning scheme.Source: E. No-cloning and no-signaling · p. 15

F. No-cloning for unitary operators

The review surveys no-cloning results and related developments, then describes universal quantum cloning machines for qubits and qudits. It presents optimality results for symmetric machines and their fidelity properties.

F. No-cloning for unitary operators

  • For unitary operators, CNOT copies a bit flip forward and a phase flip backward, so it cannot copy both operations simultaneously.The review relates these properties to quantum error correction and fault-tolerant quantum computation.Source: F. No-cloning for unitary operators · p. 16

G. Other developments and related topics

  • Related developments include classical broadcasting, no-cloning for entangled states, no-deleting and no-splitting, and applications in quantum cryptography and metrology.The review also discusses no-cloning connections to quantum channels, state distinguishability, and quantum capacity.Source: G. Other developments and related topics · pp. 16–17

III. UNIVERSAL QUANTUM CLONING MACHINES

  • A universal quantum cloning machine is designed to achieve equal fidelity for every input state, addressing the optimal fidelity achievable when the input is not known in advance.The review frames the question for qubits and for states in a d-dimensional Hilbert space.Source: §III · p. 18

A. Symmetric UQCM for qubit

  • The symmetric qubit UQCM is state-independent and optimal; the N-to-M machine is also proved optimal.The review establishes optimality of the N-to-M machine for N = 1, 2, …, 7, followed by a complete proof.Source: A. Symmetric UQCM for qubit · p. 19

B. Symmetric UQCM for qudit

  • The symmetric qudit UQCM is uniquely optimal, and its M→∞ single-copy fidelity agrees with the state-estimation result.For N = 1 and M = 2, its fidelity reduces to (d + 3)/(2d + 2).Source: B. Symmetric UQCM for qudit · pp. 20–21

C. Asymmetric quantum cloning

Asymmetric cloning distributes information unequally between copies, creating a trade-off between their fidelities. The review presents qubit and higher-dimensional formulations, extensions to multiple outputs, and a unified construction for asymmetric universal cloning.

C. Asymmetric quantum cloning

  • When a = 0, F_A = 1 and F_B = 1/2; when a^2 = b^2 = 1/3, both fidelities are 5/6.These cases illustrate the trade-off between concentrating information in one copy and symmetric cloning.Source: C. Asymmetric quantum cloning · p. 22
  • For d-dimensional cloning, the symmetric 1-to-2 case has fidelity (d + 3)/(2d + 2), and the asymmetric fidelities obey a trade-off relation.The trade-off inequality is optimal when saturated.Source: C. Asymmetric quantum cloning · p. 23
  • In the generalized parameterization, optimal cloning requires y = 0, and the parameters map to the earlier formalism as a = v − x and b = dx.The normalization is v^2 + 2(d − 1)x^2 + (d − 1)^2y^2 = 1.Source: C. Asymmetric quantum cloning · p. 24
  • Asymmetric constructions extend to multiple output groups, including 1-to-1+n and general N-to-M_A+M_B cloning.The 1-to-1+n construction is described as a generalization of symmetric UQCM; broader asymmetric cases are also studied (Iblisdir et al., 2005b).Source: C. Asymmetric quantum cloning · p. 24

D. A unified UQCM

  • Wang et al.’s unified construction replaces the symmetric operator with a linear combination of identity and permutation operators, yielding a six-parameter general asymmetric UQCM without requiring specific input positions.The resulting fidelity expressions coincide with the earlier results.Source: D. A unified UQCM · pp. 25–26
  • The final expression has exactly three independent parameters, denoted A, B, and C, corresponding respectively to α + β, γ + µ, and δ + ν.Source: D. A unified UQCM · p. 26

E. Singlet monogamy and optimal cloning

The review connects restrictions on sharing entanglement with asymmetric cloning, using singlet monogamy to describe limits on output fidelities. Its stated fidelity relation recovers the symmetric UQCM result when all outputs are treated equally and is consistent with earlier asymmetric cases.

E. Singlet monogamy and optimal cloning

  • The review uses singlet monogamy for restrictions on entanglement sharing and relates it to optimal asymmetric 1-to-N cloning.Singlet monogamy names these restrictions because the singlet is a maximally entangled state.Source: E. Singlet monogamy and optimal cloning · p. 27
  • For 1-to-N cloning, α_i parameters specify the required output asymmetry and satisfy Σ_i α_i = 1.The cloning output is defined using reduced density matrices for each output copy.Source: E. Singlet monogamy and optimal cloning · p. 27
  • In the symmetric UQCM case, equal p_0,i values recover F = (2N + d − 1)/[N(d + 1)], and earlier asymmetric cases are consistent with this approach.The equal value is p_0,i = (N + d − 1)/(dN).Source: E. Singlet monogamy and optimal cloning · p. 28

F. Mixed-state quantum cloning

This section reviews mixed-state quantum cloning and related topics, including approximate universal spin flipping, state sets that characterize universal cloning, and probabilistic schemes that trade certainty for perfect output on success. It also covers the existence of such a machine and a two-input example related to prior probabilistic cloning work and its extension to larger output families.

F. Mixed-state quantum cloning

  • Mixed-state cloning is assessed by input-independent single-copy shrinking factors, and the 2→3 construction attains 5/6, matching the pure-state optimum.The review also notes that direct symmetrization can alter two identical mixed inputs, while superbroadcasting can purify local outputs for N ≥4.Source: F. Mixed-state quantum cloning · pp. 28–29

G. Universal NOT gate

  • The universal NOT gate is approximate: a universal cloning machine constructs it, its output matches the UQCM ancilla as an anti-clone, and its optimality is proved.An exact universal NOT operation is excluded because the state-conjugation map is anti-unitary and not completely positive (Buˇzek et al., 1999).Source: G. Universal NOT gate · p. 29

H. Minimal input set, six-state cryptography and other results

  • The six states used in six-state QKD have optimal equal-fidelity 1→2 cloning equivalent to the qubit UQCM, while four tetrahedral states determine that machine.The six-state UQCM is also identified as the optimal attack on that protocol.Source: H. Minimal input set, six-state… · p. 30

I. Other developments and related topics

  • The review connects cloning with state estimation, incompatible-observable measurement, and QKD security, and surveys noisy implementations, proposed devices, and experiments.Implementation proposals include spin networks, optics, cavities, and ion traps; experiments include entangled parametric amplification and asymmetric cloning (De Chiara et al., 2004; De Martini et al., 2000; Zhao et al., 2005).Source: §I · pp. 31–33

IV. PROBABILISTIC QUANTUM CLONING

A. Probabilistic quantum cloning machine

  • For two linearly independent states, optimal probabilistic cloning succeeds with η = 1/(1 + |⟨Ψ0|Ψ1⟩|); such cloning is possible only for linearly independent inputs.The success outcome is heralded by an ancilla measurement; the Duan-Guo bound is expressed by positive semidefiniteness of X(1) − Γ.Source: A. Probabilistic quantum cloning machine · p. 34

B. A novel quantum cloning machine

  • Pati’s probabilistic scheme can produce a superposition of outputs with different copy counts for linearly independent inputs, but not when one input is a superposition of others.The Duan and Guo machine is the special case M = 1.Source: B. A novel quantum cloning machine · p. 35

C. Probabilistic quantum anti-cloning and NOT gate

  • Probabilistic anti-cloning and NOT schemes aim to preserve the input while producing its orthogonal state, with probe measurements indicating success.The review presents a 1→2 probabilistic anti-cloning and NOT gate.Source: C. Probabilistic quantum anti-cloning… · p. 35
  • Positive semidefiniteness of CC† guarantees that such a cloning machine always exists.The passage gives a simple example with n = 2, but its coefficient expression is incomplete in the supplied text.Source: C. Probabilistic quantum anti-cloning… · p. 36
  • The n = 2 example’s constraint on f matches the Duan and Guo case, which Li et al. (Li et al., 2007) extended to larger output families.The extension includes outputs containing |Ψ⟩|Ψ⊥⟩, |Ψ⟩|Ψ⊥⟩⊗2, …, |Ψ⟩|Ψ⊥⟩⊗M.Source: C. Probabilistic quantum anti-cloning… · p. 36

D. Other developments and related topics

The review connects state-dependent cloning to QKD security and compares cloning strategies for specified input ensembles. It also surveys probabilistic cloning and related developments, including theoretical proposals and experiments.

D. Other developments and related topics

  • Probabilistic cloning seeks perfect copies at a success-probability cost and is linked to quantum-state discrimination; the review also covers variants and implementation proposals.It reports experimental probabilistic cloning in an NMR system.Source: D. Other developments and related topics · p. 36

V. PHASE-COVARIANT AND STATE-DEPENDENT QUANTUM CLONING

  • For QKD security analysis, cloning models an eavesdropper keeping one copy and forwarding another, with the attack assessed by information gained and errors induced.The review motivates state-dependent machines because a universal machine may not be optimal for a protocol's states.Source: §V · p. 37

A. Quantum key distribution protocols

  • The review covers BB84, B92, and six-state QKD; the six-state protocol's optimal cloning machine is the universal quantum cloning machine.For higher-dimensional protocols, it also discusses mutually unbiased bases and a retrodiction protocol with 100% successful measurement versus 1/(g + 1) for protocols such as BB84, without quantum memory for Bob.Source: A. Quantum key distribution protocols · pp. 37–38

B. General state-dependent quantum cloning

C. Quantum cloning of two non-orthogonal states

D. Phase-covariant quantum cloning: economic quantum cloning for equatorial qubits

For equatorial qubits, the review identifies optimal BB84 cloning with phase-covariant cloning and derives optimal 1→M results. It also distinguishes economic and non-economic forms while noting that optimal general N→M cloning remains unresolved.

D. Phase-covariant quantum cloning: economic quantum cloning for equatorial qubits

E. One to many phase-covariant quantum cloning machine for equatorial qubits

G. Phase-covariant quantum cloning for qudits

The review treats phase-covariant cloning for qudits with fixed input amplitudes and arbitrary phases, then examines how symmetry identifies minimal input sets. It reports that three symmetric states determine the phase-covariant machine, while the corresponding minimal sets in higher dimensions remain open.

G. Phase-covariant quantum cloning for qudits

  • For qudits with fixed input amplitudes and arbitrary phases, the review optimizes 1→2 phase-covariant cloning fidelity under α^2 + β^2 = 1.The optimal fidelity and corresponding α, β values are given, though their expressions are not included in the supplied passages.Source: G. Phase-covariant quantum cloning for… · pp. 46–47
  • The optimal phase-covariant fidelity exceeds the corresponding universal-cloning fidelity and reduces to known results for d = 2 and d = 3.The review identifies the optimal qudit machine and fidelity with references (199, 204) and (203), respectively.Source: G. Phase-covariant quantum cloning for… · p. 47

H. Symmetry condition and minimal sets in determining quantum cloning machines

  • The four BB84 states suffice to determine phase-covariant cloning, but are not minimal; the minimal set has three states with the highest Bloch-sphere symmetry.The review characterizes the three-state minimal set as having the highest symmetry in the Bloch sphere.Source: H. Symmetry condition and minimal sets… · p. 47
  • For equatorial qubits, three symmetric states with phases 0, 2π/3, and 4π/3 determine the phase-covariant machine, whose fidelity is independent of phase.The review presents the three-state set as sufficient to determine the machine completely.Source: H. Symmetry condition and minimal sets… · pp. 47–48
  • The three-state set also determines 1→M phase-covariant cloning, and ancillary states do not increase cloning fidelity.The review says these conclusions follow by investigations similar to those for 1→2 cloning.Source: H. Symmetry condition and minimal sets… · p. 48
  • Which symmetries define minimal input sets for phase-covariant cloning or universal cloning in higher dimensions remains an open question.The review contrasts the known three-state symmetry in two dimensions with the unclear minimal sets for higher-dimensional systems.Source: H. Symmetry condition and minimal sets… · p. 48

I. Quantum cloning machines of arbitrary set of MUBs

This section examines quantum cloning under quantum-mechanical constraints, covering fidelity trade-offs for arbitrary sets of mutually unbiased bases, applications to mean-king retrodiction QKD, phase-covariant cloning developments, and local cloning and entanglement properties for entangled states.

I. Quantum cloning machines of arbitrary set of MUBs

  • For arbitrary sets of g+1 MUBs, the review derives Eve’s optimal fidelity at fixed Bob fidelity under a balanced-attack condition.The framework includes the two-basis and all d+1-basis cases; the optimal cloner for d+1 MUBs is equivalent to the universal quantum cloning machine.Source: §I · pp. 50–52
  • For d MUBs, the derived cloner coincides with phase-covariant cloning, including in the asymmetric case, while minimal input sets in general dimension remain unknown.The asymmetric fidelity agreement is supported by numerical evidence; the general-d minimal-input-set question is left open.Source: §I · pp. 52–53

J. Quantum cloning in mean king problem as a quantum key distribution protocol

  • Mean-king retrodiction QKD combines an entangled-pair step with a BB84-like retrodiction step, and Eve’s attacks can involve both channel preparation and cloning-based transmission.Bob has a 100% winning strategy in the stated qubit and higher-dimensional results.Source: J. Quantum cloning in mean king… · p. 53
  • At fixed Bob disturbance, mean-king retrodiction QKD gives Eve a lower probability of guessing correctly than BB84 or six-state QKD, and three bases outperform two.The comparison is shown for d=2 in FIG. 6.Source: J. Quantum cloning in mean king… · p. 53, Figure 6 · p. 54
  • Mean-king retrodiction generates a raw key on every run, whereas standard QKD using g+1 MUBs yields one on average every g+1 runs.Source: J. Quantum cloning in mean king… · p. 53

K. Other developments and related topics

  • The review reports phase-covariant cloning proposals for spin networks and implementations in optical systems, NMR, and diamond nitrogen-vacancy centers using three energy levels (Chiara et al., 2004; Chen et al., 2007; Pan et al., 2011).Source: K. Other developments and related topics · p. 55

VI. LOCAL CLONING OF ENTANGLED STATES, ENTANGLEMENT IN QUANTUM CLONING

  • Quantum cloning seeks operations that realize optimal cloning subject to the constraints of quantum mechanics.Source: §VI · p. 56
  • For local cloning of entangled states, the operations must additionally be local, and a no-cloning theorem for entangled states exists in principle.Source: §VI · p. 56
  • The review also studies entanglement properties in quantum cloning machines, motivated by entanglement’s crucial role in quantum information.Source: §VI · p. 56

A. Local cloning of Bell states

This section examines local cloning of selected entangled states using shared entanglement and local gates, alongside telecloning schemes for remotely distributed copies. It covers symmetric and economical telecloning, optimal phase-covariant fidelity and output states, and the entanglement resources for optimal qudit cloning.

A. Local cloning of Bell states

  • Alice and Bob can perfectly clone either shared Bell state with local CNOT gates, consuming one shared |Φ+⟩ resource, which is optimal because at least one resource state is required.The unknown state need not be identified before the two copies are produced.Source: A. Local cloning of Bell states · pp. 56–57

B. Local cloning and local discrimination

  • For prime-dimensional systems, certain maximally entangled states are locally distinguishable and perfectly locally copied using a shared |Φ+⟩ ancilla and generalized CNOT gates.The construction also perfectly copies the associated unitary operators.Source: B. Local cloning and local… · pp. 57–58

C. Entanglement of quantum cloning

VII. TELECLONING

  • Telecloning combines quantum cloning with teleportation to create optimal copies remotely; entanglement resources support this alternative to flying qubits, which may decohere in noisy channels.Related schemes include reversible and irreversible telecloning, including continuous-variable protocols.Source: §VII · p. 59

A. Teleportation

  • The generalized teleportation scheme combines quantum-information distribution with teleportation through joint Bell-type measurements, classical communication, and local recovery at spatially separated receivers.The discussion describes N senders and M receivers, with M > N.Source: A. Teleportation · p. 60

B. Symmetric 1 →M telecloning

  • The 1 →M qudit telecloning output matches optimal cloning while placing the M receiver qudits in spatially separated locations.The resource includes a port particle and output states for the receivers and ancillas.Source: B. Symmetric 1 →M telecloning · p. 61

C. Economical phase-covariant telecloning

  • For economical phase-covariant telecloning, fidelity equals the ancilla-assisted optimum when d = 2, is lower for d > 2, and the optimum is achievable probabilistically.Receivers obtain the clones only for Bell-measurement outcomes {m = 0, n}, which occur with probability 1/d.Source: C. Economical phase-covariant… · p. 62
  • The telecloning fidelity equals Fopt for any d, and its output state is equivalent to the optimal phase-covariant cloning state after tracing out the ancilla.The fidelity is identified with the optimal phase-covariant cloning fidelity.Source: C. Economical phase-covariant… · p. 62
  • For d > 2, the relevant state is partially entangled, so optimal 1→2 qudit cloning with probability 1/d uses special nonmaximally entangled configurations rather than maximally entangled states (Fan et al., 2003).Its von Neumann entropy is less than log2 d.Source: C. Economical phase-covariant… · p. 63

D. Asymmetric telecloning

Asymmetric telecloning distributes clones to different receivers with different fidelities, while general telecloning extends remote distribution to N inputs and M receivers. The review describes the measurement and recovery procedure, including a finite-POVM construction for qubits relevant to experimental implementation.

D. Asymmetric telecloning

  • Asymmetric telecloning allows different receiver fidelities, with Bob’s fidelity potentially higher when Alice trusts him more than Claire.The qubit fidelities satisfy a trade-off relation.Source: D. Asymmetric telecloning · p. 63
  • The qubit scheme uses a shared entangled resource parameterized by p and q, with p+q=1, and local recovery operations to produce the output state.The reported Bob and Claire fidelities depend on this asymmetric resource choice.Source: D. Asymmetric telecloning · p. 63
  • For the d-dimensional 1→2 case, setting p=q=1/2 gives fidelities in agreement with the symmetric-cloning result attributed to Werner (1998).Source: D. Asymmetric telecloning · p. 64

E. General telecloning

  • For general telecloning, the sender measures N input particles and N ports with a POVM, then communicates the result for local recovery by M receivers, where M≥N.The receivers and the M−N ancillas obtain the optimal clones after applying the specified local recovery unitary operator.Source: E. General telecloning · p. 64
  • The general POVM is constructed to resolve the identity; for d=2, an explicit finite POVM can replace the integral, which is needed for experimental implementation.Source: E. General telecloning · p. 65
  • Telecloning distributes a universal cloning state among spatially separated receivers, and its reverse process is remote quantum information concentration.The review describes distribution to M receivers assisted by M−N ancillas and identifies distribution and concentration as potential quantum-network functions.Source: E. General telecloning · pp. 65–66

VIII. QUANTUM CLONING FOR CONTINUOUS VARIABLE SYSTEMS

This section treats coherent-state N→M Gaussian cloning in continuous-variable systems, deriving optimal fidelity bounds and describing optical implementations. It also reviews related cloning, QKD, and experimental work.

A. Optimal bounds for Gaussian cloners of coherent states

B. Implementation of optimal Gaussian QCM with a linear amplifier and beam splitters

C. Optimal 1 →2 Gaussian QCM

D. Optimal Gaussian N →M QCM

  • The general implementation concentrates N input modes, amplifies the combined mode at gain M/N, then distributes it symmetrically across M outputs.A discrete Fourier transform provides the concentration and distribution steps.Source: D. Optimal Gaussian N →M QCM · p. 70

E. Other developments and related topics

  • Related work covers CV cloning for QKD, non-Gaussian settings, other state types, and experimental implementations.The review notes that coherent light-state cloning with linear optics achieved about 65% fidelity.Source: E. Other developments and related topics · p. 71

IX. SEQUENTIAL UNIVERSAL QUANTUM CLONING

This section presents sequential universal cloning as a way to construct multi-qubit outputs through successive ancilla–qubit interactions. It develops 1→M and N→M schemes and their ancilla requirements.

IX. SEQUENTIAL UNIVERSAL QUANTUM CLONING

  • Sequential cloning addresses the difficulty of preparing multipartite entangled states with a global unitary, though sequential decompositions are not always successful for genuine entangling operations.Source: §IX · p. 72

A. 1 →M sequential UQCM

  • The 1→M scheme derives sequential ancilla interactions from a matrix product state representation, using an ancilla of dimension 2M rather than a global unitary whose dimension grows exponentially with M.Source: A. 1 →M sequential UQCM · pp. 72–74

B. N →M optimal sequential UQCM

C. Sequential UQCM in d dimensions

The review extends sequential universal cloning to d-dimensional systems and describes physical realizations of cloning machines, including a circuit adjustable between two cloning types and a bosonic projection scheme.

C. Sequential UQCM in d dimensions

A. A unified quantum cloning circuit

B. A simple scheme of realization of UQCM and Valence-Bond Solid state

C. UQCM realized by photon stimulated emission and the experiment

The review describes photon-stimulated emission as a realization of universal cloning for qubits and qudits, and surveys an experimental phase-covariant cloning implementation in diamond. It also summarizes other experimental developments and notes that new work may emerge while a review is being prepared.

C. UQCM realized by photon stimulated emission and the experiment

  • Photon-stimulated emission realizes the qubit UQCM; the review relates the process to symmetric projection using maximally entangled states.The review also presents this process as an alternative way to obtain optimal cloning transformations.Source: C. UQCM realized by photon stimulated… · pp. 81–82

D. Higher dimension UQCM realized by photon stimulated emission

  • With one excited state and d ground states, the photonic scheme realizes the optimal universal quantum cloning machine for qudits.The review states that the qudit time evolution follows the qubit case and identifies the resulting transformation as the UQCM of qudits.Source: D. Higher dimension UQCM realized by… · pp. 82–83

E. Experimental implementation of phase-covariant quantum cloning by nitrogen-vacancy defect center in diamond

  • 85.2% average experimental fidelity is close to the 1/2 + 1/8 ≈85.4% optimal bound and is reported as better than universal cloning.The review describes the experimental results as very close to theoretical expectations.Source: E. Experimental implementation of… · p. 84
  • Nanodiamond phase-cloning experiments require active input-phase control; the review notes their integration potential and improved fluorescence collection from sub-wavelength size.The experiment uses state tomography for readout.Source: E. Experimental implementation of… · p. 84

F. Experimental developments

  • Experimental work spans NMR and optical cloning implementations, with related applications including radiometry and electro-optic quantum memory.The review also notes proposals for universal cloning in cavity QED.Source: F. Experimental developments · p. 85

XI. CONCLUDING REMARKS

  • The authors note that new results may emerge during review preparation and that feedback helped identify literature missed in earlier versions.They report trying to incorporate new results during revision.Source: §XI · p. 85

XII. APPENDIX

The appendix supplies explicit matrix forms for sequential universal quantum cloning machines, detailing the qubit case and stating that the qudit extension follows directly.

XII. APPENDIX

  • The appendix provides explicit forms of V [n]in for different sequential UQCMs.It introduces these results as detailed forms for sequential cloning of qudits.Source: §XII · p. 86

A. N →M sequential UQCM of qubits.

  • For qubits, the appendix lays out V [n] cases across the sequence, including initial, intermediate, final, and additional-copy stages.It states that all the operators V [n]in have been obtained.Source: A. N →M sequential UQCM of qubits · pp. 86–88

B. N →M sequential UQCM of qudits

  • For qudits, the appendix lists cases for n = 1, intermediate n, n = M, and n = M + l, then says the extension from the two-dimensional case is direct.The listed cases provide the structure for building V [n]in.Source: B. N →M sequential UQCM of qudits · pp. 88–89
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