4.1 Introduction

Perfect cloning of quantum states that are a priori unknown is forbidden by the laws of quantum mechanics (1–3). Perfect cloning is only possible when the input states belong to a known set of orthogonal states. For example, the Controlled‐NOT quantum gate (4), which operates as follows on two qubits (two‐level systems):

4.1

where denotes addition modulo two and represent basis states for each qubit, implements a perfect cloning transformation for qubits, when the second qubit is initially prepared in state (the first qubit is the one to be cloned and is initially in one of the two orthogonal states or ). The requirement that the input state belongs to a known class of orthogonal states is quite restrictive. It is intuitive to expect that by relaxing the conditions on the class of allowed input states, perfect cloning can be approximated with a decreasing efficiency.

This chapter describes approximate cloning transformations for different sets of input states and analyzes the corresponding optimal qualities in terms of fidelity. In Section 4.2, we review the no‐cloning theorem. In Section 4.3, we analyze the smallest nontrivial class of input states, namely, the set of two nonorthogonal states, and then consider the case of two pairs of orthogonal states. In Section 4.4, we consider another interesting set of input states, Namely, the one of all possible states lying on the equator of the Bloch sphere. In Section 4.5, we describe the least restrictive case, where the input states of the qubits are completely unknown, and report the optimal fidelities for qubits and then for systems with arbitrary finite dimension. We review fidelities of various processes and show how the fidelity increases by restricting the class of inputs. In Section 4.6, we drop the constraint that all copies should have identical output density matrices and study asymmetric cloning. In Section 4.7 we discuss probabilistic cloning, where perfect copies can be created with a certain probability. Before concluding, we finally briefly report on experimental quantum cloning in Section 4.8. ‐ An overview of approximate quantum cloning can be found in (5).

4.2 The No‐Cloning Theorem

The no‐cloning theorem states that it is not possible to perfectly clone an unknown quantum state, or a state drawn from a set of two (or more) nonorthogonal states ( 1– 3). The theorem can be easily proved by contradiction. Let us assume that such an ideal cloner exists and it can be described by a unitary operator that acts on the global system of the initial copy, in a pure state , a blank copy on which the state will be cloned, in an initially arbitrary state , and, in general, an auxiliary system (ancilla) whose dimension is not specified, initially in a state . Notice that all the states that we consider are normalized. Assuming that ideal cloning is possible for two nonorthogonal input states and , the cloning transformation would lead to

4.2

where and represent the output states of the ancilla and . Since the cloning transformation is unitary, it preserves the scalar product. The scalar product of the two possible inputs in the aforementioned expressions must be then equal to the corresponding scalar product between the outputs, that is, . Since the two possible input states are assumed to be nonorthogonal, this relation leads to

4.3

which clearly can never be satisfied, unless in the trivial case . Thus, does not exist. All cloning transformations presented in the following sections are therefore approximate cloning transformations, the optimal quality of which depends on the scenario.

Another reason for the impossibility of perfect quantum cloning is the impossibility of superluminal signaling: assume the situation where Alice and Bob are distant and share a maximally entangled state, for example, the singlet state for two qubits. Alice measures her qubit and encodes one bit of information into whether her measurement is in the ‐ or the ‐basis. If Bob would possess a perfect cloner, he could make many perfect copies of his qubit (after Alice's measurement) and measure half of them in the ‐basis, half of them in the ‐basis. In the case where his basis coincides with Alice's, all measurement outcomes are identical; in the other case half of his results are 0, half of them 1. The speed of information transfer would just depend on the speed of the cloner, and if Alice and Bob would be far enough from each other, they could communicate with superluminal speed. Note that the impossibility of superluminal signaling does not only arise in a relativistic theory but also in quantum mechanics, due to linearity of any physical transformation (CP‐map) (6).

4.3 State‐Dependent Cloning

In this section, we study approximate cloning transformations for a set of two nonorthogonal input states, parameterized as follows:

4.4

where . This set of two input states can equivalently be specified by their scalar product .

We will derive here a lower bound for the fidelity of an optimal cloning transformation that operates on input states of the form , with . This analysis was performed in (7) for and , and later generalized in (8) for any values of and . The resulting transformation is called state‐dependent cloner, because its form depends explicitly on the set of initial states, namely, on the parameter .

We will consider a unitary operator acting on the Hilbert space of qubits and define the final states and as

4.5

4.6

Unitarity gives the following constraint on the scalar product of the final states:

4.7

Notice that this ansatz does not describe the most general cloning transformation because we have not included an auxiliary system. Therefore, the fidelities derived below will be lower bounds on the optimal cloning fidelity.

As a convenient criterion for optimality of the cloning transformation, we maximize the average global fidelity of both final states and with respect to the perfectly cloned states and . The average global fidelity is defined formally as

4.8

It can be easily shown (7) that the above fidelity is maximized when the states and lie in the two‐dimensional space , which is spanned by the vectors . We will now maximize explicitly the value of the global fidelity 4.8. We can think about it in a geometrical way and define , , and as the “angles” between vectors and , and , and , respectively, as illustrated in Figure 4.1. The global fidelity 4.8 then takes the form

4.9

and is thus maximized when the angle between and is equal to that between and , that is, . The optimal situation thus corresponds to the maximal symmetry in the disposition of the vectors. This symmetry guarantees that the fidelity is the same for both input states and . By inserting the explicit definitions of the angles and – notice that due to we have – the optimal global fidelity then takes the form

4.10

We will now derive the explicit expression of a different figure of merit, namely, the single‐copy fidelity of each output copy with respect to the initial state. We first write the output states as

4.11

where

4.12

From these equations, the reduced density operator corresponding to one of the output copies can be easily derived (notice that the global states of the copies and belong to the symmetric subspace, that is, the space spanned by all states which are invariant under any permutation of the constituent subsystems, therefore each output copy is described by the same reduced density operator):

4.13

The fidelity is then calculated as

4.14

As mentioned, notice that by the symmetry of the transformation the fidelity of the output state with respect to the input leads to the same result.

Notice that the single‐copy fidelities for the cloner of nonorthogonal states 4.14 are just a lower bound. Actually, in order to find the optimal state‐dependent cloner to be compared with the phase covariant and universal ones, the fidelity should be maximized explicitly, and, in general, additional auxiliary systems interacting with the qubits should be considered in the definition of the cloning transformation . Reference (7) showed that for the case the maximization of leads to a different cloning transformation than the one considered here, where the global fidelity is maximized. However, the value of the resulting optimal fidelity is only slightly different from the fidelity reported in Eq. 4.14 for and , which was first derived in (7) and reads explicitly

4.15

As an illustration, we also report here the explicit form of the bound 4.14 for the fidelity corresponding to the case of the cloner

4.16

Figure 4.2 shows the fidelities for the and the cloners as functions of the parameter . The dashed curve corresponds to , the full curve to . As expected, the values of the fidelity are always much higher than and for the optimal phase‐covariant and universal cloners, respectively, and than and , see sections 4.4 and 4.5 .

We can describe the state of each qubit in terms of its Bloch vector representation

4.17

where is the identity matrix, is the Bloch vector (with unit length for pure states) and are the Pauli matrices. The length of the output Bloch vector can then be easily calculated. For example, in the case it takes the form

4.18

It can be seen that, differently from the phase‐covariant and universal cases, which will be analyzed in the next sections, in state‐dependent cloning the Bloch vector of the input states is not simply shrunk along the direction of the input Bloch vector, but is also rotated in the Bloch sphere.

We now slightly enlarge the class of possible input states and consider an ensemble that consists of two pairs of orthogonal states for a two‐dimensional quantum system (9). These four states can be parameterized in the Bloch sphere representation with a single parameter in the following way. The four Bloch vectors for the states with

4.19

where is the identity operator and with are the Pauli matrices, are given by

4.20

In this representation, the four vectors are lying in the ‐plane, and each of them includes an angle or with the ‐axis; see Figure 4.3. The two pairs of orthogonal states are given by and .

We could also parameterize the states with the real parameters and with :

4.21

where the relation between the parameters and is given by

4.22

We study the case of cloning and consider the most general cloning transformation as a unitary operation acting on the input, a prescribed blank qubit, and an auxiliary system, initially in an arbitrary state . In order to derive the optimal cloning transformation, due to linearity it is sufficient to define its action on the basis states of the input, namely

4.23

where the coefficients can be taken real and positive by including possible phases into the ancilla states. The above form for the cloning transformation guarantees that the two output copies are described by the same reduced density operator. We study cloning transformations that lead to the same efficiency for the four states . Since the four states are transformed into one another by renaming the basis states, that is, , the cloning transformation will be invariant under the exchange of and . This condition leads to . Moreover, unitarity of the cloning transformation dictates the condition

4.24

We will now optimize the fidelity of each output copy with respect to the input state , where and the trace is performed over the auxiliary system and one of the output copies. With our symmetric way to parameterize the states, we can easily derive the fidelity for the four input states, as we just have to calculate the fidelity once and can then use symmetry arguments in order to find the explicit form of the other three cases, for example, we can replace by to go from the fidelity for to the fidelity for . We require the fidelities for the four input states to be equal. This condition leads to

4.25

Independently of the coefficients , the fidelity will be maximal for the following choice of scalar products between the auxiliary states:

4.26

which can be reached with a two‐dimensional ancilla and, for example, the choice

4.27

Inserting this into equation 4.25, we arrive at

4.28

The optimal cloning transformation corresponds to the maximum value of the fidelity 4.28, together with the constraint 4.24 due to unitarity.

Using the method of Lagrange multipliers, we thus have to solve the system of equations

4.29

where is the Lagrange multiplier. The solution for the coefficients and turns out to be

4.30

Inserting this into equation 4.28 leads to the optimal fidelity

4.31

The explicit form of the resulting optimal cloning transformation is found immediately by inserting equations 4.30 and 4.27 into equation 4.23.

In Figure 4.4 we plot as a function of the angle . The figure demonstrates that the cloning task is performed in the worst way for the two pairs being maximally spread, that is, in the case .

We point out the following geometrical description of the cloning transformation. For states with a Bloch vector lying on the plane of the Bloch sphere, namely states given by the density operator , we can describe the cloning transformation 4.23 in terms of two shrinking factors for the ‐component of the Bloch vector, and for its ‐component, such that the output state of each copy takes the form . The explicit expression for the two shrinking factors with our choice of ancillas 4.27 is given by

4.32

In the case of the optimal transformation, according to equation 4.30, the shrinking factors depend only on the value of :

4.33

According to the symmetry of the input ensemble 4.20 that we used to perform the optimization, the shrinking factors are related as . Furthermore, the identity holds. The shrinking factors become equal for , namely . Notice that this case turns out to coincide with the optimal phase‐covariant cloner, which is discussed next.

4.4 Phase‐Covariant Cloning

In this section, we extend the set of input states to a continuous one and consider states of the form

4.34

where . Notice that this class of states corresponds to a Bloch vector lying on the plane in the Bloch sphere representation. We are interested in cloning transformations that treat each input state belonging to this class in the same way, namely whose quality does not depend on the value of the phase . This requirement corresponds to imposing the following phase‐covariant condition on the operation of the cloning map :

4.35

for all input states and for all unitary phase shift operators , where . In this equation, denotes the trace operation over all the output copies except one. Cloning transformations satisfying the above condition will be called phase covariant.

It can be shown (10) that phase‐covariant cloning transformations for input states correspond to a shrinking of the Bloch vector by a factor (in this case represents the shrinking in the plane of the Bloch representation). The simplest case of and was reported for the first time in (10) and corresponds to the optimal transformation for two pairs of orthogonal states, derived in Section 4.3 , for . We point out that this transformation coincides with the optimal eavesdropping strategy in the BB84 scheme (11).

The case of general and was studied in (12,13). The derivation is very involved and will not be reported here. In Reference (12) a cloning transformation from an arbitrary number of input copies to an arbitrary number of output copies was presented and was proved to be optimal only for . The optimal maps for the case with equal parity of and (i.e., and are either both even or both odd) were derived in (13). For the case, the optimal phase‐covariant fidelity is given by (13)

4.36

4.37

Moreover, when and have the same parity, the fidelity takes the form

4.38

where is the binomial coefficient . It is interesting to notice that, in contrast to the universal case in which the optimal maps are the same for optimization of the global or single‐ particle fidelity (14), in the phase‐covariant case the solutions are, in general, different (13).

It is possible to extend the definition of phase‐covariant cloning to higher‐dimensional systems with finite dimension , by optimizing the cloning transformations on “equatorial” states

4.39

where the 's are independent phases in the interval . The optimal fidelity for the case is given by (15)

4.40

The general case was analyzed in (16), where explicit simple solutions were obtained for a number of output copies given by , where is a positive integer. In this case, the optimal fidelity takes the explicit form

4.41

In this summation represent indices that have to fulfill the constraint . The interesting aspect of these cloning transformations is that they can be achieved “economically”, without the need of auxiliary systems in addition to the output copies (16).

4.5 Universal Cloning

4.5.1 The Case of Qubits

We now consider the least restrictive set of pure input states, namely, the one corresponding to the whole two‐dimensional Hilbert space of a qubit. We will investigate universal cloning transformations, namely, transformations whose quality does not depend on the input state. As a figure of merit, we use the single‐copy fidelity .

Universal quantum cloning is a unitary transformation acting on an extended input, which contains original qubits all in the same unknown pure state , “blank” qubits and auxiliary systems, and leading to output clones. The blanks and the auxiliary systems are initially in some prescribed quantum state. In order to guarantee that the output qubits have the same reduced density operator (symmetry condition), we require that the output state of the copies is supported on the symmetric subspace. When requiring that all input states must be treated in the same way (universality condition), it has been shown (7) that the reduced density operator , describing the state of each of the output qubits, is related to the input state, characterized by the Bloch vector , via the transformation

4.42

namely the Bloch vector is just shortened by a shrinking factor . Notice that the shrinking factor is simply related to the single‐copy fidelity as

4.43

In order to optimize the fidelity , or, equivalently, the shrinking factor , of an universal cloning transformation we follow the approach of Ref. (17), relating universal cloning to state estimation. The aim of state estimation is to find a measurement that leads to the best possible estimation of the a priori unknown quantum state . The most general measurement is a positive operator valued measure (POVM), namely, a set of positive operators , such that . Suppose that we have at our disposal copies of the state . The outcome of each instance of the measurement provides, with probability , the “candidate” for We can calculate the fidelity of state estimation by averaging over the outcomes of the measurement as follows:

4.44

where represents the reconstructed density operator corresponding to the state . For a universal state estimating procedure, the fidelity must not depend on , thus the reconstructed density operator can also be written as in Eq. 4.42, with shrinking factor . It has been shown in (18) that the optimal fidelity for state estimation of pure qubits has the form

4.45

corresponding to the optimal shrinking factor

4.46

We now want to show a connection between optimal universal cloning and optimal universal state estimation, given by the equality

4.47

To prove it, we first consider a measurement procedure performed on copies, which is composed of an optimal cloning process and a subsequent universal measurement on the output copies. This concept is illustrated in Figure 4.5. The total procedure can be regarded as a possible state estimation method. Since the state of the output copies of the optimal universal cloner is supported on the symmetric subspace, it can be conveniently decomposed as (19)

4.48

where the coefficients add up to one ( ), but are not necessarily positive. After performing the optimal universal measurement on the outputs of the cloner, we can calculate the average fidelity of the total estimation process, due to linearity of the measurement procedure as follows:

4.49

4.50

where we explicitly exploited the universality of state estimation from Eq. 4.49 to Eq. 4.50. In the limit , we have . Remembering that at the output of the cloner , the average estimation fidelity in the limit can be written as

4.51

This fidelity cannot be higher than the one for the optimal state estimation performed directly on pure inputs, thus we conclude

4.52

We can derive the opposite inequality by noticing that after performing a universal measurement procedure on identically prepared input copies , we can prepare a state of systems, supported on the symmetric subspace, where each system has the same reduced density operator, given by . As mentioned above, a universal cloning process generates outputs that are supported on the symmetric subspace. Therefore, the aforementioned method of performing state estimation followed by preparation of a symmetric state can be viewed as a universal cloning process, and, thus, it cannot lead to a higher fidelity than the optimal cloning transformation. Therefore we find the inequality

4.53

which holds for any value of , in particular for . The above inequality, together with equation 4.52, leads to the equality 4.47. ‐ Note that the equivalence between asymptotic cloning and state estimation holds for any ensemble of states, as shown in (20), by using the monogamy of entanglement and properties of entanglement breaking channels, and in (21), by analyzing channels that distribute information to many users.

An interesting property of universal cloning transformations is that the shrinking factors of universal cloning machines multiply (17), namely, the shrinking factor of a universal cloner composed of a sequence of an cloner followed by an cloner is the product of the two shrinking factors: . Moreover, since a sequence of an and an universal cloner cannot perform better than the optimal universal cloner, we can write the following upper bound for an cloner:

4.54

where we have used Eqs. 4.46, 4.47, and 4.43 on the right‐hand side. The corresponding fidelity reads

4.55

The above bound is achieved by the cloning transformations proposed in (22) for and , and in (23) for arbitrary values of and . The explicit optimal transformation for universal cloning of qubits, suggested by Bužek and Hillery (22), reads

4.56

Here, one still has the freedom of a unitary transformation of the output ancilla states.

4.5.2 Higher Dimensions

The optimal cloning transformation for pure states in arbitrary finite dimension was derived in Ref. (19). The corresponding optimal single‐copy fidelity is given by

4.57

which generalizes the optimal fidelity derived in Eq. 4.55 to arbitrary finite dimension.

Subsequently, explicit unitary realizations of the above transformations were shown in (24). It is interesting to notice that the link 4.47 between optimal universal cloning and optimal universal state estimation can be proved in a very similar way also for higher‐dimensional systems (25), thus leading to the following explicit evaluation of the optimal fidelity for state estimation of identical states in dimension ,

4.58

4.5.3 Entanglement Structure

In Eq. 4.56 the output of a universal cloner for qubits was given. It is clear that the output state is entangled. In Reference (26) the entanglement structure for the output of a cloner was studied. For the simple case of a cloner, it was shown that the 3‐qubit output is an entangled state from the ‐class By considering the concurrence, which is a good measure of entanglement for two‐qubit subsystems, it was also shown that the entanglement between clone and ancilla is higher than between the clones.

For the case and general , it is straightforward to derive an explicit expression for the concurrence between two clones or one clone and one ancilla, by calculating the respective reduced density matrices and using their symmetry properties, as derived in (27). The concurrence between two clones is found to be

4.59

As we can see, the entanglement between two clones surprisingly vanishes for . The concurrence between one clone and one ancilla can be calculated as

4.60

This expression is nonzero for all finite , that is, there is always an entanglement between a clone and an ancilla, unless .

Generalizing these results to the cloner for qubits, one can again calculate the concurrence between two clones. Again, the entanglement between two clones does not only vanish for , but already for finite , namely, for . The entanglement between one clone and one ancilla, however, has different properties: the concurrence is nonzero for any finite , and only vanishes in the limit .

It is also possible to study multipartite entanglement in the cloning output. An interesting example is the qubit cloner, for which no bipartite entanglement between the clones exists, as mentioned above. However, by studying the reduced density matrix of three clones, which consists of a mixture of projectors onto W‐states and a certain product state, it was shown (26) that there does exist genuine tripartite entanglement of the ‐type between three clones.

4.6 Asymmetric Cloning

So far we have always assumed symmetry for the output copies, that is, all reduced 1‐particle output density matrices of the cloner were supposed to be identical. If one gives up this requirement, one can study asymmetric quantum cloning. For the universal ‐cloner it was shown in (28) for qubits, and in (29) for ‐dimensional systems, that there exists a trade‐off for the quality of the copies: Increasing the fidelity of one copy requires decrease in the fidelity of the other copy. The resulting no‐cloning inequality reads

4.61

where denotes the fidelity of copy . Note that this bound is tight. For the symmetric case , this bound reduces to the upper bound given in Eq. 4.57 for . This concept has been generalized in (30) to the case of identical inputs and a number of output copies having the same fidelity, while outputs all have some different fidelity. For the scenario of asymmetric phase‐covariant cloning in dimensions, similar inequalities have been derived in (31).

Asymmetric cloning is closely related to security issues in quantum cryptography: the eavesdropper can use an asymmetric cloner in order to win partial information about the state sent from Alice to Bob, by keeping one copy and sending on the second one. In all protocols where the optimal eavesdropping strategy (optimal in the sense of maximising Eve's mutual information with Alice for a fixed disturbance) is known, it turns out that the optimal eavesdropping strategy is equivalent to optimal asymmetric cloning (32,33). Note, however, that in general the task of optimal cloning is not identical to optimal eavesdropping in quantum key distribution; for details, see (34).

4.7 Probabilistic Cloning

In the previous sections, we have always considered deterministic quantum cloning, that is, the case where the cloning machine consists of a unitary operation only. The different concept of probabilistic quantum cloning (35) allows for a unitary operation plus measurement. By selecting a certain measurement result, one may arrive at perfect clones, however with a success probability of less than 1. It was shown in (35) that the states chosen from a set can be probabilistically cloned if and only if the are linearly independent. In this case, a unitary transformation of the following form exists:

4.62

with and for . Measuring the ancilla state in the basis then leads with probability to the desired clones .

The most simple example is given by an input set of only two states, namely, . Here, the success probabilities have to obey the inequality (35)

4.63

In the more general case of input states, one arrives at bounds for the respective success probabilities by solving a certain series of inequalities.

4.8 Experimental Quantum Cloning

The first explicit idea of how to implement an approximate cloning transformation in an experiment was suggested in (36), where it was shown that optimal universal quantum cloning can be realized via stimulated emission in certain three‐level‐systems, for example, atoms in a cavity. These three‐level systems have a ground state and two degenerate excited levels, connected to the ground state by two orthogonal modes of the electromagnetic field, and . The aim is to clone general superposition states , via stimulated emission. Another experimental possibility is based on stimulated parametric down‐conversion. The latter proposal was used for an experimental demonstration of a optimal universal cloning process (37). A quality of the clones that is close to the optimal value of was reached, namely, . The interaction Hamiltonian for parametric down‐conversion reads

4.64

where is a coupling constant and is the creation operator for a vertically (horizontally) polarized photon in spatial mode , and analogously for . This Hamiltonian is invariant under joint identical polarization transformations in modes and , thus ensuring universality of the cloning process.

Polarized photons have been used in (38) to realize optimal universal quantum cloning, in (39) to demonstrate optimal phase‐covariant cloning, in (40) to implement optimal cloning of four‐dimensional quantum states, and in (41) to demonstrate experimental eavesdropping based on optimal quantum cloning.

A completely different idea was realized in (42), where a nuclear magentic resonance (NMR) experiment with three qubits was used to implement the approximate cloner via the network that was derived in (43) (in a slightly modified version). The universality was tested explicitly by studying 312 input states, covering the Bloch sphere. ‐ Phase‐covariant cloning (44) and state‐dependent cloning (45) and, more recently, probabilistic cloning (46) have also been implemented with NMR techniques.

Another physical system in which quantum cloning could be implemented is Cavity QED (47). It has been shown (48) that optimal phase‐covariant cloning can be achieved in a spin network with a certain XY Hamiltonian. This method is more robust against noise than the network approach (43).

4.9 Summary and Outlook

In this contribution, we have reviewed approximate quantum cloning transformations for various scenarios. Here, we have only discussed cloning for finite‐dimensional systems. Optimal cloning of continuous variable systems has also been studied in the literature, mainly for Gaussian cloning transformations (49), but is beyond the scope of our chapter.

The topic of approximate quantum cloning is mainly of fundamental interest: for example, limits on the cloning fidelity imply limits on the security in quantum cryptography. Thus, we have learned about differences between classical and quantum information processing by studying cloning. It is interesting to ask whether a cloning process be used as a tool in quantum information processing. Reference (50) has shown that quantum information distribution can improve the performance of certain quantum computation tasks. This distribution can be naturally implemented with different types of quantum cloning procedures: the information content of the input state is spread over the output state.

As a generalization of the concept of quantum cloning of pure input States, one can consider mixed input states. Here, one arrives at the so‐called no‐broadcasting theorem (51), which states that it is impossible to create from one mixed state, drawn from a set of two noncommuting density operators, an ‐party output state, where each single‐copy density operator is equal to the input. Further developments in this direction (52) have shown that the no‐broadcasting theorem does not hold if one increases the number of input copies ( ).

Recently, approximate quantum cloning has found new interest in the context of various information‐theoretical concepts: In the resource theory of asymmetry, where a measure for asymmetry is given by the relative entropy of a state with respect to its symmetrized version, this quantity leads to a tight bound for the achievable cloning fidelity (53). In the same context it was shown, by employing entropic inequalities, that universal cloning machines and symmetrized partial trace channels are dual to each other, that is, one can be used as approximate recovery map for the other (54). Furthermore, it has been shown in (55) that universal cloning is the optimal coding/decoding strategy for the compression of identically prepared (mixed) states. The scheme of probabilistic superreplication has been suggested in (56): here, the authors showed that pure copies from a phase‐covariant set can be transformed to approximate copies. In the limit , the fidelity approaches one, while the probability of success tends to zero. Here, the number of almost perfect output copies is restricted to due to the Heisenberg limit.

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