Optimal asymptotic cloning machines

Consider a quantum system with Hilbert space $\mathcal{H}$ and a set of unit vectors {|ψ〉}_x∈𝖷 ⊂, representing the possible input states of the copy machine. The task of optimal quantum cloning is to convert N identical copies of a state $|{{\psi }_{x}}\rangle$, chosen at random with probability ${{p}_{x}}$, into M approximate copies that are as accurate as possible. The most general cloning process is described by a quantum channel (completely positive trace-preserving map) ${{\mathcal{C}}_{N,M}}$ transforming density matrices on ${{\mathcal{H}}^{\otimes N}}$ into density matrices on ${{\mathcal{H}}^{\otimes M}}$. As a figure of merit for the quality of the copies we consider the global fidelity

$$\begin{eqnarray}&&F\left[ N\to M \right]=\mathop{\sum }\limits_{x\in\mathsf{X}}\;{{p}_{x}}\;{\mbox{Tr}}\left[ \psi _{x}^{\otimes M}{{\mathcal{C}}_{N,M}}\left( \psi _{x}^{\otimes N} \right) \right],\end{eqnarray} \tag{ 1 }$$

where ${{\psi }_{x}}$ denotes the projector ${{\psi }_{x}}:=|{{\psi }_{x}}\rangle \langle {{\psi }_{x}}|$. When the set ${{\left\{ |{{\psi }_{x}}\rangle \right\}}_{x\in\mathsf{X}}}$ is continuous, it is understood that the sum has to be replaced by the integral and the probability ${{p}_{x}}$ is replaced with a probability density $p\left( x \right){\mbox{d}}x$. The optimal cloner is defined to be the quantum channel that maximizes $F\left[ N\to M \right]$ and its fidelity will be denoted by ${{F}_{{\mbox{clon}}}}\left[ N\to M \right]$. In some cases it is interesting to consider, instead of the average fidelity of equation (1), the worst case fidelity

$$\begin{eqnarray}&&{{F}^{*}}\left[ N\to M \right]=\mathop{{\rm inf} }\limits_{x\in\mathsf{X}}{\mbox{Tr}}\left[ \psi _{x}^{\otimes M}{{\mathcal{C}}_{N,M}}\left( \psi _{x}^{\otimes N} \right) \right].\end{eqnarray} \tag{ 2 }$$

The advantage of the worst case fidelity is that it does not require us to specify a prior. The maximum value of the worst case fidelity over all possible channels will be denoted by $F_{{\mbox{clon}}}^{*}\left[ N\to M \right]$. In general, the cloner that maximizes the worst case fidelity can be different from the cloner that maximizes the average fidelity. Nevertheless, when the input states are generated by the action of a group and when the prior probabilities are uniform, one can easily prove that the optimal cloners for these two criteria coincide.

The estimation-based machines are described by measure-and-prepare (MP) channels, i.e. channels that can be realized by measuring the input copies with a positive operator-valued measure (POVM) ${{\left\{ {{P}_{y}} \right\}}_{y\in\mathsf{Y}}}$ and, conditional on outcome y, by re-preparing the system in a state ${{\rho }_{y}}$. In the case of cloning, the POVM $\left\{ {{P}_{y}} \right\}$ acts on the Hilbert space of the N input copies and the states $\left\{ {{\rho }_{y}} \right\}$ are states on the Hilbert space of the M output copies. Averaging over the measurement outcomes, the output of the MP channel ${{\mathcal{C}}_{N,M}}$ is given by ${{\mathcal{C}}_{N,M}}\left( \rho \right)={{\sum }_{y\in\mathsf{Y}}}{\mbox{Tr}}\left[ {{P}_{y}}\rho \right]{{\rho }_{y}}$. We denote by ${{F}_{{\mbox{est}}}}\left[ N\to M \right]$ (respectively, $F_{{\mbox{est}}}^{*}\left[ N\to M \right]$) the maximum of the average (respectively, worst case) fidelity over the set of MP channels. Such a maximum is known in the literature as classical fidelity threshold [16, 25–30] and can be used as a benchmark for the experimental demonstration of quantum advantages.

The key question of this paper is whether the difference between ${{F}_{{\mbox{clon}}}}\left[ N\to M \right]$ and ${{F}_{{\mbox{est}}}}\left[ N\to M \right]$ (or, alternatively, between $F_{{\mbox{clon}}}^{*}\left[ N\to M \right]$ and $F_{{\mbox{est}}}^{*}\left[ N\to M \right]$) becomes negligible in the limit $M\to \infty$. In the formalization of the problem there is a catch, because in some interesting cases both fidelities tend to zero in the limit $M\to \infty$. For example, the fidelity tends to zero whenever the set of states to be cloned contains a one-parameter family of 'clock states' $\left\{ |{{\psi }_{t}}\rangle ={{e}^{-iHt}}|\psi \rangle ,t\in \mathbb{R} \right\}$ generated by the action of a Hamiltonian H: in all such cases a non-vanishing fidelity would violate the strong converse of the standard quantum limit for information replication [34], which states that every quantum channel that produces more than O(N) output copies must necessarily have vanishing fidelity. In order not to trivialize the question of the global equivalence, it is then important to consider the difference between the two fidelities at the leading order. For this reason, we formulate our conjecture as follows:

Conjecture 1  (Global equivalence of cloning and state estimation). For every set of pure states $\left\{ |{{\psi }_{x}}\rangle \right\}$ and prior probabilities $\left\{ {{p}_{x}} \right\}$, the global fidelities of cloning and estimation satisfy the relation

$$\begin{eqnarray}&&\mathop{{\rm lim} }\limits_{M\to \infty }\frac{{{F}_{{\mbox{clon}}}}\left[ N\to M \right]-{{F}_{{\mbox{est}}}}\left[ N\to M \right]}{{{F}_{{\mbox{clon}}}}\left[ N\to M \right]}=0\qquad \forall N\in \mathbb{N}.\end{eqnarray} \tag{ 3 }$$

Of course, the conjecture can be also formulated in terms of the worst-case fidelities, if one prefers not to specify a prior probability distribution. When the equality holds, we say that asymptotic cloning and state estimation are globally equivalent, with respect to the average or to the worst-case fidelity. In the following sections we will prove the global equivalence in a variety of different settings, exploring in details the structural features of the optimal cloner and of the optimal MP protocol.

Here we derive a general upper bound on the optimal cloning fidelity in terms of the probability density of correct identification of the input state, the so-called likelihood of the estimation [3, 4]. The key idea in our argument is that GPCS can be used to construct a POVM [38], also known as the coherent state POVM. Indeed, thanks to Schur's lemma one has the identity

$$\begin{eqnarray}&&\int {\mbox{d}}g\;\psi _{g}^{\otimes M}=\frac{{{P}_{M}}}{{{d}_{M}}},\end{eqnarray} \tag{ 7 }$$

where ${\mbox{d}}g$ is the Haar measure, ${{P}_{M}}$ is the projector on the subspace of ${{\mathcal{H}}^{\otimes M}}$ spanned by the states $\left\{ |{{\psi }_{g}}{{\rangle }^{\otimes M}} \right\}$, and ${{d}_{M}}$ is a suitable normalization constant, given by

$$\begin{eqnarray}&&{{d}_{M}}={{\left( \int {\mbox{d}}g|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2M}} \right)}^{-1}}\end{eqnarray} \tag{ 8 }$$

For compact groups, the Haar measure can be normalized to one and the constant ${{d}_{M}}$ is just the trace of ${{P}_{M}}$, or, equivalently, the dimension of the subspace of ${{\mathcal{H}}^{\otimes M}}$ spanned by the states $\left\{ |{{\psi }_{g}}{{\rangle }^{\otimes M}} \right\}$. For non-compact groups in infinite dimensions, ${{d}_{M}}$ is called formal dimension and in our discussion we assume that ${{d}_{M}}>0$ for sufficiently large M. As a matter of fact, this condition is satisfied in all known cases, including the Weyl–Heisenberg coherent states (where the condition is satisfied for all M) and the squeezed vacuum states (where the condition is satisfied for all $M\geqslant 3$ [39]). Thanks to equation (7), one can define the coherent-state POVM $\left\{ {{E}_{g}} \right\}$ via the relation ${{E}_{g}}:={{d}_{M}}\psi _{g}^{\otimes M}$.

We are now in position to derive our upper bound on the cloning fidelity:

Proposition 2. For every set of coherent states $\left\{ |{{\psi }_{g}}\rangle \right\}$ and for every probability density p(g), the cloning fidelity satisfies the bound

$$\begin{eqnarray}&&{{F}_{{\mbox{clon}}}}\left[ N\to M \right]\leqslant \frac{p_{{\mbox{succ}}}^{\left( N \right)}}{{{d}_{M}}},\end{eqnarray} \tag{ 9 }$$

where $p_{{\mbox{succ}}}^{\left( N \right)}$ is the maximum probability density of correct identification of the input state, namely

$$\begin{eqnarray*}&&p_{{\mbox{succ}}}^{\left( N \right)}:=\mathop{{\rm sup} }\limits_{\left\{ {{P}_{g}} \right\}}\int {\mbox{d}}g\;p\left( g \right){{\langle {{\psi }_{g}}{{|}^{\otimes N}}{{P}_{g}}|{{\psi }_{g}}\rangle }^{\otimes N}},\end{eqnarray*}$$

the supremum running over all possible POVMs $\left\{ {{P}_{g}} \right\}$.

Proof.  Let ${{\mathcal{C}}_{N,M}}$ be the optimal cloning channel. Then, we have

$$\begin{eqnarray*}\begin{array}{rcl} {{F}_{{\mbox{clon}}}}\left[ N\to M \right] & = & \int {\mbox{d}}g\;p\left( g \right){\mbox{Tr}}\left[ \psi _{g}^{\otimes M}{{\mathcal{C}}_{N,M}}\left( \psi _{g}^{\otimes N} \right) \right] \\ {} & = & \frac{1}{{{d}_{M}}}\int {\mbox{d}}g\;p\left( g \right){\mbox{Tr}}\left[ \left( {{d}_{M}}\psi _{g}^{\otimes M} \right){{\mathcal{C}}_{N,M}}\left( \psi _{g}^{\otimes N} \right) \right] \\ {} & = & \frac{1}{{{d}_{M}}}\int {\mbox{d}}g\;p\left( g \right){\mbox{Tr}}\left[ {{P}_{g}}\;\psi _{g}^{\otimes N} \right] \\ {} & = & \frac{{{p}_{{\mbox{succ}}}}}{{{d}_{M}}}, \\ \end{array}\end{eqnarray*}$$

where $\left\{ {{P}_{g}} \right\}$ is the POVM that corresponds to measuring the coherent-state POVM $\left\{ {{E}_{g}} \right\}$ after the channel ${{\mathcal{C}}_{N,M}}$ and ${{p}_{{\mbox{succ}}}}$ is the probability density of correct identification of the input state with the POVM $\left\{ {{P}_{g}} \right\}$, averaged over all possible states. Optimizing over all possible POVM then gives the desired bound.□

Note that, in general, there is no guarantee that the upper bound of equation (9) is achievable. In the following paragraphs we will see that the bound is achievable in the limit $M\to \infty$. Moreover, we will consider the worst-case scenario, showing that in this case the bound can be achieved for every finite M.

Consider the naive MP protocol, which consists in estimating the input state with the optimal POVM $\left\{ {{P}_{g}} \right\}$ and in re-preparing M copies according to the estimate. In the case of coherent states, it is easy to see that this protocol is asymptotically optimal. By definition, its fidelity is given by

$$\begin{eqnarray*}&&F\left[ N\to M \right]=\int {\mbox{d}}g\;p(g)\int {\mbox{d}}\hat{g}\;|\langle {{\psi }_{{\hat{g}}}}|{{\psi }_{g}}\rangle {{|}^{2M}}{\mbox{Tr}}\left[ {{P}_{{\hat{g}}}}\psi _{g}^{\otimes N} \right].\end{eqnarray*}$$

Clearly, in the large M limit the overlap $|\langle {{\psi }_{{\hat{g}}}}|{{\psi }_{g}}\rangle {{|}^{2M}}$ becomes sharply peaked around the correct value $\hat{g}=g$, so that in the integral we can substitute the value of the slowly varying function ${\mbox{Tr}}\left[ {{P}_{{\hat{g}}}}\psi _{g}^{\otimes N} \right]$ with its value at $\hat{g}=g$. With this approximation, we obtain

$$\begin{eqnarray*}\begin{array}{rcl} F\left[ N\to M \right] & \approx & \int {\mbox{d}}g\;p\left( g \right)\int {\mbox{d}}\hat{g}\;|\langle {{\psi }_{{\hat{g}}}}|{{\psi }_{g}}\rangle {{|}^{2M}}{\mbox{Tr}}\left[ {{P}_{g}}\psi _{g}^{\otimes N} \right] \\ {} & = & \frac{1}{{{d}_{M}}}\int {\mbox{d}}g\;p\left( g \right)\;{\mbox{Tr}}\left[ {{P}_{g}}\psi _{g}^{\otimes N} \right] \\ {} & = & \frac{p_{{\mbox{succ}}}^{\left( N \right)}}{{{d}_{M}}}, \\ \end{array}\end{eqnarray*}$$

where in the second line we used equation (7). Hence, the naive MP protocol achieves the upper bound of equation (9) in the asymptotic limit. An explicit quantification of the error in the approximation will be provided for the worst-case fidelity in section 4.4.

In the worst-case scenario, the strong symmetry of the set of coherent states simplifies the optimization of the optimal cloning machine and of the optimal estimation strategy, enabling a complete solution, as shown in the following (Optimal cloner of coherent states).

Proposition 3 For a given set of coherent states $\{|{{\psi }_{g}}\rangle \}$, the optimal worst-case fidelity is

$$\begin{eqnarray}&&F_{{\mbox{clon}}}^{*}\left[ N\to M \right]=\frac{{{d}_{N}}}{{{d}_{M}}}\end{eqnarray} \tag{ 10 }$$

and is achieved by the optimal cloner

$$\begin{eqnarray}&&{{\mathcal{C}}_{N,M}}\left( \rho \right)=\frac{{{d}_{N}}}{{{d}_{M}}}{{P}_{M}}\left[ \rho \otimes {{I}^{\otimes \left( M-N \right)}} \right]{{P}_{M}},\end{eqnarray} \tag{ 11 }$$

where ${{P}_{M}}$ is the projector on the subspace spanned by the vectors ${{\{|{{\psi }_{g}}\rangle }^{\otimes M}}\}$.

The expert reader can recognize in equation (11) the same general form of the optimal cloner defined by Werner for pure states [10], which is extended here to coherent states associated to arbitrary groups. It is also worth noting that also the optimal cloner of coherent states of the harmonic oscillator [12, 13] can be cast in the form of equation (11).

Proof.  Following the same steps in the proof of proposition 2, one can derive the bound

$$\begin{eqnarray*}&&F_{{\mbox{clon}}}^{*}\left[ N\to M \right]\leqslant \frac{p_{{\mbox{succ}}}^{*\left( N \right)}}{{{d}_{M}}},\end{eqnarray*}$$

with $p_{{\mbox{succ}}}^{*\,\left( N \right)}={{{\rm sup} }_{\left\{ {{P}_{g}} \right\}}}\;{{{\rm inf} }_{g}}\;{{\langle {{\psi }_{g}}{{|}^{\otimes N}}{{P}_{g}}|{{\psi }_{g}}\rangle }^{\otimes N}}$, the supremum running over all possible POVMs. Now, in the presence of symmetry, the POVM that maximizes the worst case probability density $p_{{\mbox{succ}}}^{*\,\left( N \right)}$ is the maximum likelihood POVM from [40, 41]. In the case of coherent states, the general formula for the maximum likelihood POVM yields the coherent-state POVM $\{{{E}_{g}}\}$, given by ${{E}_{g}}={{d}_{N}}\psi _{g}^{\otimes N}$. Substituting in the above bound we then obtain

$$\begin{eqnarray*}&&F_{{\mbox{clon}}}^{*}\left[ N\to M \right]\leqslant \frac{{{d}_{N}}}{{{d}_{M}}}.\end{eqnarray*}$$

Clearly, the channel ${{\mathcal{C}}_{N,M}}$ defined in equation (11) achieves the bound.□

Remark (the case of uniform prior).  Note that, in the case of compact groups, where the Haar measure can be normalized, the optimal worst-case fidelity is equal to the optimal average fidelity with respect to the uniform prior $p\left( g \right){\mbox{d}}g={\mbox{d}}g$. Similarly, the worst-case success probability is equal to the optimal average probability with respect to the uniform prior, namely $p_{{\mbox{succ}}}^{*\,\left( N \right)}=p_{{\mbox{succ}}}^{\left( N \right)}$. Moreover, the average density matrix of the target states is $\rho _{AV}^{\left( M \right)}:=\int {\mbox{d}}g\;\psi _{g}^{\otimes M}={{P}_{M}}/{{d}_{M}}$, having used equation (7). Collecting these observations together, we can cast the average fidelity in the form

$$\begin{eqnarray}&&{{F}_{{\mbox{clon}}}}\left[ N\to M \right]=\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }p_{{\mbox{succ}}}^{\left( M \right)}.\end{eqnarray} \tag{ 12 }$$

The expression of equation (12) will reappear many times in this paper, for sets of states that are not necessarily coherent states. In general, equation (12) will provide an upper bound on the cloning fidelity, which becomes achievable in the asymptotic limit. Here, what is specific of coherent states is that equation (7) gives the exact value of the cloning fidelity for every N and M, not only in the asymptotic limit.

In the worst-case scenario, it is possible to prove that the best MP protocol for coherent states is, again, the naive protocol, which consists in measuring the input copies with the coherent-state POVM and re-preparing M identical copies according to the estimate [42]. For compact groups, we will now show that this protocol is optimal in the asymptotic limit.

The argument is based on a lower bound on the worst-case fidelity of the naive protocol, which reads

$$\begin{eqnarray*}\begin{array}{rcl} {{F}^{*}}\left[ N\to M \right] & = & {{d}_{N}}\int {\mbox{d}}g\;|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2M}}|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2N}} \\ {} & \geqslant & {{d}_{N}}{{\int }_{{{\mathsf{S}}_{\epsilon }}}}{\mbox{d}}g\;|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2M}}|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2N}}, \\ \end{array}\end{eqnarray*}$$

where ${{\mathsf{S}}_{\epsilon }}$ is the set of values of g such that $|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2M}}\geqslant \epsilon$, for some fixed . By definition ${{\mathsf{S}}_{\epsilon }}$ satisfies

$$\begin{eqnarray*}&&{{\int }_{{{\mathsf{S}}_{\epsilon }}}}{\mbox{d}}g\;|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2M}}\geqslant \int {\mbox{d}}g\;|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2M}}-\epsilon .\end{eqnarray*}$$

Combining this fact with the relation $|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2}}\geqslant {{\epsilon }^{1/M}}\forall g\in {{\mathsf{S}}_{\epsilon }}$, it is straightforward to obtain the bound

$$\begin{eqnarray}\begin{array}{rcl} {{F}^{*}}\left[ N\to M \right] & \geqslant & {{d}_{N}}\;{{\epsilon }^{\frac{N}{M}}}\;{{\int }_{{{\mathsf{S}}_{\epsilon }}}}{\mbox{d}}g|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2M}} \\ {} & \geqslant & {{d}_{N}}\;{{\epsilon }^{\frac{N}{M}}}\left( \;\int {\mbox{d}}g|\langle \psi |{{\psi }_{g}}\rangle {{|}^{2M}}-\epsilon \right) \\ {} & = & \frac{{{d}_{N}}\;{{\epsilon }^{\frac{N}{M}}}\left( 1-\epsilon \;{{d}_{M}} \right)}{{{d}_{M}}}, \\ \end{array}\end{eqnarray} \tag{ 13 }$$

having used equation (7) in the last equality. Now, for every compact group, the subspace spanned by the coherent states $\left\{ |{{\psi }_{g}}{{\rangle }^{\otimes M}} \right\}$ is contained in the symmetric subspace, and therefore we have the bound ${{d}_{M}}=O\left( {{M}^{d-1}} \right)$. Hence, we can set $\epsilon =N/\left( M{{d}_{M}} \right)$ and obtain the relation ${{{\rm lim} }_{M\to \infty }}{{\epsilon }^{N/M}}=1$, which, combined with equations (10) and (13), gives

$$\begin{eqnarray*}&&\mathop{{\rm lim} }\limits_{M\to \infty }\frac{F_{{\mbox{clon}}}^{*}\left[ N\to M \right]-F_{est}^{*}\left[ N\to M \right]}{F_{{\mbox{clon}}}^{*}\left[ N\to M \right]}\leqslant \mathop{{\rm lim} }\limits_{M\to \infty }1-{{\epsilon }^{N/M}}\left( 1-\epsilon \;{{d}_{M}} \right)=0,\end{eqnarray*}$$

This provides a quantitative proof of the equivalence between cloning and state estimation for all families of coherent states generated by compact groups.

We start by summarizing a few facts that will be used in the derivation of the optimal asymptotic cloners. First of all, the state of the N input copies can be expressed as

$$\begin{eqnarray*}&&|{{\psi }_{{\vec{\theta }}}}{{\rangle }^{\otimes N}}=\mathop{\sum }\limits_{\vec{n}\in {{\mathcal{P}}_{N,d}}}{{e}^{i\vec{n}\cdot \vec{\theta }}}\;\sqrt{{{p}_{N,\vec{n}}}}|N,\vec{n}\rangle ,\end{eqnarray*}$$

where $\vec{n}=\left( {{n}_{0}},...,{{n}_{d-1}} \right)$ is a partition of N into d non-negative integers, ${{\mathcal{P}}_{N,d}}$ denotes the set of such partitions, $|N,\vec{n}\rangle$ is the unit vector that is obtained by projecting the state $|0{{\rangle }^{\otimes {{n}_{0}}}}|1{{\rangle }^{\otimes {{n}_{2}}}}\cdots |d-1{{\rangle }^{\otimes {{n}_{d-1}}}}$ in the symmetric subspace, and ${{p}_{N,\vec{n}}}$ is the multinomial distribution

$$\begin{eqnarray*}&&{{p}_{N,\vec{n}}}=\frac{N!}{{{n}_{0}}!...{{n}_{d-1}}!}p_{0}^{{{n}_{0}}}...p_{d-1}^{{{n}_{d-1}}},\end{eqnarray*}$$

For the uniform probability distribution over the states $|{{\psi }_{{\vec{\theta }}}}\rangle$ we will use the notation

$$\begin{eqnarray*}&&\frac{{\mbox{d}}\vec{\theta }}{{{\left( 2\pi \right)}^{d-1}}}:=\frac{{\mbox{d}}{{\theta }_{1}}\cdots {\mbox{d}}{{\theta }_{d-1}}}{{{\left( 2\pi \right)}^{d-1}}},\end{eqnarray*}$$

Since the input state $|{{\psi }_{{\vec{\theta }}}}{{\rangle }^{\otimes N}}$ is chosen with uniform probability, the optimal cloning channel can be chosen without loss of generality to be covariant, that is, satisfying the property

$$\begin{eqnarray}&&U_{{\vec{\theta }}}^{\otimes M}{{\mathcal{C}}_{N,M}}\left( \rho \right)U_{{\vec{\theta }}}^{\dagger \otimes M}={{\mathcal{C}}_{N,M}}\left( U_{{\vec{\theta }}}^{\otimes N}\rho U_{{\vec{\theta }}}^{\dagger \otimes N} \right),\end{eqnarray} \tag{ 15 }$$

for every density matrix $\rho$ and for every $\vec{\theta }$. Covariant channels are known to be optimal also in the worst case scenario, and the maximum of the average fidelity coincides with the average of the worst case fidelity. For this reason, in the following we will carry out the analysis in the average scenario, taking for granted that all the results can be translated immediately into worst-case results.

In order to find the optimal asymptotic cloner, we first derive an upper bound on the global fidelity:

Proposition 4. For every phase-covariant set of qudit states $\left\{ |{{\psi }_{{\vec{\theta }}}}\rangle \right\}$, the cloning fidelity satisfies the bound

$$\begin{eqnarray}&&{{F}_{{\mbox{clon}}}}\left[ N\to M \right]\leqslant \left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }\;p_{{\mbox{succ}}}^{\left( N \right)},\end{eqnarray} \tag{ 16 }$$

where $\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }$ is the maximum eigenvalue of the average target state $\rho _{AV}^{\left( M \right)}:=\int {\mbox{d}}\vec{\theta }/{{\left( 2\pi \right)}^{d-1}}\;\psi _{{\vec{\theta }}}^{\otimes M}$ and $p_{{\mbox{succ}}}^{\left( N \right)}$ is the maximum probability density of correct identification of the input state, namely $p_{{\mbox{succ}}}^{\left( N \right)}:={{{\rm max} }_{\left\{ {{P}_{{\vec{\theta }}}} \right\}}}\int {\mbox{d}}\vec{\theta }/{{\left( 2\pi \right)}^{d-1}}\;{{\langle {{\psi }_{{\vec{\theta }}}}{{|}^{\otimes N}}{{P}_{{\vec{\theta }}}}\;|{{\psi }_{{\vec{\theta }}}}\rangle }^{\otimes N}}$, the maximum running over all possible POVMs $\left\{ {{P}_{{\vec{\theta }}}} \right\}$.

The proof of the upper bound is presented in appendix B and is based on two ingredients: the first ingredient is an upper bound on the fidelity derived from the optimization over all possible covariant cloners, the second ingredient is a translation of the upper bound in terms of the quantities $\shortparallel \rho _{AV}^{\left( M \right)}{{\shortparallel }_{\infty }}$ and $p_{{\mbox{succ}}}^{\left( N \right)}$. Since this translation will be used several times in the following, we discuss it explicitly here: regarding $\shortparallel \rho _{AV}^{\left( M \right)}{{\shortparallel }_{\infty }}$, this can be easily computed using Schur's lemma, which gives $\rho _{AV}^{\left( M \right)}={{\sum }_{\vec{m}\in {{\mathcal{P}}_{M,d}}}}\;{{p}_{M,\vec{m}}}\;|M,\vec{m}\rangle \langle M,\vec{m}|$, and, therefore

$$\begin{eqnarray}&&\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }={\rm max} \left\{ {{p}_{M,\vec{m}}}|\vec{m}\in {{\mathcal{P}}_{M,d}} \right\},\end{eqnarray} \tag{ 17 }$$

Regarding $p_{{\mbox{succ}}}^{\left( N \right)}$, we use the general expression of the optimal POVM for sets of pure states with group symmetry, provided in [40, 41]. In the case of multiphase covariant states, the general recipe yields the optimal POVM

$$\begin{eqnarray}&&{{P}_{\theta }}=|{{\eta }_{{\vec{\theta }}}}\rangle \langle {{\eta }_{{\vec{\theta }}}}|,\qquad |{{\eta }_{{\vec{\theta }}}}\rangle :=|0\rangle +{{e}^{i{{\theta }_{1}}}}|1\rangle +...+{{e}^{i{{\theta }_{d-1}}}}|d-1\rangle \;\end{eqnarray} \tag{ 18 }$$

and the maximum probability density $p_{{\mbox{succ}}}^{\left( N \right)}$ is given by

$$\begin{eqnarray}&&p_{{\mbox{succ}}}^{\left( N \right)}=\langle {{\eta }_{{\vec{\theta }}}}|\psi _{{\vec{\theta }}}^{\otimes M}|{{\eta }_{{\vec{\theta }}}}\rangle ={{\left( \mathop{\sum }\limits_{\vec{n}\in {{\mathcal{P}}_{N,\vec{n}}}}\sqrt{{{p}_{N,\vec{n}}}} \right)}^{2}}\qquad \forall \vec{\theta }.\end{eqnarray} \tag{ 19 }$$

In the next two paragraphs we will show two radically different cloning strategies that match the upper bound of equation (16) in the asymptotic limit. The first strategy consists in using an economical quantum channel, which coherently encodes the N input copies into the space of M systems. The second strategy consists in using a suitable MP protocol—this time, however, not the naive one.

Let us start by showing a quantum strategy that achieves the upper bound of equation (16) in the asymptotic limit. Consider the economical channel ${{\mathcal{C}}_{N,M}}$ defined by

$$\begin{eqnarray}&&{{\mathcal{C}}_{N,M}}\left( \rho \right)={{V}_{N,M}}\rho V_{N,M}^{\dagger }\qquad {{V}_{N,M}}=\mathop{\sum }\limits_{\vec{n}\in {{\mathcal{P}}_{N,d}}}|M,\vec{n}-{{\vec{n}}_{*}}+{{\vec{m}}_{*}}\rangle \langle N,\vec{n}|,\end{eqnarray} \tag{ 20 }$$

where ${{\vec{n}}_{*}}$ (respectively, ${{\vec{m}}_{*}}$) is the partition that maximizes the multinomial probability ${{p}_{N,\vec{n}}}$ (respectively, ${{p}_{M,\vec{m}}}$). The cloning fidelity of channel of this channel is

$$\begin{eqnarray}\begin{array}{rcl} F\left[ N\to M \right] & = & \int \frac{{\mbox{d}}\vec{\theta }}{{{\left( 2\pi \right)}^{d-1}}}{{\left| {{\langle {{\psi }_{{\vec{\theta }}}}{{|}^{\otimes M}}{{V}_{N,M}}|{{\psi }_{{\vec{\theta }}}}\rangle }^{\otimes N}} \right|}^{2}} \\ {} & = & {{\left( \mathop{\sum }\limits_{\vec{n}\in {{\mathcal{P}}_{N,d}}}\sqrt{{{p}_{N,\vec{n}}}\;{{p}_{M,\vec{n}-{{{\vec{n}}}_{*}}+{{{\vec{m}}}_{*}}}}} \right)}^{2}}. \\ \end{array}\end{eqnarray} \tag{ 21 }$$

Clearly, when M is large compared to N, the multinomial probability ${{p}_{M,\vec{n}-{{{\vec{n}}}_{*}}+{{{\vec{m}}}_{*}}}}$ is approximately constant in an interval of size O(N) centred around ${{\vec{m}}_{*}}$, up to an error of order ${{N}^{2}}/M$. Hence, the fidelity can be approximated as

$$\begin{eqnarray*}\begin{array}{rcl} F\left[ N\to M \right] & = & {{p}_{M,{{{\vec{m}}}_{*}}}}{{\left( \mathop{\sum }\limits_{\vec{n}\in {{\mathcal{P}}_{N,d}}}\sqrt{{{p}_{N,\vec{n}}}} \right)}^{2}}\left[ 1+O\left( \frac{{{N}^{2}}}{M} \right) \right] \\ {} & \equiv & \left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }\;p_{{\mbox{succ}}}^{\left( N \right)}\left[ 1+O\left( \frac{{{N}^{2}}}{M} \right) \right], \\ \end{array}\end{eqnarray*}$$

having used equations (17) and (19). Comparing this value with the upper bound of equation (16), we conclude that our cloner is asymptotically optimal.

Remark (large N asymptotics).  The fidelity of the economical cloner has a simple and intriguing expression when both M and N are large. In this case, the multinomial distribution ${{p}_{N,\vec{n}}}$ can be approximated as

$$\begin{eqnarray}&&{{p}_{N,\vec{n}}}={{G}_{N,\vec{n}}}\left[ 1+O\left( \frac{1}{\sqrt{{{N}^{1-\delta }}}} \right) \right]+{{\eta }_{N}},\end{eqnarray} \tag{ 22 }$$

where ${{G}_{N,\vec{n}}}$ is the Gaussian distribution

$$\begin{eqnarray*}&&{{G}_{N,\vec{n}}}:=\frac{{\rm exp} \left[ -\mathop{\sum }\limits_{j=1}^{d}\frac{{{\left( {{n}_{j}}-N{{p}_{j}} \right)}^{2}}}{2N{{p}_{j}}} \right]}{\sqrt{{{\left( 2\pi N \right)}^{d-1}}{{p}_{0}}{{p}_{1}}...{{p}_{d-1}}}},\end{eqnarray*}$$

$\delta >0$ is arbitrary, and ${{\eta }_{N}}$ is an error term vanishing faster than the inverse of every polynomial. The Gaussian can be expressed conveniently as

$$\begin{eqnarray}&&{{G}_{N,\vec{n}}}:=\sqrt{{\rm det} \left( \frac{A}{2\pi N} \right)}{\rm exp} [-\frac{{{{\vec{x}}}^{T}}A\vec{x}}{2N}],\end{eqnarray} \tag{ 23 }$$

where $\vec{x}=\left( {{x}_{j}} \right)_{j=1}^{d-1}$ is the vector of the independent variables ${{x}_{j}}={{n}_{j}}-N{{p}_{j}}$, and A is the positive non-singular matrix with entries ${{A}_{jk}}:=1/{{p}_{j}}\;{{\delta }_{jk}}+1/{{p}_{0}}$. Replacing the summation in equation (21) with a Gaussian integral, one gets

$$\begin{eqnarray*}&&{{F}_{{\mbox{clon}}}}[N\to M]={{\left( \frac{\sqrt{4MN}}{M+N} \right)}^{d-1}}\left[ 1+O\left( \frac{1}{\sqrt{{{N}^{1-\delta }}}} \right) \right],\end{eqnarray*}$$

for arbitrary $\delta >0$. What is intriguing here is that the asymptotic fidelity depends only on the dimension of the Hilbert space, and not on the actual values of the probabilities $\left\{ {{p}_{j}} \right\}_{j=0}^{d-1}$ (as long as they are non-zero). Apparently, the asymptotic limit washes out many of the details of the family of input states, so that the fidelity depends only on a very coarse-grained information, namely, the number of parameters needed to describe the states. Note that, when some probabilities are zero, the asymptotic formula for the optimal fidelity still holds, provided that one replaces d with the number of non-zero probabilities.

We now show that the optimal cloning fidelity can be achieved asymptotically by a suitable MP protocol. For the first time, here we need to consider an MP protocol that is different from the naive one: as we anticipated in the introduction, the naive MP protocol is asymptotically suboptimal. Intuitively, the reason why re-preparing M identical copies is suboptimal is that the state $|{{\psi }_{{\vec{\theta }}}}{{\rangle }^{\otimes M}}$ is very sensitive to small variations of $\vec{\theta }$. Hence, a small error in the estimate of the input state results in the re-preparation of a state that is almost orthogonal to the target state. In order to avoid this drawback, one idea is to re-prepare a number of copies K that is smaller than M, but still sufficiently large to mimic, in some way, the target state of M copies. Driven by the above intuition, we consider an MP protocol of the following special form:

• (i)  
  estimate θ from the N input copies, using the optimal POVM of equation (18)
• (ii)  
  prepare $K={{M}^{1-\epsilon }}$ copies of the estimated state, for some $\epsilon \in \left( 0,1 \right)$
• (iii)  
  produce M clones via the economical K-to-M cloning of equation (20).

Let us see that the protocol achieves the optimal cloning fidelity for every $\epsilon >0$. First of all, note that the fidelity can be written as

$$\begin{eqnarray*}&&{{F}_{K}}[N\to M]=\int \frac{{\mbox{d}}\vec{\tau }}{{{(2\pi )}^{d-1}}}{{p}_{N}}(\vec{\tau }|0)\;{{f}_{K,M}}(\vec{\tau }),\end{eqnarray*}$$

with $\vec{\tau }:=\left( 1,{{\tau }_{1}},...,{{\tau }_{d-1}} \right)$, ${{p}_{N}}\left( \vec{\tau }|0 \right):=\langle {{\eta }_{{\vec{\tau }}}}|\;{{\psi }^{\otimes N}}\;|{{\eta }_{{\vec{\tau }}}}\rangle$, and ${{f}_{K,M}}\left( \vec{\tau }\; \right):={{\left| {{\langle \psi {{|}^{\otimes M}}{{V}_{K,M}}|{{\psi }_{{\vec{\tau }}}}\rangle }^{\otimes K}} \right|}^{2}}$. For large M and K, the Gaussian approximation to the multinomial distribution gives

$$\begin{eqnarray*}&&{{f}_{M,K}}\left( \vec{\tau }\; \right)={{\left( \frac{2\sqrt{MK}}{M+K} \right)}^{d-1}}{\rm exp} [-\frac{2MK}{M+K}{{\vec{\tau }}^{\,T}}{{A}^{-1}}\vec{\tau }]\left[ 1+O\left( \frac{1}{\sqrt{{{K}^{1-\delta }}}} \right) \right]+{{\eta }_{K}},\end{eqnarray*}$$

with arbitrary $\delta >0$ and ${{\eta }_{K}}$ vanishing faster than the inverse of every polynomial. Now, when M and K are large compared to N, the Gaussian ${{f}_{M,K}}\left( \vec{\tau }\; \right)$ decays rapidly when $\vec{\tau }$ moves away from the peak $\vec{\tau }=\left( 0,...,0 \right)$. Hence, the slowly varying function ${{p}_{N}}\left( \vec{\tau }|0 \right)$ can be treated as a constant in the integral. Using this fact, we obtain

$$\begin{eqnarray}\begin{array}{rcl} {{F}_{K}}\left[ N\to M \right] & = & {{p}_{N}}(0|0)\left[ \int \frac{{\mbox{d}}\vec{\tau }}{{{\left( 2\pi \right)}^{d-1}}}{{f}_{K,M}}\left( {\vec{\tau }} \right) \right]\left[ 1+O\left( \frac{1}{\sqrt{K}}+\frac{N}{K} \right) \right]+{{\eta }_{K}} \\ {} & = & {{p}_{N}}\left( 0|0 \right)\sqrt{{\rm det} \left[ \frac{A}{2\pi \left( M+K \right)} \right]}\left[ 1+O\left( \frac{1}{\sqrt{{{K}^{1-\delta }}}}+\frac{N}{K} \right) \right]. \\ \end{array}\end{eqnarray} \tag{ 24 }$$

Now, by definition we have ${{p}_{N}}\left( 0|0 \right)=\langle \eta |{{\psi }^{\otimes N}}|\eta \rangle \equiv p_{{\mbox{succ}}}^{\left( N \right)}$ (cf equation (19)) and

$$\begin{eqnarray*}&&\sqrt{{\rm det} \left[ \frac{A}{2\pi \left( M+K \right)} \right]}=\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }\left[ 1+O\left( \frac{K}{M}+\frac{1}{\sqrt{{{M}^{1-\delta }}}} \right) \right].\end{eqnarray*}$$

Hence, we obtained

$$\begin{eqnarray*}&&{{F}_{K}}\left[ N\to M \right]=p_{{\mbox{succ}}}^{\left( N \right)}\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }\left[ 1+O\left( \frac{1}{\sqrt{{{K}^{1-\delta }}}}+\frac{N}{K}+\frac{K}{M} \right) \right].\end{eqnarray*}$$

Comparing with the bound of equation (16), it is clear that our MP protocol is asymptotically optimal. In summary, this establishes the global equivalence for arbitrary multiphase-covariant states.

Here we prove that the naive MP protocol is suboptimal, even in the asymptotic limit. This protocol has the same structure of the optimal protocol that we introduced in the previous paragraph, except for the fact that in the naive protocol one has K = M instead of $K={{M}^{1-\epsilon }}$. Making this substitution, we can compute the cloning fidelity using equation (24), which now gives

$$\begin{eqnarray*}\begin{array}{rcl} {{F}_{K=M}}\left[ N\to M \right] & = & {{p}_{N}}\left( 0|0 \right)\;\sqrt{{\rm det} \left[ \frac{A}{4\pi M} \right]}\left[ 1+O\left( \frac{1}{\sqrt{{{M}^{1-\delta }}}}+\frac{N}{M} \right) \right] \\ {} & = & \frac{{{F}_{{\mbox{clon}}}}\left[ N\to M \right]}{\sqrt{{{2}^{d-1}}}}\left[ 1+O\left( \frac{1}{\sqrt{{{K}^{1-\delta }}}}+\frac{{{N}^{2}}}{K} \right) \right]. \\ \end{array}\end{eqnarray*}$$

Clearly, this relation shows that re-preparing M identical copies is a suboptimal strategy, even in the asymptotic limit. Moreover, the gap between the fidelity of the optimal cloner and the fidelity of the naive MP protocol increases exponentially fast with the dimension d. Finally, note that the ratio between the fidelity of the naive MP protocol and the optimal cloning fidelity is independent of the specific values of the probabilities $\left\{ {{p}_{j}} \right\}$ that define the set of input states in equation (14), as long as the probabilities are non-zero. Again, we see that the asymptotic limit washes away the information about the set of input states, so that the ratio of the fidelities depends only on the number of parameters.

Let us denote by ${\mbox{Spec}}\left( H \right)$ the (integer) spectrum of H and expand the input state $|\psi \rangle$ as

$$\begin{eqnarray*}&&|\psi \rangle =\mathop{\sum }\limits_{E\in {\mbox{Spec}}\left( H \right)}\;\sqrt{{{p}_{E}}}\;|E\rangle ,\end{eqnarray*}$$

where $|E\rangle$ is an eigenvector of H for the eigenvalue E and ${{p}_{E}}\geqslant 0$ is the probability that a measurement on the eigenspaces of H gives outcome E. Without loss of generality, we choose H so that

$$\begin{eqnarray*}&&\langle \psi |H|\psi \rangle =\mathop{\sum }\limits_{E\in {\mbox{Spec}}\left( H \right)}\;{{p}_{E}}\;E=0.\end{eqnarray*}$$

Consider now the N-copy Hamiltonian, given by ${{H}^{\left( N \right)}}=\sum _{n=1}^{N}{{H}_{n}}$ where ${{H}_{n}}$ is the Hamiltonian H acting on the nth copy. The N-copy state can be expanded as

$$\begin{eqnarray*}&&|\psi {{\rangle }^{\otimes N}}=\mathop{\sum }\limits_{E\in {\mbox{Spec}}\left( {{H}^{\left( N \right)}} \right)}\;\sqrt{{{p}_{N,E}}}\;|N,E\rangle \end{eqnarray*}$$

where $|N,E\rangle$ is an eigenvector of ${{H}^{\left( N \right)}}$ for the eigenvalue E and ${{p}_{N,E}}$ is a probability. For large N the probability distribution ${{p}_{N,E}}$ converges to a Gaussian centred around the E = 0 and with variance proportional to N, denoted by

$$\begin{eqnarray}&&{{g}_{N,E}}=\frac{1}{\sqrt{2\pi \langle {{H}^{2}}\rangle N}}{\rm exp} \left[ \frac{-{{E}^{2}}}{2\langle {{H}^{2}}\rangle N} \right],\end{eqnarray} \tag{ 26 }$$

where $\langle {{H}^{2}}\rangle =\langle \psi |{{H}^{2}}|\psi \rangle$. More precisely, we can put the Gaussian approximation in the form

$$\begin{eqnarray*}&&{{p}_{N,E}}=c\;{{g}_{N,E}}\left[ 1+O\left( \frac{1}{\sqrt{{{N}^{1-\delta }}}} \right) \right]+{{\eta }_{N}},\end{eqnarray*}$$

where $c>0$ is a proportionality constant taking into account the conversion from a finite probability distribution to a continuous one, $\delta >0$ is arbitrary and ${{\eta }_{N}}$ is an error vanishing faster than the inverse of every polynomial. This form of the Gaussian approximation can be obtained by expressing the energy in terms of partitions of N into $|{\mbox{Spec}}\left( H \right)|$ non-negative integers [34], applying the Gaussian approximation of equation (22) to the multinomial distribution of the partitions, and finally taking the marginal over all partitions that give the same energy.

As one can easily expect, the upper bound on the fidelity derived for multiphase covariant cloners can be generalized to arbitrary phase covariant sets in the following way.

Proposition 5. For every phase covariant set of states ${{\left\{ |{{\psi }_{\theta }}\rangle \right\}}_{\theta \in \left[ -\pi ,\pi \right)}}$, the cloning fidelity satisfies the bound

$$\begin{eqnarray}&&{{F}_{{\mbox{clon}}}}\left[ N\to M \right]\leqslant \left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }\;p_{{\mbox{succ}}}^{\left( N \right)},\end{eqnarray} \tag{ 27 }$$

where $\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }$ is the maximum eigenvalue of the average target state $\rho _{AV}^{\left( M \right)}:=\int {\mbox{d}}\theta /2\pi \;\psi _{\theta }^{\otimes M}$ and $p_{{\mbox{succ}}}^{\left( N \right)}$ is the maximum probability density of correct identification of the input state, namely $p_{{\mbox{succ}}}^{\left( N \right)}:={{{\rm max} }_{\left\{ {{P}_{\theta }} \right\}}}\int {\mbox{d}}\theta /2\pi \;{{\langle {{\psi }_{\theta }}{{|}^{\otimes N}}{{P}_{\theta }}|{{\psi }_{\theta }}\rangle }^{\otimes N}},$ the maximum running over all possible POVMs $\left\{ {{P}_{\theta }} \right\}$.

Following the path delineated for multiphase states, it is easy to come up with an economical cloner that saturates the upper bound of equation (27) for large M: just choose the economical cloner ${{\mathcal{C}}_{N,M}}$ defined by

$$\begin{eqnarray}&&{{\mathcal{C}}_{N,M}}\left( \rho \right)={{V}_{N,M}}\rho V_{N,M}^{\dagger },\qquad {{V}_{N,M}}:=\mathop{\sum }\limits_{E\in {\mbox{Spec}}\left( {{H}^{\left( N \right)}} \right)}|M,E+{{E}_{0}}\rangle \langle N,E|,\end{eqnarray} \tag{ 28 }$$

where ${{E}_{0}}$ is the eigenvalue of ${{H}^{\left( M \right)}}-{{H}^{\left( N \right)}}$ with minimum modulus. This cloner has fidelity

$$\begin{eqnarray}&&F\left[ N\to N \right]={{\left( \mathop{\sum }\limits_{E\in {\mbox{Spec}}\left( {{H}^{\left( N \right)}} \right)}\sqrt{{{p}_{M,E+{{E}_{0}}}}\;{{p}_{N,E}}} \right)}^{2}}.\end{eqnarray} \tag{ 29 }$$

For large M, we can use the fact that ${{p}_{M,E}}$ is almost constant in ${\mbox{Spec}}({{H}^{(N)}})$. In this way, we obtain

$$\begin{eqnarray*}&&F\left[ N\to N \right]={{p}_{M,{{E}_{*}}}}{{\left( \mathop{\sum }\limits_{E\in {\mbox{Spec}}\left( {{H}^{\left( N \right)}} \right)}\sqrt{{{p}_{N,E}}} \right)}^{2}}\left[ 1+O\left( \frac{{{N}^{2}}}{M} \right) \right]\end{eqnarray*}$$

where ${{E}_{*}}$ is the value that maximizes ${{p}_{M,E}}$. Now, by definition we have ${{p}_{M,{{E}_{*}}}}=\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }$. On the other hand, the quantity ${{\left( {{\sum }_{E}}\sqrt{{{p}_{N,E}}} \right)}^{2}}$ can be identified with $p_{{\mbox{succ}}}^{\left( N \right)}$, using Holevo's classic result on phase estimation [4]. In conclusion, we obtained

$$\begin{eqnarray*}&&F\left[ N\to M \right]=\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }p_{{\mbox{succ}}}^{\left( N \right)}\left[ 1+O\left( \frac{{{N}^{2}}}{M} \right) \right],\end{eqnarray*}$$

proving the achievability of the bound in equation (27).

Remark (large N asymptotics).  When both M and N are large, the Gaussian approximation provides the following simple expression for the fidelity of our economical cloner

$$\begin{eqnarray*}&&{{F}_{{\mbox{clon}}}}[N\to M]=\frac{\sqrt{4MN}}{M+N}\left[ 1+O\left( \frac{1}{\sqrt{{{N}^{1-\delta }}}} \right) \right],\end{eqnarray*}$$

with arbitrary $\delta >0$. Note that the value of the fidelity is independent of the Hamiltonian H and of the state $|\psi \rangle$, as long as the variance of H is non-zero on $|\psi \rangle$. Moreover, note that the fidelity approaches to 1 whenever the number of extra-copies $M-N$ is negligible compared to N, whereas it approaches 0 whenever N is negligible compared to M. This fact is a concrete illustration of the standard quantum limit for cloning established in [34], which can be derived from general arguments about quantum metrology.

Here we show that the upper bound of equation (27) can be asymptotically achieved by an MP protocol. In order to reach the bound, we follow the same prescription used in subsection 5.4 for multiphase-covariant cloning: (i) estimate the state using the optimal POVM, (ii) if the estimate is τ, then prepare K copies of $|{{\psi }_{\tau }}\rangle$, and (iii) generate M approximate copies using the optimal K-to-M economical cloner. Again, we show that choosing $K=\lceil{{M}^{1-\epsilon }}\rceil$ for some $\epsilon \in \left( 0,1 \right)$ allows one to achieve the maximum cloning fidelity in the limit $M\to \infty$. For a given K, the fidelity of our protocol can be written as

$$\begin{eqnarray*}&&{{F}_{K}}[N\to M]=\int _{-\pi }^{\pi }\frac{{\mbox{d}}\tau }{2\pi }{{p}_{N}}(\tau |0)\;{{f}_{K,M}}(\tau )\;,\end{eqnarray*}$$

with ${{p}_{N}}(\tau |0):=\;\langle {{\eta }_{\tau }}|\;{{\psi }^{\otimes N}}\;|{{\eta }_{\tau }}\rangle$ and ${{f}_{K,M}}\left( \tau \right):={{\left| {{\langle \psi {{|}^{\otimes M}}{{V}_{K,M}}|{{\psi }_{\tau }}\rangle }^{\otimes K}} \right|}^{2}}$. For large M and K, the Gaussian approximation gives

$$\begin{eqnarray*}&&{{f}_{K,M}}\left( \tau \right)=\frac{2\sqrt{MK}}{M+K}\;{\rm exp} [-\frac{2MK\langle {{H}^{2}}\rangle {{\tau }^{2}}}{\left( M+K \right){{c}^{2}}}]\left[ 1+O\left( \frac{1}{\sqrt{{{K}^{1-\delta }}}} \right) \right]+{{\eta }_{K}}.\end{eqnarray*}$$

When M and K are large compared to N, the Gaussian ${{f}_{M,K}}\left( \tau \right)$ decays rapidly when τ moves away from the peak $\tau =0$, and, therefore, the slowly varying function ${{p}_{N}}\left( \tau \right)$ can be treated as a constant in the integral. Using this fact, we obtain

$$\begin{eqnarray*}\begin{array}{rcl} {{F}_{K}}\left[ N\to M \right] & = & {{p}_{N}}(0|0)\left[ \int _{-\infty }^{\infty }\frac{{\mbox{d}}\tau }{2\pi }{{f}_{K,M}}\left( \tau \right) \right]\left[ 1+O\left( \frac{1}{\sqrt{{{K}^{1-\delta }}}}+\frac{N}{K} \right) \right]+{{\eta }_{K}} \\ {} & = & {{p}_{N}}(0|0)\;\sqrt{\frac{{{c}^{2}}}{2\pi \langle {{H}^{2}}\rangle \left( M+K \right)}}\left[ 1+O\left( \frac{1}{\sqrt{{{K}^{1-\delta }}}}+\frac{N}{K} \right) \right] \\ {} & = & p_{{\mbox{succ}}}^{\left( N \right)}\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }\left[ 1+O\left( \frac{1}{\sqrt{{{K}^{1-\delta }}}}+\frac{N}{K}+\frac{K}{M} \right) \right], \\ \end{array}\end{eqnarray*}$$

having used the fact that, by construction of the protocol, ${{p}_{N}}(0|0)=p_{{\mbox{succ}}}^{\left( N \right)}$ and, by direct inspection, $\shortparallel \rho _{AV}^{\left( M \right)}{{\shortparallel }_{\infty }}={{p}_{M,{{E}_{*}}}}=c{{g}_{M,0}}\left[ 1+O\left( 1/\sqrt{{{M}^{1-\delta }}} \right) \right]$. In conclusion, our MP protocol achieves asymptotically the optimal fidelity. As we noted in the multiphase covariant case, the naive protocol consisting in measuring the optimal POVM and then re-preparing M copies of the estimated state is strictly suboptimal, even in the asymptotic limit: indeed, setting K=M one obtains

$$\begin{eqnarray*}&&{{F}_{K=M}}\left[ N\to M \right]=\frac{{{F}_{{\mbox{clon}}}}\left[ N\to M \right]}{\sqrt{2}}\left[ 1+O\left( \frac{1}{\sqrt{{{M}^{1-\delta }}}}+\frac{{{N}^{2}}}{M} \right) \right].\end{eqnarray*}$$

Consider a general two-qubit maximally entangled state $|{{\psi }_{g}}\rangle \in {{\mathcal{H}}_{A}}\otimes {{\mathcal{H}}_{B}},{{\mathcal{H}}_{A}}\simeq {{\mathcal{H}}_{B}}\simeq {{\mathbb{C}}^{2}}$. We can imagine that the state $|{{\psi }_{g}}\rangle$ is shared by two parties, Alice and Bob, holding qubits A and B, respectively. The state can be parametrized as

$$\begin{eqnarray*}&&|{{\psi }_{g}}\rangle =\frac{({{U}_{g}}\otimes I)|I\rangle \rangle }{\sqrt{2}},\qquad g\in SU(2),\end{eqnarray*}$$

using the 'double-ket notation' $|\Psi \rangle \rangle :={{\sum }_{m,n}}\langle m|\Psi |n\rangle |m\rangle |n\rangle$ for a generic operator $\Psi$ [49]. Using this notation, the state of N identical copies can be decomposed in a convenient way. First of all, rearranging the Hilbert spaces in the tensor product, the input state $|{{\psi }_{g}}{{\rangle }^{\otimes N}}$ can be considered as a vector in $\mathcal{H}_{A}^{\otimes N}\otimes \mathcal{H}_{B}^{\otimes N}$. Then, with a suitable choice of basis, the Hilbert spaces $\mathcal{H}_{A}^{\otimes N}$ and $\mathcal{H}_{B}^{\otimes N}$ can be decomposed as direct sum of tensor product pairs as

$$\begin{eqnarray*}&&\mathcal{H}_{x}^{\otimes N}=\oplus {\rm min} (N)N/2j=j\left( \mathcal{R}_{x}^{(j,N)}\otimes \mathcal{M}_{x}^{(j,N)} \right)\qquad x=A,B\end{eqnarray*}$$

where j is the quantum number of the total angular momentum, $j_{{\rm min} }^{\left( N \right)}=0$ for even N and $j_{{\rm min} }^{\left( N \right)}=\frac{1}{2}$ for odd N, $\mathcal{R}_{x}^{\left( j,N \right)}$ is a representation space, of dimension ${{d}_{j}}=2j+1$, and $\mathcal{M}_{x}^{\left( j,N \right)}$ is a multiplicity space, of dimension

$$\begin{eqnarray*}&&m_{j}^{\left( N \right)}=\frac{2{{d}_{j}}}{N+{{d}_{j}}+1}\left( \frac{N}{N/2+j} \right)\end{eqnarray*}$$

(see e.g. [50]). Relative to this decomposition, we can write $U_{g}^{\otimes N}$ as a block diagonal matrix with the blocks labelled by j, namely

$$\begin{eqnarray}&&U_{g}^{\otimes N}=\oplus {\rm min} \left( N \right)N/2j=j\left[ U_{g}^{\left( j,N \right)}\otimes I_{{{m}_{j}}}^{\left( N \right)} \right],\end{eqnarray} \tag{ 30 }$$

where $U_{g}^{\left( j,N \right)}$ is the unitary operator representing the action of the element $g\in SU\left( 2 \right)$ on the Hilbert space $\mathcal{R}_{x}^{\left( j,N \right)}$ and $I_{{{m}_{j}}}^{\left( N \right)}$ denotes the identity on $\mathcal{M}_{x}^{\left( j,N \right)}$.

Now, using equation (30), the input state $|{{\psi }_{g}}{{\rangle }^{\otimes N}}$ can be cast in the form

$$\begin{eqnarray}&&|{{\psi }_{g}}{{\rangle }^{\otimes N}}=\frac{|U_{g}^{\otimes N}\rangle \rangle }{\sqrt{{{2}^{N}}}}\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&\qquad \ \ \;=\,\frac{1}{\sqrt{{{2}^{N}}}}\oplus {\rm min} \left( N \right)N/2j=j\left( |U_{g}^{\left( j,N \right)}\rangle \rangle \otimes |I_{{{m}_{j}}}^{\left( N \right)}\rangle \rangle \right)\end{eqnarray} \tag{ 32 }$$

with $|U_{g}^{\left( j,N \right)}\rangle \rangle \in \mathcal{R}_{A}^{\left( j,N \right)}\otimes \mathcal{R}_{B}^{\left( j,N \right)}$ and $|I_{{{m}_{j}}}^{\left( N \right)}\rangle \rangle \in \mathcal{M}_{A}^{\left( j,N \right)}\otimes \mathcal{M}_{B}^{\left( j,N \right)}$. Hence, we obtained the decomposition

$$\begin{eqnarray}&&|{{\psi }_{g}}{{\rangle }^{\otimes N}}=\oplus {\rm min} \left( N \right)N/2j=j\sqrt{{{p}_{N,j}}}\;|\psi _{g}^{\left( j,N \right)}\rangle \end{eqnarray} \tag{ 33 }$$

where

$$\begin{eqnarray}&&|\psi _{g}^{\left( j,N \right)}\rangle :=\frac{|U_{g}^{\left( j,N \right)}\rangle \rangle }{\sqrt{{{d}_{j}}}}\otimes \frac{|I_{{{m}_{j}}}^{\left( N \right)}\rangle \rangle }{\sqrt{m_{j}^{\left( N \right)}}}\end{eqnarray} \tag{ 34 }$$

and

$$\begin{eqnarray}&&{{p}_{N,j}}:=\frac{{{d}_{j}}m_{j}^{\left( N \right)}}{{{2}^{N}}}=\frac{2d_{j}^{2}}{N+{{d}_{j}}+1}{{B}_{N,j}},\end{eqnarray} \tag{ 35 }$$

${{B}_{N,j}}$ being the binomial distribution ${{B}_{N,j}}:=\;\left( \frac{N}{N/2+j} \right)/{{2}^{N}}$.

Following the strategy developed in the previous examples, we start our search of the optimal asymptotic cloners by proving an upper bound on the global fidelity. The upper bound has the familiar form that appeared in propositions 2, 4 and 5, although its proof, provided in appendix C, requires a higher degree of technicality:

Proposition 6. The global fidelity of the optimal N-to-M cloner of maximally entangled states is upper bounded as

$$\begin{eqnarray}&&{{F}_{{\mbox{clon}}}}\left[ N\to M \right]\leqslant \left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }\;p_{{\mbox{succ}}}^{\left( N \right)},\end{eqnarray} \tag{ 36 }$$

where $\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }$ is the maximum eigenvalue of the average target state $\rho _{AV}^{\left( M \right)}:=\int {\mbox{d}}g\;\psi _{g}^{\otimes M}$ and $p_{{\mbox{succ}}}^{\left( N \right)}$ is the maximum probability density of correct identification of the input state, namely $p_{{\mbox{succ}}}^{\left( N \right)}:={{{\rm max} }_{\left\{ {{P}_{g}} \right\}}}\int {\mbox{d}}g\;{{\langle {{\psi }_{g}}{{|}^{\otimes N}}{{P}_{g}}\;|{{\psi }_{g}}\rangle }^{\otimes N}}\;$ the maximum running over all possible POVMs $\left\{ {{P}_{g}} \right\}$.

Similar to the proofs of the previous bounds, the proof here is based on two ingredients: the first is an upper bound on the fidelity derived from the optimization over all possible covariant cloners, the second is a translation of the upper bound in terms of the quantities $\shortparallel \rho _{AV}^{\left( M \right)}{{\shortparallel }_{\infty }}$ and $p_{{\mbox{succ}}}^{\left( N \right)}$. Regarding $\shortparallel \rho _{AV}^{\left( M \right)}{{\shortparallel }_{\infty }}$, this can be easily computed using the Schur's lemma (cf appendix C), which gives

$$\begin{eqnarray}&&\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }=\frac{{{p}_{M,j_{{\rm min} }^{\left( M \right)}}}}{d_{j_{{\rm min} }^{\left( M \right)}}^{2}}.\end{eqnarray} \tag{ 37 }$$

Regarding $p_{{\mbox{succ}}}^{\left( N \right)}$, we recall the expression of the optimal POVM for the estimation of a maximally entangled state from N copies, derived in [50], and given by

$$\begin{eqnarray}&&{{P}_{g}}:=|{{\eta }_{{\hat{g}}}}\rangle \langle {{\eta }_{{\hat{g}}}}|,\qquad |{{\eta }_{g}}\rangle :=U_{g}^{\otimes N}|\eta \rangle \qquad |\eta \rangle :=\oplus {\rm min} \left( N \right)N/2j=j{{d}_{j}}|{{\psi }^{(j,N)}}\rangle .\end{eqnarray} \tag{ 38 }$$

Using this fact, we have

$$\begin{eqnarray}&&p_{{\mbox{succ}}}^{(N)}:=\langle {{\eta }_{g}}|\psi _{g}^{\otimes N}|{{\eta }_{g}}\rangle =\sum {\rm min} \left( N \right)N/2j=j\sqrt{{{p}_{N,j}}}\;{{d}_{j}}2\qquad \forall g\in SU\left( 2 \right).\end{eqnarray} \tag{ 39 }$$

The detailed derivation of equation (36) is provided in appendix C.

Here we exhibit an economical cloner that achieves the upper bound of equation (36) in the limit of large M. For simplicity, we will assume that N and M are either both even or both odd. In this case, our cloner consists simply in embedding the input state $|{{\psi }_{g}}{{\rangle }^{\otimes N}}$ into the output space of M copies. This operation is described by the isometry ${{V}_{N,M}}$ defined by the relation

$$\begin{eqnarray}&&{{V}_{N,M}}\;|\psi _{g}^{\left( j,N \right)}\rangle =|\psi _{g}^{\left( j,M \right)}\rangle ,\end{eqnarray} \tag{ 40 }$$

for every $g\in SU\left( 2 \right)$ and for every $j\in \left\{ j_{{\rm min} }^{\left( N \right)},...,N/2 \right\}$. Interestingly, this economical cloner can be achieved using only local operations, because Alice and Bob just have to embed their input systems into the the output space of M clones, and this can be done without any communication between them. Cloning of bipartite states under the restriction of local operations and classical communication (LOCC) has been recently considered by Kumagai and Hayashi [51], who investigated the asymptotic scenario where both N and M are large. The problem in [51] was to copy a single, known bipartite state—a task that is made non-trivial by the LOCC restriction. Comparing this problem with ours, it is interesting to note that the copy machine provided in equation (40) does not require knowledge of which maximally entangled state is copied and does not require any exchange of classical information.

We now show that our cloner is asymptotically optimal. Indeed, its fidelity is given by

$$\begin{eqnarray}&&F\left[ N\to M \right]=\sum {\rm min} \left( N \right)N/2j=j\sqrt{{{p}_{N,j}}\;{{p}_{M,j}}}2.\end{eqnarray} \tag{ 41 }$$

Now, we know from equation (35) that the ratio ${{p}_{M,j}}/d_{j}^{2}$ is proportional to the binomial ${{B}_{M,j}}$. When M is large compared to N, the binomial is almost constant in the interval $\left[ j_{{\rm min} }^{(N)},N/2 \right]$ and the fidelity can be approximated as

$$\begin{eqnarray*} &&F\left[ N\to M \right]=\frac {p_{M,j_{\rm{min}}^{(M)}}}{d_{j_{\rm{min}}^{(M)}}^2}\left(\sum\limits_{j=j_{min}^\left( N \right)}^{N/2}\sqrt{{p}_{N,j}}{d}_{j}\right)^2 \left[1+O\left(\frac{N^2}{M}\right)\right] \\ &&=\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty } p_{\rm{succ}}^\left( N \right)\left[1+O\left(\frac{N^2}{M}\right)\right] \end{eqnarray*}$$

having used equations (37) and (39). This means that, asymptotically, our economical cloner saturates the upper bound of equation (36).

Remark (large N asymptotics).  For large N, the Gaussian approximation for the binomial allows one to approximate the probability distribution ${{p}_{N,j}}$ in equation (35) as

$$\begin{eqnarray}&&{{p}_{N,j}}=\sqrt{\frac{2}{\pi {{N}^{3}}}}\;2{{\left( 2j+1 \right)}^{2}}{\rm exp} [-\frac{2{{j}^{2}}}{N}]\left[ 1+O\left( \frac{1}{\sqrt{{{N}^{1-\delta }}}} \right) \right]+{{\eta }_{N}}\end{eqnarray} \tag{ 42 }$$

for arbitrary $\delta >0$ and ${{\eta }_{N}}$ vanishing faster than the inverse of any polynomial. Approximating the summation in equation (41) with a Gaussian integral, one can find the close form expression

$$\begin{eqnarray*}&&F\left[ N\to M \right]={{\left( \frac{\sqrt{4MN}}{M+N} \right)}^{3}}\left[ 1+O\left( \frac{1}{\sqrt{{{N}^{1-\delta }}}} \right) \right].\end{eqnarray*}$$

Intriguingly, the above fidelity is equal to the fidelity of the economical cloner for multiphase covariant states in dimension d = 4. In both cases, the fidelity of the economical cloner has the form

$$\begin{eqnarray*}&&F\left[ N\to M \right]={{\left( \frac{\sqrt{4MN}}{M+N} \right)}^{f}}\left[ 1+O\left( \frac{1}{\sqrt{{{N}^{1-\delta }}}} \right) \right],\end{eqnarray*}$$

where f is the number of free parameters needed to specify the input state. This observation, along with the observations made for phase- and multiphase- covariant cloning, suggests that the optimal economical cloners satisfy a universality property, which forces their fidelity to depend only on the numbers of free parameters, and not on the specific details of the states to be cloned.

Now we are going to show how to reach the optimal cloning fidelity with a suitable MP protocol. By now, the choice of the protocol should be obvious: estimate the state using the optimal POVM of equation (38), prepare $K={{M}^{1-\epsilon }}$ copies according to the estimate, and clone from K to M using the cloner of equation (40). The calculation of the fidelity, done in appendix D, gives the value

$$\begin{eqnarray}&&{{F}_{K}}\left[ N\to M \right]=p_{{\mbox{succ}}}^{\left( N \right)}\;\sqrt{\frac{8}{\pi {{\left( M+K \right)}^{3}}}}\left[ 1+O\left( \frac{1}{\sqrt{{{K}^{1-\delta }}}}+\frac{N}{K} \right) \right].\end{eqnarray} \tag{ 43 }$$

Now, choosing $K={{M}^{1-\epsilon }}$, in the large M limit we have

$$\begin{eqnarray*}\begin{array}{rcl} \sqrt{\frac{8}{\pi {{\left( M+K \right)}^{3}}}} & = & \frac{2{{B}_{M,0}}}{M}\left[ 1+O\left( \frac{K}{M} \right) \right] \\ {} & = & \left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }\left[ 1+O\left( \frac{K}{M} \right) \right], \\ \end{array}\end{eqnarray*}$$

and, therefore

$$\begin{eqnarray*}&&{{F}_{K}}\left[ N\to M \right]=p_{{\mbox{succ}}}^{\left( N \right)}\left\|\rho_{AV}^{\left( M \right)}\right\|_{\infty }\left[ 1+O\left( \frac{1}{\sqrt{{{K}^{1-\delta }}}}+\frac{N}{K}+\frac{K}{M} \right) \right].\end{eqnarray*}$$

This relation shows that asymptotically, the fidelity of our protocol is arbitrarily close to the fidelity of the optimal cloner. Once more, note that re-preparing M identical copies is strictly suboptimal, even in the asymptotic limit. Indeed, setting K = M in equation (43) we obtain

$$\begin{eqnarray*}&&{{F}_{M}}\left[ N\to M \right]=\frac{{{F}_{{\mbox{clon}}}}\left[ N\to M \right]}{\sqrt{{{2}^{3}}}}\left[ 1+O\left( \frac{1}{\sqrt{{{M}^{1-\delta }}}}+\frac{{{N}^{2}}}{M} \right) \right].\end{eqnarray*}$$

Note that the ratio between the fidelity of the naive MP protocol and that of the optimal cloner has the same value of the ratio in the case of multiphase-covariant cloners in dimension d=4. Again, it appears that the optimal economical cloners enjoy a universality property, where the relevant quantities depend only on the number of free parameters needed to describe the input state.

Here we give a precise quantitative meaning to the the statement that economical channels are far from MP channels. As a distance measure for channels, we use the trace distance, which for two channels $\mathcal{C}$ and $\tilde{\mathcal{C}}$ is defined as

$$\begin{eqnarray}&&\shortparallel \mathcal{C}-\tilde{\mathcal{C}}{{\shortparallel }_{1}}:=\mathop{{\rm max} }\limits_{|\psi \rangle \in {{\mathcal{H}}_{{\mbox{in}}}},\shortparallel |\psi \rangle \shortparallel =1}\left\|\mathcal{C}\left( \psi \right)-\tilde{\mathcal{C}}{{\left( \psi \right)}}\right\|_{1},\end{eqnarray} \tag{ 44 }$$

where ${{\mathcal{H}}_{{\mbox{in}}}}$ is the input Hilbert space of channels $\mathcal{C}$ and $\tilde{\mathcal{C}}$ and $\shortparallel A{{\shortparallel }_{1}}:={\mbox{Tr}}|A|$ denotes the trace norm of the operator A. The operational meaning of the trace distance between two channels is provided by Helstrom's theorem on minimum error discrimination: if one tries to discriminate between the two channels $\mathcal{C}$ and $\tilde{\mathcal{C}}$, given with prior probabilities p and $1-p$, respectively, one can achieve the average probability of error

$$\begin{eqnarray*}&&{{p}_{{\mbox{err}}}}=\frac{1-\shortparallel p\;\mathcal{C}-\left( 1-p \right)\tilde{\mathcal{C}}{{\shortparallel }_{1}}}{2},\end{eqnarray*}$$

by choosing the best input state in ${{\mathcal{H}}_{{\mbox{in}}}}$. This means that when the trace distance is close to one, the two channels $\mathcal{C}$ and $\tilde{\mathcal{C}}$ are almost perfectly distinguishable, even without using the assistance of additional ancillas. A simple bound on the trace-distance between an economical channel and an MP channel is given by the following:

Proposition 7. Let $\mathcal{C}$ and $\tilde{\mathcal{C}}$ be an economical channel and an MP channel, transforming density matrices on ${{\mathcal{H}}_{{\mbox{in}}}}$ into density matrices on ${{\mathcal{H}}_{{\mbox{out}}}}$, respectively. The trace distance between $\mathcal{C}$ and $\tilde{\mathcal{C}}$ is lower bounded as

$$\begin{eqnarray}&&\shortparallel \mathcal{C}-\tilde{\mathcal{C}}{{\shortparallel }_{1}}\geqslant 2\left( 1-d_{{\mbox{in}}}^{-1} \right).\end{eqnarray} \tag{ 45 }$$

The proof is provided in appendix E. Note that the lower bound tends to the maximum possible value $\shortparallel \mathcal{C}-\tilde{\mathcal{C}}{{\shortparallel }_{1}}=2$ when ${{d}_{{\mbox{in}}}}$ is large, meaning that for large input spaces economical channels and MP protocols produce almost orthogonal output states. Due to the above bound, there is no way to approximate an economical cloning channel with an MP protocol, even in the macroscopic limit $M\to \infty$. Hence, proving the global asymptotic equivalence with state estimation means proving a non-trivial statement about the performances of two radically different types of processes.