Superbroadcasting of continuous variables mixed states

We consider the problem of broadcasting quantum information encoded in the average value of the field from N to M>N copies of mixed states of radiation modes. We derive the broadcasting map that preserves the complex amplitude, while optimally reducing the noise in conjugate quadratures. We find that from two input copies broadcasting is feasible, with the possibility of simultaneous purification (superbroadcasting). We prove similar results for purification (M<=N) and for phase-conjugate broadcasting.


I. INTRODUCTION
Quantum cloning is impossible [1]. This means that one cannot produce a number of independent physical systems prepared in identical states out of a smaller amount of systems prepared in the same state. Since the formulation of the no-cloning theorem the search for quantum devices that can perform cloning with the highest possible fidelity gave rise to a whole branch in the literature. Optimal cloners have been found, for qubits [2,3,4], for general finite-dimensional systems [5], for restricted sets of input states [6,7], and for infinite-dimensional systems such as harmonic oscillators-the so called continuous variables cloners [8]. However, for the case of mixed states, a different type of cloning transformation can be considered-the so-called broadcasting-in which the output copies are in a globally correlated state whose local "reduced" states are identical to the input states. This possibility has been considered in Ref. [9], where it has been shown that broadcasting a single copy from a noncommuting set of density matrices is always impossible. Later, such a result has been considered in the literature as the generalization of the no-cloning theorem to mixed states. However, more recently, for qubits an effect called superbroadcasting [10] has been discovered, which consists in the possibility of broadcasting the state while even increasing the purity of the local state, for at least N ≥ 4 input copies, and for sufficiently short input Bloch vector (and even for N = 3 input copies for phase-covariant broadcasting instead of universal covariance [11]).
In the present paper, we analyze the broadcasting of continuous variable mixed states by a signal-preserving map. More precisely, this means that we consider a set of states obtained by displacing a fixed mixed state by a complex amplitude in the harmonic oscillator phase space, while the broadcasting map is covariant with respect to the (Weyl-Heisenberg) group of complex displacements. We will focus mainly on displaced thermal states (which are equivalent to coherent states that have suffered Gaussian noise), however, all results of the present paper hold in terms of noise of conjugated quadratures for the set of states obtained by displacing any fixed state.
As we will see, superbroadcasting is possible for continuous variable mixed states, namely one can produce a larger number of copies, which are purified locally on each use, and with the same signal of the input. For displaced thermal states, for example, superbroadcasting can be achieved for at least N = 2 input copies, with thermal photon number n in ≥ 1 3 , whereas, for sufficiently large n in at the input, one can broadcast to an unbounded number M of output copies. For purification (i.e. M ≤ N), quite surprisingly the purification rate is n out /n in = N −1 , independently on M. The particular case of 2 to 1 for noisy coherent states has been reported in Ref. [12]. We will prove also similar results for broadcasting of phase-conjugated copies of the input.
The paper is organized as follows. In Section II we introduce the problem of covariant broadcasting, deriving the general form of a covariant channel (trace-preserving CP map), and introduce a special channel that broadcast from N to M > N copies. In Section III we prove that such a channel is optimal for broadcasting any noisy displaced state.
In Section IV we consider the same problem for purification (i.e. M < N). In Section V we derive superbroadcasting for the output copies with a conjugate phase with respect to the originals. In Section VI we show the optimality by a simpler derivation, namely by exploiting the bounds from the theory of linear amplification (which is then based on supplementary assumptions). In Sec. VII we show a simple experimental scheme to achieve optimal broadcasting/purification. Section VIII closes the paper with a summary of results and some concluding remarks.

II. COVARIANT BROADCASTING FOR THE WEYL-HEISENBERG GROUP
We consider the problem of broadcasting N input copies of displaced (generally) mixed states of harmonic oscillators (with boson annihilation operators denoted by a 0 , a 1 , ..., a N −1 ) to M output copies (with boson annihilation operators b 0 , b 1 , ..., b M −1 ). In order to preserve the signal, the broadcasting map B must be covariant, i. e. in formula where D c (α) = exp(αc † − α * c) denotes the displacement operator, and Ξ represents an arbitrary N-partite state. It is useful to consider the Choi-Jamio lkowski bijective correspondence of completely positive (CP) maps B from H in to H out and positive operators R B acting on H out ⊗ H in , which is given by the following expressions where |Ω = ∞ n=0 |ψ n |ψ n is a maximally entangled vector of H ⊗2 in , and X τ denotes transposition of X in the basis |ψ n . In terms of the operator R B the covariance property (1) can be written as In order to deal with this constraint we introduce the multisplitter operators U a and U b , that perform the unitary transformations Notice that such transformations perform a Fourier transform over all input and output modes. Moreover, we will make use of the squeezing transformation S a 0 b 0 defined as follows Hence, upon introducing an operator B on modes b 1 , ..., b M −1 , a 0 , ..., a N −1 , the operator R B can be written in the form Notice that R B ≥ 0 is equivalent to B ≥ 0. The further condition that B is trace-preserving , one obtains the condition where a/a i denote all the input modes apart from a i , and similarly for b/b i .
We will now consider the map corresponding to Applying the corresponding map B to a generic N-partite state Ξ we get which is equivalent to Using the expression in Eq. (10) we obtain where and taking the complex conjugate of Eq. (4) we have Now, we can easily evaluate obtaining Hence, Eq. (13) can be rewritten as As an example, we will now consider N displaced thermal states from which we want to obtain M states, the purest as possible. Thanks to the covariance property, it is sufficient to focus attention on the output of ρ ⊗N 0 . For a tensor product of and recalling the following expression for the thermal states we obtain where The above state is permutation-invariant and separable, with thermal local state at each mode with average thermal photon More generally, for any state Ξ, the choice (10) gives M identical clones whose state can be written as Since for any mode c one has it is easy to verify that the superbroadcasting condition (output total noise in conjugate quadratures smaller than the input one), is equivalent to require smaller photon number at the output than at the input, namelȳ This can be true for any N > 1, and to any M ≤ ∞, since Actually, the solution given in Eq. (24) is optimal. To prove this, in the following we will show that the expectation of the total number of photons Tr ] of the M clones of ρ cannot be smaller than Mn ′′ . Since the multisplitter preserves the total number of photons we have to consider the trace We can write then, from Eq. (9) and positivity of B, one has In fact, one can easily check that the choice of B in Eq. (10) saturates the bound (32).
Also the more general solution given in Eq. (25) is optimal, in the sense that it represents the state of M identical clones with minimal photon number, which is given by Notice that forn = 0 one has N coherent states at the input, andn ′′ = M −N M N , namely one recovers the optimal cloning for coherent states of Ref. [14].
From Eq. (26), one can see that our optimization maximally reduces the total noise in conjugate quadratures. Alternatively, one might minimize the output entropy, which would be informationally more satisfactory. This case, however, turns out to be a non trivial task, and is beyond the scope of this article.

IV. PURIFICATION
For M < N one can look for the optimal "purification" map with M output systems.
The result can be obtained as in section II, provided that we replace the operator S a 0 b 0 in where now µ = for all α. Consequently, R B has the form and trace preservation is equivalent to Now, we consider the map with The corresponding output for given input state Ξ is given by The integral gives a thermal state for the mode b 0 with average photon numbern ′ such that Hence, the single-site reduced state is a thermal state with a number of thermal photons which is rescaled with respect to the input by a factor N, independently of the number of output copies. The same analysis as in section III shows that this is the minimum output number compatible with complete positivity of the map B, and then is optimal.
For a generic input state Ξ the local output state is given by Notice that both Eq. (23) and Eq. (42) given ′′ =n N also for M = N, and this result can be proved as follows. The difference from the previous proof resides in the fact that the squeezing operator S a 0 b 0 is ill defined in this case. However, once we unitarily transform a/a 0 , the squeezing operator on modes a 0 and b 0 is not needed, and it is sufficient to remark that the representation D b 0 ( √ N α) ⊗ D a 0 ( √ N α) * is abelian, and its joint eigenvectors can be written as Consequently, the covariance condition for the map B is given by with the trace-preserving constraint expressed by We consider the following form for ∆ a/a 0 ,b/b 0 (γ) which gives and then we can prove optimality by the same technique used in the other cases. The output of Ξ is given by which is separable, and its local states are thermal states with n ′′ =n N . (51)

V. PHASE-CONJUGATING BROADCASTING
We now consider the problem of broadcasting with simultaneous phase-conjugate output.
This means that we look for the optimal transformation where the average of the output field of each copy is the complex conjugate with respect to the value of the input one. The covariance property of such a map is the following for all α, and in terms of R C this corresponds to We will use the same multisplitters defined in Eq. (4), and introduce the following beamsplitter [U a 0 b 0 , a n ] = [U a 0 b 0 , b n ] = 0, n > 0 Analogously to the previous sections, the covariance condition translates in the following form for R C : where C is an operator on modes b 1 , . . . , b M −1 , a 0 , . . . , a N −1 , and the trace-preserving condition requires that which finally gives We now consider the map corresponding to Applying such a map to a generic N-partite state Ξ we get where ξ τ = Tr a/a 0 [U † a Ξ τ U a ]. Moreover, one has and Eq. (60) gives where ξ = Tr a/a 0 [U a ΞU † a ], and For Ξ = ρ ⊗N 0 we have simply ξ = ρ 0 , and this implies that a simple scheme to achieve this map is the following. First, the N input states interact through an N-splitter, then the system labeled 0 carrying all the information about the coherent signal is measured by heterodyne detection, and for any outcome γ a coherent state with amplitude M N γ * is generated. Finally, the prepared state is sent through an M-splitter along with M − 1 modes in the vacuum state.
The output state C (ρ ⊗N 0 ) is now given by which is equal to and its single-site reduced state is simply a thermal state with Notice that this is independent of the number of output copies, and is the same average number as the one for superbroadcasting in the limit M → ∞. More generally, the local output for generic input state Ξ is The proof of optimality is analogous to the proof for the superbroadcasting map. It is sufficient to replace U a 0 b 0 with S a 0 b 0 in Eqs. (29) and (30).

VI. A PROOF OF THE OPTIMALITY IN TERMS OF LINEAR AMPLIFIERS
We are interested in a transformation that provides M (generally correlated) modes b 0 , b 1 , ..., b M −1 from N uncorrelated modes a 0 , a 1 , ..., a N −1 , such that the unknown complex amplitude is preserved and the output has minimal phase-insensitive noise. In formula, we have input uncorrelated modes for all i = 0, 1, .., N − 1, where Heisenberg uncertainty relation is taken into account. The output modes should satisfy b i = α , and we look for the minimal Γ. The minimal Γ can be obtained by applying a fundamental theorem for phase-insensitive linear amplifiers [15]: the sum of the uncertainties of conjugated quadratures of a phase-insensitive amplified mode with (power) gain G is bounded as follows.
where A and B denotes the input and the amplified mode, respectively. Our transformation can be seen as a phase-insensitive amplification from the mode A = 1 Hence, the bound can be rewritten as In the present case, since modes a i are uncorrelated, one has i=0 γ i , and so the bound Eq. (71) is written as On the other hand, one has Eqs. (74) and (75) together give the bound for the minimal noise Γ The example in the previous sections corresponds to γ =n + 1 2 and Γ =n ′′ + 1 2 . A similar derivation gives a bound for purification, where N > M. In such a case G < 1, and Eq.
(71) is replaced with and one obtains the bound We would like to stress that the derivation of all bounds in the present section relies on the theorem of the added noise in linear amplifiers, namely only linear transformations of modes are considered. Hence, in principle, these bounds might be violated by more exotic and nonlinear transformations. Therefore, the derivation of Eq. (32) is stronger, since it has general validity.
By a similar derivation, using the bound for phase-conjugated amplifiers ∆X 2 B + ∆Y 2 B ≥ G(∆X 2 A + ∆Y 2 A ) + G−1 2 , one can obtain the bound for phase-conjugation broadcasting

VII. EXPERIMENTAL IMPLEMENTATION
The optimal broadcasting can be easily implemented by means of an inverse N-splitter which concentrates the signal in one mode and discards the other N − 1 modes. The each. In Fig. 1 we sketch the scheme for 2 to 3 superbroadcasting.
FIG. 1: Experimental scheme to achieve optimal superbroadcasting from 2 to 3 copies. This setup involves just a beam splitter, a phase-insensitive amplifier and a tritter, which in turn can be implemented by two suitably balanced beam splitters. The phase-insensitive amplifier can be implemented by a beam splitter and heterodyne-assisted feed-forward. The output copies carrying the same signal as the input ones are locally more pure, the noise being shifted to classical correlations between them.
phase and amplified intensity. For achieving the optimal purification, one simply uses an inverse N-splitter which concentrates the signal in one mode and discards the other N − 1 modes. Then by N-splitting with N − 1 vacuum modes, one obtained N purified signals (although classically correlated).

VIII. CONCLUSION
In conclusion, we proved that broadcasting of M copies of a mixed radiation state starting from N < M copies is possible, even with lowering the total noise in conjugate quadratures.
Since the noise cannot be removed without violating the quantum data processing theorem, the price to pay for having higher purity at the output is that the output copies are correlated.
Essentially noise is moved from local states to their correlations, and our superbroadcasting channel does this optimally. We obtained similar results also for purification (i.e. M ≤ N), along with the case of simultaneous broadcasting and phase-conjugation, with the output copies carrying a signal which is complex-conjugated of the input one. Despite the role that correlations play in this effect, no entanglement is present in the output (as long as the single input copy has a positive P -function), as it can be seen by the analytical expression of the output states. Moreover, a practical and very simple scheme for experimental achievement of the maps has been shown, involving mainly passive media and only one parametric amplifier.
The superbroadcasting effect has a relevance form the fundamental point of view, opening new perspectives in the understanding of correlations and their interplay with noise, but may be also promising from a practical point of view, for communication tasks in the presence of noise.