Introduction

Design and analysis of the optimal protocols for processing, storing and transmitting information is the subject of the fundamental research field of information theory pioneered in the last century by C. E. Shannon1. In reality, information needs to be recorded necessarily onto a physical medium and transmitted via a physical channel. Therefore, in addition to information theory, communication protocols should obey the laws of physics. The progress in microminiaturization of data-processing systems leads to use of information carriers that cannot be described by a classical theory and behave according to quantum mechanics (for example, photons, electrons, atoms, etc.). The task of quantum information and communication theory is to study the laws of information transmission and processing in quantum mechanical systems2,3,4,5.

A large part of quantum communication theory is devoted to the transmission of electromagnetic radiation via bosonic Gaussian channels4,6,7,8. The latter are formally defined as completely positive and trace preserving operations mapping Gaussian input states into Gaussian output states. The most relevant channels are also invariant under phase space rotations and are called phase-insensitive. For example, the transmission of optical quantum states through realistic physical devices3 (like for example, optical fibres, free space communication lines, dielectric media, etc.) can be described by phase-insensitive Gaussian channels.

In the spirit of classical communication theory1, one may ask what is the minimum amount of ‘disorder’ achievable at the output of a Gaussian channel. For quantum systems there are two main figures of merit which can be used to quantify the idea of disorder9,10,11,12: the von Neumann entropy and the concept of majorization. The entropy of a state ρ is defined as S(ρ)=−Tr[ρ log(ρ)] and one can say that a state ρ1 is more disordered than ρ2 if S(ρ1)>S(ρ2). A different (and stronger) way of saying that ρ1 is more disordered than ρ2 is given by the following equation:

where the vectors and consist of the eigenvalues of the respective states arranged in decreasing order. If the condition (1) is satisfied then one says that ρ2 majorizes ρ1 and this is usually indicated by the expression ρ2ρ1. The previous definition has a very intuitive operational interpretation since it can be shown that ρ2ρ1 if and only if ρ1 can be obtained from ρ2 by a proper convex combination of unitary operations9,10,11,12. These considerations extend also to the infinite-dimensional case13 relevant for the quantum description of electromagnetic modes.

According to the previous ideas of disorder, for a phase-insensitive bosonic Gaussian channel it was conjectured14 that:

(1) the minimum-output entropy is achieved by coherent input states,

(2) the output states resulting from coherent input states majorize all other output states.

A graphical representation of the last property is given in Fig. 1. Both conjectures have broad implications in many research areas like classical and quantum optics, telecommunication engineering, mathematical and statistical physics, and for this reason they attracted the attention of many scientists. In particular, the validity of conjecture (1) and (2) has a number of important corollaries and relations ranging from entanglement theory15,16,17,18, channel capacities6,14,18,19,20,21,22, entropic inequalities19,20,23,24 to quantum discord25,26.

Figure 1: Graphical representation of the majorization conjecture (2).
figure 1

A coherent state |α› ‹ α| and an arbitrary state ρ are both transmitted through the same phase-insensitive Gaussian channel Φ. The respective output states always satisfy the majorization relation Φ(|α› ‹ α|)Φ(ρ). This means that coherent input states produce less ‘noise’ at the output of the communication channel.

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In the last decade, many analytical and numerical evidences supporting both conjectures were presented14,19,20,21,22,23,24,27,28,29,30,31 but a general proof was missing. Only very recently the first one was finally proved18,32 under the assumption of a finite mean energy. In this work we prove the second conjecture (2) and highlight some of its implications. Moreover it is easy to show that ρ2ρ1 implies S(ρ1)≥S(ρ2), therefore the statement (2) is stronger than the conjecture (1) and the result presented in this work can also be seen as a proof of the minimum-output entropy conjecture, without any energy constraint. Thus both gaps in the theory are now definitely closed.

Results

Gaussian channels

Every quantum channel4,33 can be described as a global unitary operation applied to the tensor product of the state of the system ρS and the state of an appropriate environment ρE:

Single-mode phase-insensitive channels21 can be classified in three main classes , n and . Physically, represents a thermal channel, which can be realized by a beamsplitter of transmissivity, η, mixing the input signal with a thermal state, ρE, with mean photon number, N:

where a and aE are the annihilation operators of the system and of the environment, respectively. Then, n is the classical additive noise channel where the input state is displaced according to a random Gaussian distribution of variance n and, finally, is the quantum amplifier where the state of the environment is in a thermal state ρE:

More precisely, these channels can be defined according to their action on the quantum characteristic function in the following way8:

Any of the previous phase-insensitive channels, which we denote by the symbol Φ, can always be decomposed22,32 into a pure-loss channel followed by a quantum-limited amplifier:

for appropriate values of κ and η. In what follows we will make use of this decomposition and for simplicity we introduce the symbols k and η for indicating the respective quantum-limited channels with N=0.

Complementary channels

One can associate to every channel (2) the respective complementary channel defined as

and physically representing the flow of information from the input state to the environment4,8. An important property of complementary channels is that, whenever the system and the environment are in a pure state, the nonzero spectra of the output states Φ(|ψS› ‹ ψS|) and are equal. This is a simple consequence of the Schmidt decomposition of the global pure state (for an explicit proof see ref. 33).

In what follows the complementary channel of the quantum-limited amplifier will play an important role. In this case ρE is the vacuum and U is the two-mode squeezing operation8 acting in the Heisenberg picture according to equation (4) and

From equation (4), tracing out the environment, one obtains k defined in equation (7) with N=0. From equation (10) instead, tracing out the system, we get the complementary channel acting on the characteristic function as

Importantly, the complementary channel does not have the same structure as the amplifier given in equation (7), since the complex variable μ appears conjugated in the RHS of equation (11). In quantum optics this effect is known as phase conjugation or time reversal and corresponds to the positive (but not completely positive) map

which at the level of density operators behaves as transposition T(ρ)=ρT in the Fock basis and therefore it preserves the eigenvalues. This means that each time we are interested in spectral properties of the output state (as in the proof of Lemma 1), we can neglect the effect of the phase conjugation operator T.

Minimization of strictly concave functionals

Before giving the proof of the majorization conjecture we consider an important minimization problem.

Let F: → be a unitary invariant and strictly concave functional acting on the infinite-dimensional Hilbert space of density matrices of a single bosonic mode. We assume that F can take values in [0, +∞], having in mind applications to the von Neumann entropy. Unitary invariance means that F(UρU†)=F(ρ) for every unitary matrix U, while strict concavity means that

and the equality is obtained only for ρ1=ρ2. The problem that we want to address is the minimization of such functionals at the output of a phase-insensitive channel, where the optimization is performed over all possible input states:

An important case is when the functional is replaced by the von Neumann entropy F(ρ)=S(ρ)=−Tr[ρ log(ρ)], and the minimization problem reduces to the minimum-output entropy conjecture (1)14,30. We recall that this conjecture claims that the minimum is achieved by input coherent states of the form

and was recently proved32. With the next lemma, we are going to show that this extremal property of coherent states is more general and can be applied to every functional of the kind that we have previously introduced.

Lemma 1: Let Φ be a phase-insensitive bosonic channel. Then, for every non-negative unitary invariant and strictly concave functional, F, and for every quantum state, ρ, we have

where |α› is any coherent state. Moreover the equality is achieved only if ρ is a coherent state.

Proof: for a pure-loss channel , the proof is simple. Indeed coherent states are mapped to pure coherent states under the action of . Since F is concave and unitary invariant, when applied to pure states it necessarily achieves its minimum. So, in this case, inequality (16) is satisfied (the uniqueness property of coherent states is considered in the last part of this proof and is a consequence of Lemma 2 of the Methods section).

For a general phase-insensitive channel Φ we can use the decomposition of equation (8). A direct consequence of this decomposition is that we just need to prove the lemma for the minimal-noise amplification channel Φ=κ since coherent states remain coherent after a beam splitter. Let be the conjugate channel of κ. Again, can be itself decomposed according to the structure of equation (8). Indeed, from a direct application of equations (7, 11, 12) one can verify that

where T is the phase conjugation operator and η=1−1/κ.

Let be the set of all pure input states minimizing the functional F at the output of the channel κ. We need to show that coincides the set of coherent states. Let us take an optimal state |ψ› ε. From the property of complementary channels and of the phase conjugation operator mentioned before we have that, κ(|ψ› ‹|ψ|), and have the same spectrum. Since F is unitary invariant, it necessarily depends only on the eigenvalues and we have . Therefore,

where {|ψj› } is the ensemble of states obtained after the beam splitter:

From the concavity of F we have

By hypothesis |ψ› ε and so F[κ(ψ› ‹ ψ|)]≤F[κ(ψj› ‹ ψj|)] for each j. This can be true only if the inequality (19) is saturated and, from the hypothesis of strict concavity, we get

From the definition of the quantum amplifier given in equation (7), it is evident that equal output states are possible only for equal input states: |ψj› =|ψ′› for every j. As a consequence equation (18), reduces to

But now comes into play an important property of the beamsplitter which is known from the field of quantum optics34,35,36, namely that only coherent states remain pure under the action of a beamsplitter (Lemma 2 in the Methods section). Therefore, since equation (21) is valid for every choice of |ψ› ε, then necessarily contains only coherent states. Moreover, for every Gaussian channel a displacement of the input state corresponds to a (possibly different) displacement of the output state7,8, which obviously does not change the entropy. Since coherent states are equivalent up to displacement operations it means that coincides with the whole set of coherent states.

Majorization at the output of the channel

We can finally state our main result which proves the validity of the majorization conjecture (2). A graphical representation of this property is given in Fig. 1.

Proposition 1: Let Φ be a phase-insensitive bosonic channel. Then, for every input state ρ,

where |α› is any coherent state.

Proof: Let be the class of real non-negative strictly concave functions f defined on the segment [0, 1]. Consider the following functional

where fε. Then F is well-defined with values in [0, +∞], since all the terms in the series are non-negative. Moreover, it is unitary invariant and it can be shown that the strict concavity of f as a function of real numbers implies the strict concavity of F with respect to quantum states12. Therefore the previous lemma can be applied and we get, for every state ρ and every strictly concave function f,

A well-known theorem9,10,11 in the finite dimensional case states that ρ2ρ1 if and only if for every concave function f. Moreover, a similar result is valid also for strictly concave functions fε and in infinite dimensions (see Lemma 3 in the Methods section). This concludes the proof.

As a final remark, as the von Neumann entropy is a strictly concave functional11, we get an alternative proof (with respect to the one given in ref. 32) of the minimal-output entropy conjecture. By applying Lemma 1 with the choice F(ρ)=−Tr[ρ log(ρ)], we get a slightly stronger version of the conjecture (1): the minimum output entropy of a phase-insensitive channel is achieved only by coherent input states.

Notice that, differently from the proof presented in ref. 32, this result does not require the assumption that the mean energy of the input should be finite and proves also that coherent states are the unique optimizers. Moreover, choosing f(x)=x−xp, p>1, leads to the proof of the similar statement for the minimal-output Renyi entropies of all orders p>1.

Discussion

The main result of this paper is that every output state of a phase-insensitive bosonic Gaussian channel is majorized by the output associated to a coherent input state (proof of the majorization conjecture). We also prove that coherent input states are the unique minimizers of arbitrary non-negative strictly concave output functionals and, in particular, of the von Neumann entropy (minimum-output entropy conjecture). As compared with the proof of the minimal output entropy conjecture given in ref. 32, our result does not require the finiteness of the mean energy and proves the uniqueness of coherent states.

Our work, while closing two longstanding open problems in quantum communication theory, has a large variety of implications and consequences. For example, by using Lemma 1 and Proposition 1 one can: compute the entanglement of formation of non-symmetric Gaussian states (see the last section of ref. 18), evaluate the classical capacity of Gaussian channels18 and compute the exact quantum discord25 for a large class of channels26. Moreover, from Proposition 1, we conclude that coherent input states minimize every Schur-concave output function like Renyi entropies of arbitrary order19,20,23,24. Finally, it is a simple implication that the pure entangled state |Ψout› obtained from a unitary dilation of a phase-insensitive Gaussian channel is more entangled than the output state |Ψout› ′ obtained with a coherent input. What is more, from the well-known relationship between entanglement and majorization9, we also know that |Ψout› ′ can be obtained from |Ψout› with local operations and classical communication. The previous facts are just some important examples while a detailed analysis of all the possible implications will be the subject of future works.

Methods

Some useful facts

To make our analysis self-contained, in this section we present two properties (Lemma 1 and Lemma 2), which are used in the proof of the majorization conjecture.

Lemma 2: Coherent states are the only input pure states which produce a pure output for a beamsplitter.

This property is more or less implicit in several quantum optics papers34,35,36. Here we present a complete proof, following an argument similar to one used in ref. 34, but using the formalism of quantum characteristic functions.

Proof: Let be the beamsplitter of transmissivity η, 0<η<1, and

Then the complementary channel, which is the beamsplitter of transmissivity 1−η satisfies a similar relation

as the outputs of complementary channels have identical nonzero spectra. Therefore we have

where U is the unitary implementing the minimal dilation of η. The corresponding canonical transformation of the annihilation operators a, aE for the system and the environment is

and the environment mode aE, aE† is in the vacuum state. In phase space, this produces a symplectic transformation in the variables of the characteristic functions:

By letting zE=0 and z=0, respectively, we obtain

Thus, after the change of variables , and denoting , we get

The function ω(z), as well as the characteristic function χ(z), is continuous and satisfies . The only solution of equation (33) satisfying these conditions is the exponential function for some complex α. Thus we obtain

which is the characteristic function of a coherent state |α› .

Lemma 3: Given two (finite or infinite dimensional) vectors λ and λ ′ whose elements are non-negative and normalized , the following two relations are equivalent:

for every function fε, where is the class of real non-negative strictly concave functions defined on the segment [0, 1].

Proof: It is well-known in finite-dimensional majorization theory9,10,11,12 that if λ ′ λ then inequality (36) is satisfied for every concave function and so, in particular, for strictly concave functions. From the infinite dimensional generalization of the Horn–Schur theorem13 one can extend this result to all functions fε, using the fact that the two series appearing in inequality (36) converge unconditionally to a value in [0, +∞].

To prove the converse implication suppose that the majorization relation (35) is not valid, then we construct an fε which violates the condition (36). As shown in ref. 11, a simple concave (but nonstrictly concave) function can be found in a constructive way by using the following ansatz:

If λ′ λ then there exists a smallest integer n for which . It is easy to show11 that, by choosing , the function f0 violates the condition (36), that is, there is a positive and finite δ such that

However, this does not conclude our proof because f0 is not strictly concave. For this reason we take a slightly different function

which is strictly concave for every ∈>0 and belongs to the class . Now, for an arbitrary vector λ, by using the positivity and the normalization of the elements {λi} we get the following convergence:

From the last continuity relation together with equation (38) we get:

The last term can be made positive by choosing ∈<δ. Summarizing, we have shown that whenever the majorization relation (35) is not satisfied, there exists a small but finite ∈ such that f ∈ violates the inequality (36). Therefore the two conditions equation (35) and equation (36) are equivalent.

Additional information

How to cite this article: Mari, A. et al. Quantum state majorization at the output of bosonic Gaussian channels. Nat. Commun. 5:3826 doi: 10.1038/ncomms4826 (2014).