Gaussian optimizers for entropic inequalities in quantum information
We survey the state of the art for the proof of the quantum Gaussian optimizer conjectures of quantum information theory. These fundamental conjectures state that quantum Gaussian input states are the solution to several optimization problems involving quantum Gaussian channels. These problems are the quantum counterpart of three fundamental results of functional analysis and probability: the Entropy Power Inequality, the sharp Young’s inequality for convolutions and the theorem “Gaussian kernels have only Gaussian maximizers”. Quantum Gaussian channels play a key role in quantum communication theory: they are the quantum counterpart of Gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime. The quantum Gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space. The restriction of the quantum-limited Gaussian attenuator to input states diagonal in the Fock basis coincides with the thinning, the analogue of the rescaling for positive integer random variables. Quantum Gaussian channels provide then a bridge between functional analysis and discrete probability.
Gaussian optimizers for entropic inequalities in quantum information
quantum information theory \kwdquantum Gaussian channels \kwdentropic inequalities \kwdthinning \kwdEntropy Power Inequality \kwdsharp Young’s inequality for convolutions
- 1 Introduction
- 2 Gaussian optimizers in functional analysis
- 3 Quantum Gaussian systems
- 4 The minimum output entropy conjecture
5 Gaussian optimizers for entropic inequalities in quantum information
- 5.1 Quantum Gaussian channels have Gaussian maximizers
- 5.2 The Entropy Photon-number Inequality
- 5.3 The sharp Young’s inequality for the beam-splitter
- 6 The thinning
- 7 Quantum conditioning and the quantum Entropy Power Inequality
- 8 Conclusions and perspectives
Gaussian functions play a key role in both functional analysis and probability, and are the solution to several optimization problems involving Gaussian kernels. The most prominent among these problems is determining the norms of the Gaussian integral kernels that send a function to the function with . In the seminal paper “Gaussian kernels have only Gaussian maximizers” [lieb1990Gaussian], Lieb proved that these norms are achieved by Gaussian functions. A closely related fundamental result is the sharp Young’s inequality for convolutions [beckner1975inequalities, brascamp2002best, fournier1977sharpness, christ_near-extremizers_2011, bennett_brascamp-lieb_2008], stating that for any , the ratio with and is maximized by Gaussian functions, where denotes the convolution of with . This inequality has several fundamental applications, such as a proof of the Entropy Power Inequality [lieb1978proof, dembo1991information, cover2006elements], of the Brunn-Minkowski inequality [gardner2002brunn, cover2006elements] and Lieb’s solution [lieb1978proof, lieb2014proof] of Wehrl’s conjecture [wehrl1979relation, anderson1993information], stating that coherent states minimize the Wehrl entropy. The theorem “Gaussian kernels have only Gaussian maximizers” and the sharp Young’s inequality for convolutions are among the most important inequalities of functional analysis (see e.g. the book [lieb2001analysis]).
The Entropy Power Inequality [dembo1991information, stam1959some, shannon2001mathematical] states that the Shannon differential entropy of the sum of two independent random variables with values in and given Shannon differential entropies is minimum when the two random variables are Gaussian, and is a fundamental tool of information theory [cover2006elements]. The Entropy Power Inequality was introduced by Shannon to provide an upper bound to the information capacity of non-Gaussian channels [shannon2001mathematical], and was later used to bound the information capacity region of the Gaussian broadcast channel [bergmans1974simple] and the secret information capacity of the Gaussian wiretap channel [leung1978gaussian]. The Entropy Power Inequality was also employed to prove the convergence in relative entropy for the central limit theorem [barron1986entropy].
Quantum information theory [nielsen2010quantum, hayashi2016quantum, wilde2017quantum, holevo2013quantum] is the theory of the transmission and the processing of the information stored in quantum systems. Most of nowadays communications are made with electromagnetic signals traveling through optical fibers or free space. Quantum Gaussian channels [caves1994quantum, braunstein2005quantum, weedbrook2012gaussian, holevo2015gaussian] are the quantum counterpart of Gaussian integral kernels, and an -mode quantum Gaussian channel provides the mathematical model for the propagation of modes of the electromagnetic radiation along an optical fibre or free space in the quantum regime. For this reason, quantum Gaussian channels play a key role in quantum communication theory.
The subject of this review is the generalization of all the above inequalities for the convolution and for Gaussian integral kernels to quantum Gaussian channels. The solutions to the resulting quantum optimization problems are conjectured to be quantum Gaussian states, the quantum counterpart of Gaussian probability measures. This Gaussian optimizer problem arose in quantum information theory for the determination of the classical information capacity of phase-covariant quantum Gaussian channels [holevo2001evaluating, giovannetti2004minimum, giovannetti2004minimumb, de2014classical]. Indeed, proving that the coherent states constitute an optimal coding requires to prove a minimum output entropy conjecture, stating that the coherent input states minimize the output entropy of -mode phase-covariant quantum Gaussian channels (Theorem 4.1). This conjecture implies that both the minimum output entropy and the classical capacity of phase-covariant quantum Gaussian channels are additive with respect to the tensor product, i.e., that entanglement does not increase the communication rate. While the minimum output entropy of any classical channel is trivially additive, this property does not hold in general for quantum channels [hastings2009superadditivity]. The proof of the minimum output entropy conjecture has then been a fundamental result, which required more than ten years [giovannetti2010generalized, garcia2012majorization, garcia2014holy, giovannetti2015solution, mari2014quantum, giovannetti2014ultimate] (see the review [holevo2015gaussian]; see also [holevo2016constrained] for the capacity of non phase-covariant quantum Gaussian channels).
Proving that the coherent states constitute an optimal coding for the Gaussian broadcast channel requires a constrained version of the minimum output entropy conjecture. This constrained version states that quantum Gaussian input states minimize the output entropy of -mode quantum Gaussian channels among all the input states with a given entropy [guha2007classical, guha2007classicalproc, guha2008multiple, guha2008capacity, qi2016capacities] (Conjecture 5.1). The constrained minimum output entropy conjecture also implies the converse theorems for the triple trade-off coding with the quantum-limited attenuator and amplifier [wilde2012information, wilde2012quantum, qi2016capacities]. The conjecture has been generalized to the Entropy Photon-number Inequality [guha2008entropy, guha2008multiple], stating that quantum Gaussian input states minimize the output entropy of the beam-splitter among all the couple of input states each with a given entropy 5.13. Moreover, it has been realized [de2016gaussiannew] that the constrained minimum output entropy conjecture would follow from the generalization of the theorem “Gaussian kernels have Gaussian maximizers” to -mode quantum Gaussian channels (Conjecture 5.4). Since the beam-splitter is the quantum counterpart of the convolution, the Entropy Photon-number Inequality is the quantum counterpart of the Entropy Power Inequality. Based on this relation, we conjecture for the first time in this review the validity of a sharp Young’s inequality for the beam-splitter (Conjecture 5.14).
The proof of all the above quantum inequalities has been completed only in some particular cases, and is currently an active field of research. The constrained minimum output entropy conjecture has been proven only for one-mode quantum Gaussian channels [de2015passive, de2016gaussian, de2017gaussian, de2016gaussiannew, qi2017minimum] or for input states diagonal in some joint product basis [de2017multimode]. These results are based on a new majorization theorem for one-mode quantum Gaussian channels [de2015passive] (Theorem 5.2). The majorization result has been extended to single-jump lossy quantum channels [de2016passive], but unfortunately it fails for multi-mode quantum Gaussian channels [de2016passive]. The proof of the constrained minimum output entropy conjecture for one-mode quantum Gaussian channels made possible the proof of the fundamental relation between the von Neumann and the Wehrl entropy, stating that for any , -mode quantum Gaussian states have the minimum Wehrl entropy among all the -mode quantum states with a given von Neumann entropy [de2017wehrl]. For generic , the theorem “Gaussian kernels have Gaussian maximizers” has been proven only for one-mode quantum Gaussian channels [de2016pq], while for -mode channels the theorem has been proven only for [giovannetti2015majorization, holevo2015gaussian] and [frank2017norms, holevo2017quantum]. A proof of the Entropy Photon-number Inequality has been attempted with the quantum analogue of the heat semigroup technique of the proof of the Entropy Power Inequality by Blachman and Stam. This technique led instead to the proof of the quantum Entropy Power Inequality [konig2014entropy, konig2016corrections, konig2013limits, de2014generalization, de2015multimode, de2017gaussian] (Theorem 7.1), which provides a lower bound to the output entropy of the beam-splitter in terms of the entropies of the two inputs. This bound is strictly lower than the output entropy achieved by Gaussian input states, hence the quantum Entropy Power Inequality is strictly weaker than the Entropy Photon-number Inequality, that is still an open conjecture. The same heat semigroup technique led to the proof of the quantum conditional Entropy Power Inequality [koenig2015conditional, de2018conditional] and of the quantum Entropy Power Inequality for the quantum additive noise channels both in the unconditioned [huber2017geometric] and conditional [huber2018conditional] versions. The quantum conditional Entropy Power Inequality (Theorem 7.2) determines the minimum quantum conditional von Neumann entropy of the output of the beam-splitter or of the squeezing among all the input states where the two inputs are conditionally independent given the memory and have given quantum conditional entropies. This inequality has been exploited to prove an uncertainty relation for the conditional Wehrl entropy [de2017uncertainty]. These Entropy Power Inequalities have stimulated the proof of similar inequalities in different contexts, such as the qubit swap channel [audenaert2016entropy, carlen2016quantum] and information combining [hirche2017bounds]. The implications among the main results and conjectures for quantum Gaussian channels are summarized in Figure 1.
As a possible approach towards the proof of the unsolved entropic inequalities for quantum Gaussian channels, we mention that sharp functional inequalities in the commutative setting have been recently studied using the theory of optimal transport [villani_optimal_2009]. These methods led to e.g. quantitative stability results for isoperimetric [figalli_mass_2010], Sobolev and log-Sobolev [figalli_sharp_2013, fathi_quantitative_2016] inequalities. Ideas from optimal transport are also implicit in the solution of Shannon’s problem on the monotonicity of entropy [artstein_solution_2004]. Recently, transportation distances have been proposed in the quantum fermionic setting [carlen_analog_2014, carlen_gradient_2017] and have then been extended to quantum Gaussian systems [rouze2017concentration, rouze2017relating] (see also [datta2017contractivity]).
An interesting particular case of the inequalities for quantum Gaussian channels is when the input states are diagonal in the Fock basis [guha2016thinning, johnson2017bruijn]. This provides a link between quantum Gaussian channels and classical discrete probability theory. The restriction of the one-mode quantum-limited attenuator to input states diagonal in the Fock basis is the linear map acting on discrete classical probability distributions on known as thinning [de2015passive]. The quantum-limited attenuator is the quantum Gaussian channel that models the attenuation of electromagnetic signals. The thinning has been introduced by Rényi [renyi1956characterization] as a discrete analogue of the rescaling of a continuous real random variable, and has been involved with this role in discrete versions of the central limit theorem [harremoes2007thinning, yu2009monotonic, harremoes2010thinning], of the Entropy Power Inequality [yu2009concavity, johnson2010monotonicity] and of Young’s inequality [lanconelli2013holder]. Most of these results require the ad hoc hypothesis of the ultra log-concavity (ULC) of the input state. In particular, the Restricted Thinned Entropy Power Inequality [johnson2010monotonicity] states that the Poisson input probability distribution minimizes the output Shannon entropy of the thinning among all the ULC input probability distributions with a given Shannon entropy. The results on quantum Gaussian channels presented in this review led to the proof of new entropic inequalities for the thinning that apply to any probability distribution, regardless of whether they satisfy the ULC assumption. Quantum Gaussian states correspond to geometric probability distributions. The inequalities on quantum Gaussian channels imply that geometric input probability distributions both achieve the norms of the thinning [de2016pq] (Theorem 6.5) and minimize its output entropy among all the input probability distributions with a given entropy [de2016gaussian] (Theorem 6.4).
The review is structured as follows. In 2, we present the classical results for Gaussian optimizers in functional analysis. In 3, we introduce Gaussian quantum systems and channels, and the necessary notions of quantum mechanics. In 4 we present the minimum output entropy conjecture and its proof. In 5, we present all the conjectures on Gaussian optimizers in quantum information including the new sharp Young’s inequality for the beam-splitter, together with the state of the art in their proofs. In 6, we present the thinning and its relation with the results for quantum Gaussian channel. In 7, we present the quantum Entropy Power Inequality and its proof. Moreover, we introduce the quantum conditional entropy, and present the conditioned version of the quantum Entropy Power Inequality. We conclude in 8.
2 Gaussian optimizers in functional analysis
2.1 Gaussian kernels have only Gaussian maximizers
For any , the norm of a function is
Given , let us consider a Gaussian integral kernel from to :
where is a Gaussian function on , i.e., the exponential of a quadratic polynomial. The norm of is
In the seminal paper “Gaussian kernels have only Gaussian maximizers” [lieb1990Gaussian], Lieb proved that under certain fairly broad assumptions on , and , this operator is well defined, and the supremum in (2.3) is attained on a Gaussian function . If , any function that attains the supremum in (2.3) is a Gaussian function. The proof of this fundamental result is based on the multiplicativity of the norm of generic integral kernels with respect to the tensor product.
Theorem 2.1 ([lieb1990Gaussian, holevo2006multiplicativity]).
The norms of integral kernels are multiplicative, i.e., for any two (not necessarily Gaussian) integral kernels and ,
Moreover, if the ratios and are maximized by the unique functions and , the ratio is maximized by the unique function .
2.2 The sharp Young’s inequality for convolutions
The convolution operation can be considered as a degenerate Gaussian integral kernel given by a Dirac delta function centered in the origin. Indeed, the convolution of with is
The sharp Young’s inequality for convolutions states that the supremum
is finite iff
and in this case it is achieved by Gaussian functions. This result has been first proven by Beckner [beckner1975inequalities] and by Brascamp and Lieb [brascamp2002best] using a rearrangement inequality for integrals [brascamp2002general]. A completely different proof based on the heat semigroup has been provided by Toscani [toscani2014heat].
2.3 The Entropy Power Inequality
Let be a random variable with values in and whose probability law is absolutely continuous with respect to the Lebesgue measure, so that it admits a probability density . The Shannon differential entropy [cover2006elements] of is
and quantifies the noise contained in . Let be a symmetric strictly positive real matrix, and let be the centered Gaussian random variable with covariance matrix and density
The Shannon differential entropy of is proportional to the logarithm of the determinant of the covariance matrix:
Let us consider the sum of two independent random variables and with values in . The Entropy Power Inequality [dembo1991information, stam1959some, shannon2001mathematical] states that, if and have Shannon differential entropy fixed to the values and , respectively, the Shannon differential entropy of is minimum when and have a Gaussian probability distribution with proportional covariance matrices. The covariance matrix of the sum of two independent random variables is equal to the sum of their covariance matrices:
so that the Entropy Power Inequality has the form
Two different proofs of the Entropy Power Inequality are known. The first is due to Blachman and Stam [stam1959some, blachman1965convolution], and is based on perturbing the inputs and with the heat semigroup. The second is due to Lieb [lieb1978proof], and is based on the sharp Young’s inequality for convolutions and on the properties of the Rényi entropies. For any , the -Rényi entropy of the random variable with values in and density is
The Rényi entropies are a generalization of the Shannon differential entropy, which is recovered in the limit :
3 Quantum Gaussian systems
In this Section, we introduce the elements of quantum information and quantum Gaussian systems that are needed for presenting the entropic inequalities for quantum Gaussian channels. For a more comprehensive introduction, we refer the reader to the books [holevo2013quantum, serafini2017quantum] and the review [holevo2015gaussian].
3.1 Quantum systems
Let be a separable complex Hilbert space with not necessarily finite dimension. We adopt the bra-ket notation, where a vector is denoted as and the scalar product between the vectors and is denoted as , is linear in and antilinear in .
For any , the -Schatten norm of a linear compact operator on is
where is the adjoint operator of . The -Schatten norm play the role of the norm of functional analysis. The operators with finite -Schatten norm are called trace-class operators. The -Schatten norm of a continuous linear operator is defined as the supremum of the spectrum of .
Quantum states are the noncommutative counterpart of probability measures. A quantum state is a positive trace-class operator with unit trace. Any quantum state can be diagonalized in an orthonormal basis:
where denote the rank-one projectors onto the orthonormal vectors , and are the eigenvalues of . Since is positive and has unit trace, is a probability measure on . The quantum state is called pure if it is a rank-one projector, and mixed otherwise. With a small abuse of nomenclature, we call pure state both the normalized vector and the associated rank-one projector . From (3.2), any quantum state can be expressed as a convex combination of orthogonal pure states.
The von Neumann entropy of the quantum state in (3.2) is the Shannon entropy of its eigenvalues:
and is the quantum counterpart of the Shannon differential entropy. As the Shannon entropy and contrarily to the Shannon differential entropy, the von Neumann entropy is always positive, and vanishes iff is pure. If is a quantum state of the quantum system with Hilbert space , we use indistinctly the notations or for the entropy of .
As in the case of classical probability measures, we can define for any the -Rényi entropy of the quantum state as
The Rényi entropies are a generalization of the von Neumann entropy, which is recovered in the limit :
The observables of a quantum system are the self-adjoint operators on the Hilbert space, and the expectation value of the observable on the state is
If and are quantum systems with Hilbert spaces and , the joint system has Hilbert space . A pure state is called product state if for some and . A fundamental difference with respect to classical probability is that not all pure states are product states. A pure state that is not a product state is called entangled state. Let be a quantum state of the joint quantum system . We define the reduced or marginal states on and as
where and denote the partial trace over the system and , respectively. In other words, is the quantum state of such that
for any bounded operator on , and analogously for .
3.2 Quantum channels
Quantum channels [nielsen2010quantum, hayashi2016quantum, wilde2017quantum, holevo2013quantum] are the noncommutative counterpart of the Markov operators of probability theory. A quantum channel from the quantum system with Hilbert space to the quantum system with Hilbert space is a linear completely positive trace-preserving map from the trace-class operators on to the trace-class operators on . Precisely, a map is said to be
positive if for any trace-class operator ;
completely positive if the map is positive for any , where denotes the identity map on the operators on the Hilbert space ;
trace-preserving if for any trace-class operator .
These properties ensure that for any the map sends the quantum states on to quantum states on . Since any joint probability measure of two random variables is a convex combination of product probability measures, the complete positivity of any Markov operator is a trivial consequence of its positivity. On the contrary, the existence of entanglement makes complete positivity a nontrivial requirement for quantum channels. For example, the transposition is a linear positive trace-preserving map that is not completely positive.
Any quantum channel from to can be realized by an isometry followed by a partial trace, i.e., there exists an Hilbert space and an isometry with such that for any trace-class operator on ,
The Hilbert space and the isometry are unique up to isometries on . The expression (3.9) is called the Stinespring dilation of . The quantum channel from to defined on trace-class operators as
is called the complementary channel of [holevo2013quantum]. We mention that the Stinespring dilation has a classical analogue that has found some applications in the commutative setting, e.g., in infinite-dimensional stochastic analysis [hausenblas2001note].
Let be a quantum channel. The dual channel of is the linear map from bounded operators on to bounded operators on such that for any trace-class operator on and any bounded operator on
For any , the norm of a quantum channel is defined as
A fundamental question is whether the norm of a channel is multiplicative with respect to the tensor product, i.e., whether
This property holds for any classical integral kernel [lieb1990Gaussian, holevo2006multiplicativity], but it is known to fail for generic quantum channels [holevo2006multiplicativity].
3.3 Quantum Gaussian systems
A -mode Gaussian quantum system is the mathematical model for harmonic oscillators, or modes of the electromagnetic radiation. For the sake of simplicity, we present one-mode Gaussian quantum systems first.
The Hilbert space of a one-mode Gaussian quantum system is , the irreducible representation of the canonical commutation relation (see [serafini2017quantum] or [holevo2013quantum], Chapter 12 for a more complete presentation)
The operator is called ladder operator, plays the role of a noncommutative complex variable and acts on in a suitable dense domain in as
The quantum states of a one-mode Gaussian quantum system are the quantum counterparts of the probability measures on . Since each mode is associated to a complex noncommutative variable, the number of real classical components is twice the number of quantum modes. We define the Hamiltonian
that counts the number of excitations, or photons. The vector annihilated by is the vacuum and is denoted by . From the vacuum we can build the eigenstates of the Hamiltonian, called Fock states:
where denotes the scalar product in . An operator diagonal in the Fock basis is called Fock-diagonal.
A quantum Gaussian state is a quantum state proportional to the exponential of a quadratic polynomial in and . The most important Gaussian states are the thermal Gaussian states, where the polynomial is proportional to the Hamiltonian . They correspond to a geometric probability distribution for the energy:
For , we recover the vacuum state . The average energy of is
and the von Neumann entropy is
As Gaussian probability measures maximize the Shannon differential entropy among all the probability measures with a given covariance matrix, thermal quantum Gaussian states maximize the von Neumann entropy among all the quantum states with a given average energy.
The Hilbert space of a -mode Gaussian quantum system is the tensor product of Hilbert spaces of a one-mode Gaussian quantum system, i.e., the irreducible representation of the canonical commutation relations
where each ladder operator is associated to one mode. An -mode thermal quantum Gaussian state is the tensor product of identical one-mode thermal quantum Gaussian states.
3.4 Quantum Gaussian channels
Quantum Gaussian channels are the quantum channels that preserve the set of quantum Gaussian states. The most important families of quantum Gaussian channels are the beam-splitter, the squeezing, the quantum Gaussian attenuators and the quantum Gaussian amplifiers. The beam-splitter and the squeezing are the quantum counterparts of the classical linear mixing of random variables, and are the main transformations in quantum optics. Let and be one-mode quantum Gaussian systems with ladder operators and , respectively. The beam-splitter of transmissivity is implemented by the unitary operator
and performs a linear rotation of the ladder operators (see e.g. [ferraro2005gaussian], Section 1.4.2):
The physical beam-splitter is a passive element, and does not require energy for functioning. Indeed, the mixing unitary operator preserves the Hamiltonian (3.16):
The two-mode squeezing [barnett2002methods] of parameter is implemented by the unitary operator
and acts on the ladder operators as
The squeezing is an active operation that requires energy. Indeed, the squeezing unitary operator does not preserve the Hamiltonian (3.16).
We define for any joint quantum state on and any the quantum channel from to
where denotes the partial trace over the system . implements the beam-splitter for and the squeezing for .
The quantum Gaussian attenuators model the attenuation and the noise affecting electromagnetic signals traveling through optical fibers or free space. The quantum Gaussian attenuator can be implemented mixing the input state with the thermal Gaussian state with average energy through a beam-splitter of transmissivity :
The quantum Gaussian attenuators constitute a multiplicative semigroup with composition law
The quantum Gaussian amplifiers model the amplification of electromagnetic signals. The quantum Gaussian amplifier can be implemented performing a two-mode squeezing of parameter on the input state and the thermal Gaussian state with average energy :
Also the quantum Gaussian amplifiers constitute a semigroup with composition law
The attenuator and the amplifier are called quantum-limited if , i.e., if they mix the input state with the vacuum. Indeed, the vacuum as state of the environment adds the least possible noise to the input state. In this case, since is a pure state, the expressions (3.28) and (3.30) are the Stinespring dilations of the corresponding channels.
4 The minimum output entropy conjecture
The minimum output entropy of a quantum channel plays a key role in the determination of its classical information capacity. The following theorem has been a fundamental result in quantum communication theory.
For any , the vacuum state minimizes the output entropy of the -mode quantum Gaussian attenuators and of the -mode quantum Gaussian amplifiers, i.e., for any -mode quantum state , any , and
Therefore, the minimum output entropy of the quantum attenuators and amplifiers is additive.
We stress that Theorem 4.1 is trivial for classical Gaussian channels, i.e., for Gaussian integral kernels that send probability measures on to probability measures on . Indeed, for the concavity of the entropy it is sufficient to prove Theorem 4.1 for pure input states. In the classical case, the only pure probability measures are the Dirac delta functions, and they all achieve the same output entropy. As we will see, the proof of Theorem 4.1 exploits tools of quantum information theory that do not have a classical counterpart: the complementary channel and the decomposition of any Gaussian channel as a quantum-limited attenuator followed by a quantum-limited amplifier.
The proof of Theorem 4.1 is based on majorization theory [marshall2010inequalities].
Definition 1 (majorization).
We say that the quantum state majorizes the quantum state , and write , iff can be obtained applying to a convex combination of unitary operators, i.e., iff there exists a probability measure on the set of unitary operators such that
The link between majorization and the entropy is provided by the following property.
Let and be quantum states such that . Then, for any unitarily invariant convex functional on the set of quantum states. In particular,
for any ;
Theorem 4.1 is a consequence of this more fundamental result.
Theorem 4.3 (majorization for quantum Gaussian channels).
For any and for all the -mode quantum Gaussian attenuators and amplifiers, the output generated by the vacuum input state majorizes the output generated by any other input state, i.e., for any , and and for any -mode quantum state ,
Corollary 4.4 ( norms of quantum Gaussian channels).
For any and any , the vacuum input state achieves the norm of the -mode quantum Gaussian attenuators and amplifiers, i.e., for any , and
Therefore, the norms of the quantum Gaussian attenuators and amplifiers are multiplicative.
Theorem 4.1 was first proven by Giovannetti, Holevo and García-Patrón [giovannetti2015solution]. Shortly later, Mari, Giovannetti and Holevo realized that the same proof implies the more general Theorem 4.3, first for one-mode quantum Gaussian channels [mari2014quantum], and then for multi-mode quantum Gaussian channels [giovannetti2015majorization, holevo2016proof]. We present here a sketch of the proof. For more details, the reader can also consult the review [holevo2015gaussian].
The first step to prove Theorem 4.3 is the following observation.
For any , any -mode quantum Gaussian attenuator or amplifier can be decomposed as an -mode quantum-limited attenuator followed by a -mode quantum-limited amplifier.
Theorem 4.3 is trivial for the quantum-limited attenuator, since the vacuum is a fixed point. Thanks to Proposition 4.5, it is sufficient to prove Theorem 4.3 for the quantum-limited amplifier. It is easy to see that it is sufficient to prove Theorem 4.3 for pure input states. The next step exploits the following properties:
Let be a quantum channel and be its complementary channel. Then, for any pure input state , the quantum states and have the same spectrum.
From Proposition 4.6, the optimal input states for and must coincide.
The complementary channel of the quantum-limited amplifier is a quantum-limited attenuator followed by the same quantum-limited amplifier followed by the transposition, i.e., for any ,
where is the transposition operation.
From Propositions 4.6 and 4.7, the optimal input states for the quantum-limited amplifier must coincide with the optimal input states for a suitable quantum-limited attenuator composed with the same quantum-limited amplifier. Since the optimal input states must be pure, they must be left pure by the quantum-limited attenuator. The claim then follows from the following property.
For any and any , the vacuum is the only -mode quantum state such that is pure.
5 Gaussian optimizers for entropic inequalities in quantum information
The problem of determining the information capacity region of the quantum Gaussian degraded broadcast channel has led to a constrained minimum output entropy conjecture [guha2007classical], which is a generalization of Theorem 4.1 with a constrained input entropy.
Conjecture 5.1 (constrained minimum output entropy conjecture).
For any , quantum Gaussian input states minimize the output entropy of the -mode Gaussian quantum attenuators and amplifiers among all the input states with a given entropy. In other words, let be a generic -mode quantum state, and let be the one-mode thermal Gaussian state with entropy , so that is the -mode thermal Gaussian state with the same entropy as . Then, for any , and ,
where the function has been defined in (3.20).
Conjecture 5.1 has been proven only in the one-mode case () by De Palma, Trevisan and Giovannetti [de2016gaussian, de2016gaussiannew], and has been extended to one-mode gauge-contravariant quantum Gaussian channels by Qi, Wilde and Guha [qi2017minimum]. The proof by De Palma et al. is based on the following fundamental majorization result for one-mode quantum Gaussian channels [de2015passive], which extends Theorem 4.3.
For any , and and any one-mode quantum state ,
where is the passive rearrangement of , i.e., the passive state with the same spectrum as .
We recall that a passive state is a quantum state that minimizes the average energy among all the quantum states with the same spectrum [pusz1978passive, lenard1978thermodynamical, gorecki1980passive]. If is diagonalized in the orthonormal eigenbasis as
is given by
where is the Fock basis.
From Theorem 5.2, in the case of one mode the constrained minimization of the output entropy of Conjecture 5.1 can be restricted to passive input states. Unfortunately, an analogue majorization theorem does not hold for more than one mode [de2016passive].
Conjecture 5.1 has first been proven for the one-mode quantum-limited attenuator [de2016gaussian]. The proof is based on the following isoperimetric inequality, that constitutes the infinitesimal version of the conjecture.
Theorem 5.3 (isoperimetric inequality for the one-mode quantum-limited attenuator).
Among all the input states with a given entropy, quantum Gaussian input states maximize the derivative of the output entropy of the one-mode quantum-limited attenuator with respect to the attenuation parameter. In other words, let be a one-mode quantum state, and the one-mode thermal Gaussian state with the same entropy as . Then,
The adjective “isoperimetric” is due to the formal analogy between entropy and volume [dembo1991information]. Up to a change of signs, the left hand side in (5.5) plays the role of a perimeter and the function that of an isoperimetric profile.
Thanks to Theorem 5.2, it is sufficient to prove Theorem 5.3 for passive states. The proof is then performed through the Lagrange multipliers. Since the Hilbert space of a one-mode Gaussian quantum system has infinite dimension, a generic passive state has infinite parameters. This issue is solved restricting to a finite dimensional subspace with bounded maximum energy, and then proving that the maximum of the left-hand side of (5.5) for passive input states supported in the subspace tends to the right-hand side in the limit of infinite maximum energy.
The generalization of Theorem 5.3 to all the one-mode quantum Gaussian attenuators and amplifiers would have implied Conjecture 5.1 for . However, for any one-mode quantum Gaussian channel other than the quantum-limited attenuator, the infinite dimension of the Hilbert space is really an issue. Indeed, for any quantum state with a support of finite dimension is infinite for any and is infinite for any , and nothing can be proven restricting to a finite dimensional subspace. If one tries to use the Lagrange multipliers directly for the infinite dimensional problem, the Gaussian state is not the only solution [qi2017minimum], so that a new approach is needed. This approach is based on the norms and is presented in 5.1 below.
5.1 Quantum Gaussian channels have Gaussian maximizers
The theorem “Gaussian kernels have Gaussian maximizers” has been conjectured to apply also to quantum Gaussian channels.
Conjecture 5.4 (quantum Gaussian channels have Gaussian maximizers).
For any and any , quantum Gaussian input states achieve the norm of the -mode Gaussian quantum attenuators and amplifiers. In other words, for any , and ,
where is the one-mode thermal Gaussian state with average energy as in (3.18). Therefore, the norms of the quantum Gaussian attenuators and amplifiers are multiplicative.
The suprema in (5.4) are
finite and achieved for a finite if ;
finite and asymptotically achieved in the limit if ;
infinite and asymptotically achieved in the limit if .
Conjecture 5.4 can be extended to any linear and completely positive map that preserves the set of unnormalized quantum Gaussian states, i.e., the operators proportional to a quantum Gaussian state. These maps include all quantum Gaussian channels and all the probabilistic maps resulting from the conditioning on the outcome of a Gaussian measurement performed on a subsystem [holevo2013quantum, serafini2017quantum]. The generalized conjecture states that quantum Gaussian input states achieve the norms of all such maps. In this more general setup, the analogue of the optimization in the right-hand side of (5.4) cannot be restricted to the thermal Gaussian states, but has to be performed over all quantum Gaussian states.
Conjecture 5.4 has been proven only in some particular cases. As we have seen in Corollary 4.4, the majorization result Theorem 4.3 implies Conjecture 5.4 for any in the case . De Palma, Trevisan and Giovannetti proved Conjecture 5.4 in the case of one-mode quantum-limited channels, i.e., and [de2016pq]. Frank and Lieb proved Conjecture 5.4 for any in the case [frank2017norms], and Holevo extended the result to any -mode quantum Gaussian channel (still for ) [holevo2017quantum].
The proof of Conjecture 5.4 for one-mode quantum-limited Gaussian channels
Lemma 5.5 ([audenaert2009note]).
For any , any quantum channel and any operator ,
The proof of Conjecture 5.4 is then based on the following new logarithmic Sobolev inequality, that constitutes the infinitesimal version of Conjecture 5.4 (in the same way as Gross’ logarithmic Sobolev inequality is the infinitesimal version of Nelson’s Hypercontractive theorem [gross1993logarithmic]).
Theorem 5.6 (logarithmic Sobolev inequality for the quantum-limited Gaussian attenuator).
Let us fix . Let be a one-mode quantum state, and let be the thermal Gaussian state such that has the same entropy as . Then,
Thanks to Theorem 5.2, it is sufficient to prove Theorem 5.6 for passive input states. As in the case of Theorem 5.3, the proof is then performed through the Lagrange multipliers, restricting to a finite dimensional subspace with bounded maximum energy. Conjecture 5.4 for the one-mode quantum-limited attenuator follows integrating (5.8) thanks to the semigroup property of the attenuator (3.29).
Conjecture 5.4 for the one-mode quantum-limited amplifier follows from the following duality Lemma for the Schatten norms.
For any and any positive operator ,
Lemma 5.7 implies the following duality for the norms of quantum channels.
For any quantum channel and any ,
The norms of the quantum-limited amplifier can then be determined from the norms of the quantum-limited attenuator thanks to the following property.
The dual of the quantum-limited Gaussian amplifier is proportional to a quantum-limited Gaussian attenuator, i.e., for any ,