New lower bounds to the output entropy of multi-mode quantum Gaussian channels

05/31/2018
by   Giacomo De Palma, et al.
0

We prove that quantum thermal Gaussian input states minimize the output entropy of the multi-mode quantum Gaussian attenuators and amplifiers that are entanglement breaking and of the multi-mode quantum Gaussian phase contravariant channels among all the input states with a given entropy. This is the first time that this property is proven for a multi-mode channel without restrictions on the input states. A striking consequence of this result is a new lower bound on the output entropy of all the multi-mode quantum Gaussian attenuators and amplifiers in terms of the input entropy. We apply this bound to determine new upper bounds to the communication rates in two different scenarios. The first is classical communication to two receivers with the quantum degraded Gaussian broadcast channel. The second is the simultaneous classical communication, quantum communication and entanglement generation or the simultaneous public classical communication, private classical communication and quantum key distribution with the Gaussian quantum-limited attenuator. In this second scenario, our bound is the first that does not rely on still unproven conjectures.

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1 Introduction

Attenuation and noise unavoidably affect electromagnetic communications through wires, optical fibers and free space. Quantum effects become relevant for low-intensity signals as in the case of satellite communications, where the receiver can be reached by only few photons for each bit of information [chen2012optical]. Quantum Gaussian channels provide the mathematical model for the attenuation and the noise affecting electromagnetic signals in the quantum regime [chan2006free, braunstein2005quantum, holevo2013quantum, weedbrook2012gaussian, holevo2015gaussian, serafini2017quantum].

The maximum achievable communication rate of a channel depends on the minimum noise achievable at its output, which is quantified by the output entropy [wilde2017quantum, holevo2013quantum]. The determination of the maximum rates allowed by quantum mechanics for the communication to two receivers with the quantum degraded Gaussian broadcast channel [guha2007classicalproc, guha2007classical] relies on a minimum output entropy conjecture [guha2008entropy, guha2008capacity] (Conjecture 1). This fundamental conjecture states that thermal quantum Gaussian input states minimize the output entropy of the quantum Gaussian attenuators, amplifiers and phase contravariant channels among all the input states with a given entropy. The same conjecture is necessary also to determine the triple trade-off region of the Gaussian quantum-limited attenuator [wilde2012information, wilde2012quantum]. This region is constituted by all the achievable triples of rates for simultaneous classical communication, quantum communication and entanglement generation or for simultaneous public classical communication, private classical communication and quantum key distribution. So far, Conjecture 1 has been proven only in the special case of one-mode channels [de2015passive, de2016gaussian, de2016pq, de2016gaussiannew, qi2017minimum]. The best current lower bound to the output entropy of multi-mode quantum Gaussian channels is provided by the quantum Entropy Power Inequality [konig2014entropy, konig2016corrections, de2014generalization, de2015multimode, de2017gaussian, de2018conditional, huber2018conditional, huber2017geometric] (see Theorem 3). However, this lower bound is strictly lower than the output entropy generated by Gaussian input states, hence it is not sufficient to prove the conjecture (see the review [de2018gaussian] for a complete presentation of the state of the art).

We prove the minimum output entropy conjecture for the multi-mode quantum Gaussian attenuators and amplifiers that are entanglement breaking and for all the multi-mode phase contravariant quantum Gaussian channels (Corollary 5). This is the first time that the minimum output entropy conjecture is proven for a multi-mode channel without restrictions on the input states. Surprisingly, the implications of this result go beyond the quantum Gaussian channels that are entanglement breaking. Indeed, combining Corollary 5 with the quantum integral Stam inequality of Ref. [de2018conditional], we prove a new lower bound to the output entropy of all the multi-mode quantum Gaussian attenuators and amplifiers (Theorem 6). This new lower bound is strictly better than the previous best lower bound provided by the quantum Entropy Power Inequality (see Figure 1 for a comparison).

We apply Theorem 6 to determine a new upper bound to the rates for classical communication to two receivers with the quantum degraded Gaussian broadcast channel (Corollary 10) and a new outer bound to the triple trade-off region of the Gaussian quantum-limited attenuator (Corollary 13). These bounds improve the best previous bounds based on the quantum Entropy Power Inequality (see Figure 2 and Figure 3 for a comparison).

The manuscript is structured as follows. We present quantum Gaussian channels in section 2 and the minimum output entropy conjecture in section 3. In section 4 we prove the minimum output entropy conjecture for the quantum Gaussian channels that are entanglement breaking, and in section 5 we prove the new lower bound to the output entropy of the quantum Gaussian attenuators and amplifiers. We apply this result to prove a new upper bound to the rates for classical communication to two receivers with the quantum degraded Gaussian broadcast channel in section 6 and to prove a new outer bound to the triple trade-off region of the quantum-limited attenuator in section 7. We conclude in section 8.

2 Quantum Gaussian channels

A one-mode quantum Gaussian system is the mathematical model for a harmonic oscillator or a mode of the electromagnetic radiation. The Hilbert space of a one-mode quantum Gaussian system is the irreducible representation of the canonical commutation relation [serafini2017quantum], [holevo2013quantum, Chapter 12]

(1)

where is the ladder operator. We define the Hamiltonian

(2)

that counts the number of excitations or photons. The vector annihilated by

is the vacuum and is denoted by . 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 (2):

(3)

where is the average energy:

(4)

We notice that is the vacuum state of the system. The von Neumann entropy of is

(5)

An -mode Gaussian quantum system is the union of one-mode Gaussian quantum systems, and its Hilbert space is the

-th tensor power of the Hilbert space of a one-mode Gaussian quantum system. Let

be the ladder operators of the modes. The Hamiltonian of the -mode Gaussian quantum system is the sum of the Hamiltonians of each mode:

(6)

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, the quantum Gaussian amplifiers and the quantum heat semigroup. 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

(7)

and performs a linear rotation of the ladder operators [ferraro2005gaussian, Section 1.4.2]:

(8)

The squeezing [barnett2002methods] of parameter is implemented by the unitary operator

(9)

and acts on the ladder operators as

(10)

The quantum Gaussian attenuators model the attenuation and the noise affecting electromagnetic signals traveling through optical fibers or free space. The one-mode quantum Gaussian attenuator [holevo2007one, case (C) with and ] can be implemented mixing the input state with the one-mode thermal Gaussian state through a beam-splitter of transmissivity :

(11)

If the attenuator is called quantum-limited, and we denote

(12)

The quantum Gaussian amplifiers model the amplification of electromagnetic signals. The one-mode quantum Gaussian amplifier [holevo2007one, case (C) with and ] can be implemented performing a squeezing of parameter on the input state and the one-mode thermal Gaussian state :

(13)

The one-mode Gaussian phase contravariant channel [holevo2007one, case (D) with and ] is the weak complementary of : for any one-mode quantum state ,

(14)

The displacement operator with is the unitary operator that displaces the ladder operators:

(15)

The quantum Gaussian additive noise channel [holevo2007one, case () with ]

is the quantum Gaussian channel generated by a convex combination of displacement operators with a Gaussian probability measure:

(16)

3 The minimum output entropy conjecture

Conjecture 1 (minimum output entropy conjecture).

For any , quantum Gaussian thermal input states minimize the output entropy of the -mode Gaussian quantum attenuators, amplifiers, phase contravariant channels and additive noise channels among all the input states with a given entropy. In other words, let be a state of an -mode Gaussian quantum system with finite entropy, and let

(17)

where has been defined in (5), such that . Then,

(18)
Remark 2.

Conjecture 1 has been proven only in some special cases:

  • , i.e., when is pure [giovannetti2015solution, mari2014quantum, holevo2015gaussian];

  • , i.e., one-mode channels (see [de2016gaussian] for the quantum-limited attenuator, [de2016gaussiannew] for all the quantum attenuators, amplifiers and additive noise channels and [qi2017minimum] for the phase contravariant quantum Gaussian channel);

  • When is diagonal in some joint product basis [de2017multimode].

The current best lower bound to the output entropy of multi-mode quantum Gaussian channels valid for any input state is provided by the quantum Entropy Power Inequality [konig2014entropy, konig2016corrections, de2014generalization, de2015multimode, de2017gaussian, huber2018conditional, huber2017geometric]:

Theorem 3.

For any and any state of an -mode Gaussian quantum system with finite average energy,

(19)
(20)
(21)
(22)
Proof.

The claim (19) follows from the quantum Entropy Power Inequality for the beam-splitter [de2014generalization, Eq. (5)] and the representation (11) for the quantum Gaussian attenuator. The claim (20) and (21) follow from the quantum Entropy Power Inequality for the squeezing [de2014generalization, Eq. (7)] and the representations (13) and (14) for the quantum phase contravariant Gaussian channel. The claim (22) follows from [huber2017geometric, Theorem 3]. ∎

4 Gaussian states minimize the output entropy of entanglement breaking quantum Gaussian channels

In this Section, we prove Conjecture 1 for the phase contravariant quantum Gaussian channels and for the quantum Gaussian attenuators and amplifiers that are entanglement breaking. This result is a corollary of the following.

Theorem 4.

Let and be quantum systems with Hilbert spaces and , and let be an entanglement breaking quantum channel such that for any quantum state on

(23)

with increasing and convex. Then, for any and any quantum state on ,

(24)
Proof.

We prove the claim by induction on . The claim is true for . Let us then assume (24) for a given . Let be a quantum state on , and let

(25)

Since is entanglement breaking, it admits a representation as a measure-prepare channel [kholevo2005notion], i.e., there exist a complete separable metric space , a quantum-classical channel that maps quantum states on to Borel probability measures on and a classical-quantum channel that maps Borel probability measures on to quantum states on such that

(26)

We define the probability measure on taking values on quantum states on

(27)

and the probability measure on taking values on quantum states on

(28)

such that

(29)

We have

(30)

where we have used the inductive hypothesis (24) and Jensen’s inequality applied to the convex function . We then have

(31)

(a) follows from the chain rule for the entropy; (b) follows from the data processing inequality for the channel

; (c) follows from the hypothesis (23) and from (4); (d) follows from the data processing inequality for the channel (we recall that is increasing); (e) follows from Jensen’s inequality applied to the convex function ; (f) follows from the chain rule for the entropy. We have then proven that the claim (24) for implies the claim (24) for , and by induction the claim is true for any . ∎

The following Corollary 5 proves Conjecture 1 for all the channels that are entanglement breaking. This is the first time that Conjecture 1 is proven for multi-mode channels without restrictions on the input states.

Corollary 5 (minimum output entropy conjecture for entanglement breaking channels).

Conjecture 1 holds for:

  • Any quantum Gaussian attenuator with ;

  • Any quantum Gaussian amplifier with ;

  • Any quantum Gaussian phase contravariant channel ;

  • Any quantum Gaussian additive noise channel with .

Proof.

Conjecture 1 holds for . From [holevo2013quantum, Sec. 12.6.2], the conditions , and imply that , and are entanglement breaking, respectively, and is entanglement breaking for any . From [de2017multimode, Lemma 15], the functions

(32)

are increasing and convex for any , and . The claim then follows from Theorem 4. ∎

5 The new lower bound to the output entropy of quantum Gaussian channels

A striking consequence of Corollary 5 is the following improved lower bound for the output entropy of the multi-mode quantum Gaussian channels that are not entanglement breaking. We compare in Figure 1 this bound with the previous best bound provided by the quantum Entropy Power Inequality and with the output entropy achieved by quantum thermal Gaussian input states.

Figure 1: Output entropy of the Gaussian quantum-limited attenuator with attenuation parameter as a function of the input entropy. The plot compares the output entropy achieved by thermal Gaussian input states with the lower bounds provided by Theorem 6 and by the quantum Entropy Power Inequality.
Theorem 6.

For any and any state of an -mode Gaussian quantum system with finite average energy,

(33)
Proof.

Quantum Gaussian attenuators. We fix and define for any

(34)

From [de2018conditional, Eq. (113)] we have , hence

(35)

We set

(36)

such that and the channel is entanglement breaking. Then, putting together (6) and Corollary 5 we get

(37)

where for any

(38)

Let

(39)

From [de2017multimode, Lemma 15], is convex, hence

(40)

Finally, we set and get from (6) and (40)

(41)

and the claim follows.

Quantum Gaussian amplifiers. The proof for the quantum Gaussian amplifiers is analogous to the proof for the quantum Gaussian attenuators. We fix and define for any

(42)

From [de2018conditional, Eq. (113)] we have , hence

(43)

We set

(44)

such that and the channel is entanglement breaking. Then, putting together (6) and Corollary 5 we get

(45)

where for any

(46)

Let

(47)

From [de2017multimode, Lemma 15], is convex, hence

(48)

Finally, we set and get from (6) and (48)

(49)

and the claim follows.

Quantum Gaussian additive noise channels. We fix and define for any

(50)

From the proof of Theorem 5 of Ref. [huber2018conditional] we have , hence

(51)

We set

(52)

such that and the channel is entanglement breaking. Then, putting together (51) and Corollary 5 we get

(53)

where for any

(54)

Let

(55)

From [de2017multimode, Lemma 15], is convex, hence

(56)

Finally, we set and get from (53) and (56)

(57)

and the claim follows. ∎

Remark 7.

Since states with infinite average energy are unphysical, for all practical purposes the hypothesis of finite average energy in Theorem 6 is not restrictive.

6 Bound to the capacity region of the quantum degraded Gaussian broadcast channel

Let , , , be one-mode Gaussian quantum systems. The quantum degraded Gaussian broadcast channel [guha2007classicalproc, guha2007classical] maps a state of to a state of the joint quantum system with

(58)

where is the unitary operator defined in (7) and . The channel can be understood as follows. encodes the information into the state of the electromagnetic radiation , and sends it through a beam-splitter of transmissivity . and receive the transmitted and the reflected part of the signal, respectively, whose joint state is . This channel is called degraded since the state received by can be obtained applying a quantum-limited attenuator to the state received by [guha2007classical]:

(59)

The simplest communication strategy is time sharing, which consists in communicating only with for a fraction of the time and only with for the remaining fraction of the time. Superposition coding [yard2011quantum, guha2007classical, savov2015classical] is a more sophisticated strategy that achieves higher rates communicating with and simultaneously. Let be the maximum average energy per mode of the input states. Superposition coding allows to achieve with the quantum degraded Gaussian broadcast channel (58) any rate pair satisfying [guha2007classical, Sec. IV]

(60)

Assuming Conjecture 1 for the quantum-limited attenuator, the capacity region of the quantum degraded Gaussian broadcast channel coincides with the region identified by (60) [guha2007classical], i.e., any achievable rate pair satisfies (60).

Despite Conjecture 1 still lacks a proof, the known lower bounds to the output entropy of the multi-mode quantum-limited attenuators still imply bounds to the capacity region of the quantum degraded Gaussian broadcast channel. The first of these bounds has been determined from the quantum Entropy Power Inequality [de2014generalization]. The following Theorem 8 shows that any lower bound to the output entropy of the multi-mode quantum-limited attenuators in terms of the input entropy implies a bound to the capacity region of the quantum degraded Gaussian broadcast channel. We then combine Theorem 8 with Theorem 6 to obtain a new bound to this capacity region.

Theorem 8.

Let us suppose that for any , any and any input state of an -mode Gaussian quantum system with finite average energy

(61)

where the function is increasing and convex. Then, any achievable rate pair for the quantum degraded Gaussian broadcast channel satisfies

(62)

where is the maximum allowed average energy per mode of the input.

Proof.

The capacity region of the quantum degraded Gaussian broadcast channel is the closure of the union over of regions of the form [guha2007classical]

(63)
(64)

where is an ensemble of pure encoding states on copies of the quantum system and

(65)
(66)
(67)
(68)

and the average state satisfies the energy constraint

(69)

Since for any and , we have from (63)

(70)

The energy constraint (69) implies

(71)

where is the Hamiltonian on copies of , hence

(72)

where we have used that quantum thermal Gaussian states maximize the entropy among all the states with the same average energy. From (59) we have for any

(73)

Since the state has finite average energy, has finite average energy for any , and we have from the hypothesis (61)

(74)

Since is convex and increasing, we have from Jensen’s inequality and (70)