Passive states optimize the output of bosonic Gaussian quantum channels
Abstract
An ordering between the quantum states emerging from a single mode gaugecovariant bosonic Gaussian channel is proven. Specifically, we show that within the set of input density matrices with the same given spectrum, the element passive with respect to the Fock basis (i.e. diagonal with decreasing eigenvalues) produces an output which majorizes all the other outputs emerging from the same set. When applied to pure input states, our finding includes as a special case the result of A. Mari, et al., Nat. Comm. 5, 3826 (2014) which implies that the output associated to the vacuum majorizes the others.
I Introduction
The minimum von Neumann entropy at the output of a quantum communication channel can be crucial for the determination of its classical communication capacity [holevo2013quantum].
Most communication schemes encode the information into pulses of electromagnetic radiation, that travels through metal wires, optical fibers or free space and is unavoidably affected by attenuation and noise. Gaugecovariant quantum Gaussian channels [holevo2013quantum] provide a faithful model for these effects, and are characterized by the property of preserving the thermal states of electromagnetic radiation.
It has been recently proved [mari2014quantum, giovannetti2015solution, giovannetti2015majorization, holevo2015gaussian] that the output entropy of any gaugecovariant Gaussian quantum channel is minimized when the input state is the vacuum. This result has permitted the determination of the classical information capacity of this class of channels [giovannetti2014ultimate]. However, it is not sufficient to determine the capacity region of the quantum Gaussian broadcast channel [guha2007classicalproc, guha2007classical] and the triple tradeoff region of the quantumlimited Gaussian attenuator [wilde2012quantum, wilde2012information]. Indeed, solving these problems would require to prove that Gaussian thermal input states minimize the output von Neumann entropy of a quantumlimited Gaussian attenuator among all the states with a given entropy. This still unproven result would follow from a stronger conjecture, the Entropy Photonnumber Inequality (EPnI) [guha2008capacity, guha2008entropy], stating that Gaussian states minimize the output von Neumann entropy of a beamsplitter among all the couples of input states, each one with a given entropy. So far, it has been possible to prove only a weaker version of the EPnI, the quantum Entropy Power Inequality [konig2014entropy, konig2013limits, de2014generalization, de2015multimode], that provides a lower bound to the output entropy of a beamsplitter, but is never saturated.
Actually, Ref.’s [mari2014quantum, giovannetti2015majorization, holevo2015gaussian] do not only prove that the vacuum minimizes the output entropy of any gaugecovariant quantum Gaussian channel. They also prove that the output generated by the vacuum majorizes the output generated by any other state, i.e. applying a convex combination of unitary operators to the former, we can obtain any of the latter states. This paper goes in the same direction, and proves a generalization of this result valid for any onemode gaugecovariant quantum Gaussian channel. Our result states that the output generated by any quantum state is majorized by the output generated by the state with the same spectrum diagonal in the Fock basis and with decreasing eigenvalues, i.e. by the state which is passive [pusz1978passive, lenard1978thermodynamical, gorecki1980passive] with respect to the number operator (see [vinjanampathy2015quantum, goold2015role, binder2015quantum] for the use of passive states in the context of quantum thermodynamics). This can be understood as follows: among all the states with a given spectrum, the one diagonal in the Fock basis with decreasing eigenvalues produces the less noisy output. Since all the states with the same spectrum have the same von Neumann entropy, our result implies that the input state with given entropy minimizing the output entropy is certainly diagonal in the Fock basis, and then reduces the minimum output entropy quantum problem to a problem on discrete classical probability distributions.
Thanks to the classification of onemode Gaussian channels in terms of unitary equivalence [holevo2013quantum], we extend the result to the channels that are not gaugecovariant with the exception of the singular cases and , for which we show that an optimal basis does not exist.
We also point out that the classical channel acting on discrete probability distributions associated to the restriction of the quantumlimited attenuator to states diagonal in the Fock basis coincides with the channel already known in the probability literature under the name of thinning. First introduced by Rényi [renyi1956characterization] as a discrete analogue of the rescaling of a continuous random variable, the thinning has been recently involved in discrete versions of the central limit theorem [harremoes2007thinning, yu2009monotonic, harremoes2010thinning] and of the Entropy Power Inequality [yu2009concavity, johnson2010monotonicity]. In particular, the Restricted Thinned Entropy Power Inequality [johnson2010monotonicity] states that the Poissonian probability distribution minimizes the output Shannon entropy of the thinning among all the ultra logconcave input probability distributions with a given Shannon entropy. The techniques of this proof could be useful to prove that Gaussian thermal states minimize the output von Neumann entropy of a quantumlimited attenuator among all the input states diagonal in the Fock basis with a given von Neumann entropy (but without the ultra logconcavity constraint). Then, thanks to the main result of this paper it would automatically follow that Gaussian thermal states minimize the output von Neumann entropy of a quantumlimited attenuator among all the input states with a given von Neumann entropy, not necessarily diagonal in the Fock basis.
The paper is organized as follows. In Section II we introduce the Gaussian quantum channels, and in Section III the majorization. The Fock rearrangement is defined in Section IV, while Section V defines the notion of Fock optimality and proves some of its properties. The main theorem is proved in Section VI, and the case of a generic not gaugecovariant Gaussian channel is treated in Section VII. Finally, Section VIII links our result to the thinning operation.
Ii Basic definitions
In this section we recall some basic definitions and facts on Gaussian quantum channels. The interested reader can find more details in the books [holevo2013quantum, barnett2002methods, holevo2011probabilistic].
Definition II.1 (Trace norm).
The trace norm of an operator is
(II.1) 
If is finite, we say that is a traceclass operator.
Definition II.2 (Quantum operation).
A quantum operation is a linear completely positive map on traceclass operators continuous in the trace norm.
Remark II.3.
A tracepreserving quantum operation is a quantum channel.
We consider the Hilbert space of the harmonic oscillator, i.e. the irreducible representation of the canonical commutation relation
(II.2) 
has a countable orthonormal basis
(II.3) 
called the Fock basis, on which the ladder operators act as
(II.4a)  
(II.4b) 
We can define a number operator
(II.5) 
satisfying
(II.6) 
Definition II.4 (HilbertSchmidt norm).
The HilbertSchmidt norm of an operator is
(II.7) 
Definition II.5.
(HilbertSchmidt dual) Let be a linear map acting on trace class operators and continuous in the trace norm. Its HilbertSchmidt dual is the map on bounded operators continuous in the operator norm defined by
(II.8) 
for any traceclass operator and any bounded operator .
Definition II.6 (Characteristic function).
The characteristic function of a traceclass operator is
(II.9) 
where denotes the complex conjugate.
It is possible to prove that any traceclass operator is completely determined by its characteristic function.
Theorem II.7 (Noncommutative Parceval relation).
The characteristic function provides an isometry between the HilbertSchmidt product and the scalar product in , i.e. for any two traceclass operators and ,
(II.10) 
Proof.
See e.g. Theorem 5.3.3 of [holevo2011probabilistic]. ∎
Definition II.8 (Gaussian gaugecovariant quantum channel).
A gaugecovariant quantum Gaussian channel with parameters and can be defined by its action on the characteristic function: for any traceclass operator ,
(II.11) 
The channel is called quantumlimited if . If , it is a quantumlimited attenuator, while if it is a quantumlimited amplifier.
Lemma II.9.
Any gaugecovariant quantum Gaussian channel is continuous also in the HilbertSchmidt norm.
Lemma II.10.
Any gaugecovariant quantum Gaussian channel can be written as a quantumlimited amplifier composed with a quantumlimited attenuator.
Proof.
See [garcia2012majorization, giovannetti2015solution, mari2014quantum, holevo2015gaussian]. ∎
Lemma II.11.
The HilbertSchmidt dual of the quantumlimited attenuator of parameter is times the quantumlimited amplifier of parameter , hence its restriction to traceclass operators is continuous in the tracenorm.
Lemma II.12.
The quantumlimited attenuator of parameter admits the explicit representation
(II.12) 
for any traceclass operator . Then, if is diagonal in the Fock basis, is diagonal in the same basis for any also.
Proof.
The channel admits the Kraus decomposition (see Eq. (4.5) of [ivan2011operator])
(II.13) 
where
(II.14) 
Using (II.4a), we have
(II.15) 
and the claim easily follows. ∎
Lemma II.13.
The quantumlimited attenuator of parameter with can be written as the exponential of a Lindbladian , i.e. , where
(II.16) 
for any traceclass operator .
Lemma II.14.
Let
(II.18) 
be a selfadjoint HilbertSchmidt operator. Then, the projectors
(II.19) 
satisfy
(II.20) 
Proof.
Easily follows from an explicit computation. ∎
Lemma II.15 (Ky Fan’s Maximum Principle).
Let be a positive HilbertSchmidt operator with eigenvalues in decreasing order, i.e. , and let be a projector of rank . Then
(II.21) 
Proof.
(See also [bhatia2013matrix, fan1951maximum]). Let us diagonalize as in (II.18). The proof proceeds by induction on . Let have rank one. Since
(II.22) 
we have
(II.23) 
Suppose now that (II.21) holds for any rank projector. Let be a projector of rank . Its support then certainly contains a vector orthogonal to the support of , that has rank . We can choose normalized (i.e. ), and define the rank projector
(II.24) 
By the induction hypothesis on ,
(II.25) 
Since is in the support of , and
(II.26) 
we have
(II.27) 
and this concludes the proof. ∎
Lemma II.16.
Let and be positive HilbertSchmidt operators with eigenvalues in decreasing order and , respectively. Then,
(II.28) 
Iii Majorization
We recall here the definition of majorization. The interested reader can find more details in the dedicated book [marshall2010inequalities], that however deals only with the finitedimensional case.
Definition III.1 (Majorization).
Let and be decreasing summable sequences of positive numbers. We say that weakly submajorizes , or , iff
(III.1) 
If they have also the same sum, we say that majorizes , or .
Definition III.2.
Let and be positive traceclass operators with eigenvalues in decreasing order and , respectively. We say that weakly submajorizes , or , iff . We say that majorizes , or , if they have also the same trace.
From an operational point of view, majorization can also be defined with:
Theorem III.3.
Given two positive operators and with the same finite trace, the following conditions are equivalent:

;

For any continuous nonnegative convex function with ,
(III.2) 
For any continuous nonnegative concave function with ,
(III.3) 
can be obtained applying to a convex combination of unitary operators, i.e. there exists a probability measure on unitary operators such that
(III.4)
Proof.
See Theorems 5, 6 and 7 of [wehrl1974chaotic]. Notice that Ref. [wehrl1974chaotic] uses the opposite definition of the symbol “” with respect to most literature (and to Ref. [marshall2010inequalities]), i.e. there means that is majorized by . ∎
Iv Fock rearrangement
In order to state our main theorem, we need to define
Definition IV.1 (Fock rearrangement).
Let be a positive traceclass operator with eigenvalues in decreasing order. We define its Fock rearrangement as
(IV.1) 
If coincides with its own Fock rearrangement, i.e. , we say that it is passive [pusz1978passive, lenard1978thermodynamical, gorecki1980passive] with respect to the Hamiltonian . For simplicity, in the following we will always assume to be the reference Hamiltonian, and an operator with will be called simply passive.
Remark IV.2.
The Fock rearrangement of any projector of rank is the projector onto the first Fock states:
(IV.2) 
We define the notion of passivepreserving quantum operation, that will be useful in the following.
Definition IV.3 (Passivepreserving quantum operation).
We say that a quantum operation is passivepreserving if is passive for any passive positive traceclass operator .
We will also need these lemmata:
Lemma IV.4.
For any selfadjoint traceclass operator ,
(IV.3) 
where the are the projectors onto the first Fock states defined in (IV.2).
Proof.
We have
(IV.4)  
where we have used that
(IV.5) 
Since is traceclass, it is also HilbertSchmidt, the sum in (IV.4) converges, and its tail tends to zero for . ∎
Lemma IV.5.
A positive traceclass operator is passive iff for any finiterank projector
(IV.6) 
Proof.
First, suppose that is passive with eigenvalues in decreasing order, and let have rank . Then, by Lemma II.15
(IV.7) 
Suppose now that (IV.6) holds for any finiterank projector. Let us diagonalize as in (II.18). Putting into (IV.6) the projectors defined in (II.19),
(IV.8) 
where we have again used Lemma II.15. It follows that for any
(IV.9) 
and
(IV.10) 
It is then easy to prove by induction on that
(IV.11) 
i.e. is passive. ∎
Lemma IV.6.
Let be a sequence of positive traceclass operators with passive for any . Then also is passive, provided that its trace is finite.
Proof.
Follows easily from the definition of Fock rearrangement. ∎
Lemma IV.7.
Let be a quantum operation. Suppose that is passive for any passive finiterank projector . Then, is passivepreserving.
Lemma IV.8.
Let and be positive traceclass operators.

Suppose that for any finiterank projector
(IV.16) Then .

Let be passive, and suppose that . Then (IV.16) holds for any finiterank projector .
Lemma IV.9.
Let and be positive traceclass operators with . Then, for any positive traceclass operator ,
(IV.19) 
Lemma IV.10.
Let and be two sequences of positive traceclass operators, with and for any . Then
(IV.24) 
provided that both sides have finite traces.
Lemma IV.11.
The Fock rearrangement is continuous in the HilbertSchmidt norm.
Proof.
Let and be traceclass operators, with eigenvalues in decreasing order and , respectively. We then have
(IV.27) 
where we have used Lemma II.16. ∎
V Fockoptimal quantum operations
We will prove that any gaugecovariant Gaussian quantum channel satisfies this property:
Definition V.1 (Fockoptimal quantum operation).
We say that a quantum operation is Fockoptimal if for any positive traceclass operator
(V.1) 
i.e. Fockrearranging the input always makes the output less noisy, or among all the quantum states with a given spectrum, the passive one generates the least noisy output.
Remark V.2.
If is tracepreserving, weak submajorization in (V.1) can be equivalently replaced by majorization.
We can now state the main result of the paper:
Theorem V.3.
Any onemode gaugecovariant Gaussian quantum channel is passivepreserving and Fockoptimal.
Proof.
See Section VI. ∎
Corollary V.4.
Any linear combination with positive coefficients of gaugecovariant quantum Gaussian channels is Fockoptimal.
In the remainder of this section, we prove some general properties of Fockoptimality that will be needed in the main proof.
Lemma V.5.
Let be a passivepreserving quantum operation. If for any finiterank projector
(V.2) 
then is Fockoptimal.
Lemma V.6.
A quantum operation is passivepreserving and Fockoptimal iff
(V.5) 
for any two finiterank projectors and .
Proof.
Suppose first that is passivepreserving and Fockoptimal, and let and be finiterank projectors. Then
(V.6) 
We can now prove the two fundamental properties of Fockoptimality:
Theorem V.7.
Let be a quantum operation with the restriction of its HilbertSchmidt dual to traceclass operators continuous in the trace norm. Then, is passivepreserving and Fockoptimal iff is passivepreserving and Fockoptimal.
Proof.
Theorem V.8.
Let and be passivepreserving and Fockoptimal quantum operations with the restriction of to traceclass operators continuous in the trace norm. Then, their composition is also passivepreserving and Fockoptimal.
Proof.
Let and be finiterank projectors. Since is Fockoptimal and passivepreserving,
(V.10) 
and by Lemma IV.8
(V.11) 
Since is Fockoptimal and passivepreserving,
(V.12) 
From Theorem V.7 also is passivepreserving, and is passive. Lemma IV.9 implies then
(V.13) 
and the claim follows from Lemma V.6 combining (V.8) with (V.8). ∎
Lemma V.9.
Let be a quantum operation continuous in the HilbertSchmidt norm. Suppose that for any its restriction to the span of the first Fock states is passivepreserving and Fockoptimal, i.e. for any positive operator supported on the span of the first Fock states
(V.14) 
Then, is passivepreserving and Fockoptimal.
Proof.
Let and be two generic finiterank projectors. Since the restriction of to the support of is Fockoptimal and passivepreserving,
(V.15) 
Then, from Lemma IV.8
(V.16) 
From Lemma IV.4,
(V.17) 
and since , the Fock rearrangement (see Lemma IV.11) and the HilbertSchmidt product are continuous in the HilbertSchmidt norm, we can take the limit in (V.16) and get
(V.18) 
The claim now follows from Lemma V.6. ∎
Lemma V.10.
Let and be Fockoptimal and passivepreserving quantum operations. Then, also is Fockoptimal and passivepreserving.
Proof.
Easily follows from Lemma V.6. ∎
Vi Proof of the main theorem
First, we can reduce the problem to the quantumlimited attenuator:
Lemma VI.1.
If the quantumlimited attenuator is passivepreserving and Fockoptimal, the property extends to any gaugecovariant quantum Gaussian channel.
Proof.
From Lemma II.10, any quantum gaugecovariant Gaussian channel can be obtained composing a quantumlimited attenuator with a quantumlimited amplifier. From Lemma II.11, the HilbertSchmidt dual of a quantumlimited amplifier is proportional to a quantumlimited attenuator, and from Lemma V.7 also the amplifier is passivepreserving and Fockoptimal. Finally, the claim follows from Theorem V.8. ∎
By Lemma V.9, we can restrict to quantum states supported on the span of the first Fock states. Let now
(VI.1) 
where is the generator of the quantumlimited attenuator defined in (II.16). From the explicit representation (II.12), it is easy to see that remains supported on the span of the first Fock states for any . In finite dimension, the quantum states with nondegenerate spectrum are dense in the set of all quantum states. Besides, the spectrum is a continuous function of the operator, and any linear map is continuous. Then, without loss of generality we can suppose that has nondegenerate spectrum. Let
(VI.2) 
be the vectors of the eigenvalues of in decreasing order, and let
(VI.3) 
their partial sums, that we similarly collect into the vector . Let instead
(VI.4) 
be the eigenvalues of (recall that it is diagonal in the Fock basis for any ), and
(VI.5) 
their partial sums. Notice that and then . Combining (VI.4) with the expression for the Lindbladian (II.16), with the help of (II.4a) and (II.4b) it is easy to see that the eigenvalues satisfy
(VI.6) 
implying
(VI.7) 
for their partial sums. The proof of Theorem V.3 is a consequence of:
Lemma VI.2.
The spectrum of can be degenerate at most in isolated points.
Lemma VI.3.
is continuous in , and for any such that has nondegenerate spectrum it satisfies
(VI.8) 
Lemma VI.4.
Lemma VI.4 implies that the quantumlimited attenuator is passivepreserving. Indeed, let us choose passive. Since is diagonal in the Fock basis, is the sum of the eigenvalues corresponding to the first Fock states . Since is the sum of the greatest eigenvalues, . However, Lemma VI.4 implies for . Thus , so the operator is passive for any , and the channel is passivepreserving.
Then from the definition of majorization and Lemma VI.4 again,
(VI.10) 
for any , and the quantumlimited attenuator is also Fockoptimal.
Via Proof of Lemma vi.2
The matrix elements of the operator are analytic functions of . The spectrum of is degenerate iff the function
(VI.11) 
vanishes. This function is a symmetric polynomial in the eigenvalues of . Then, for the Fundamental Theorem of Symmetric Polynomials (see e.g Theorem 3 in Chapter 7 of [cox2015ideals]), can be written as a polynomial in the elementary symmetric polynomials in the eigenvalues of . However, these polynomials coincide with the coefficients of the characteristic polynomial of , that are in turn polynomials in its matrix elements. It follows that can be written as a polynomial in the matrix elements of the operator . Since each of these matrix element is an analytic function of , also is analytic. Since by hypothesis the spectrum of is nondegenerate, cannot be identically zero, and its zeroes are isolated points.
ViB Proof of Lemma vi.3
The matrix elements of the operator are analytic (and hence continuous and differentiable) functions of . Then for Weyl’s Perturbation Theorem is continuous in , and also is continuous (see e.g. Corollary III.2.6 and the discussion at the beginning of Chapter VI of [bhatia2013matrix]). Let have nondegenerate spectrum. Then, has nondegenerate spectrum for any in a suitable neighbourhood of . In this neighbourhood, we can diagonalize with
(VI.12) 
where the eigenvalues in decreasing order are differentiable functions of (see Theorem 6.3.12 of [horn2012matrix]), and