Passive states optimize the output of bosonic Gaussian quantum channels
An ordering between the quantum states emerging from a single mode gauge-covariant 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.
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. Gauge-covariant 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 gauge-covariant 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 trade-off region of the quantum-limited 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 quantum-limited Gaussian attenuator among all the states with a given entropy. This still unproven result would follow from a stronger conjecture, the Entropy Photon-number 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 gauge-covariant 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 one-mode gauge-covariant 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 one-mode Gaussian channels in terms of unitary equivalence [holevo2013quantum], we extend the result to the channels that are not gauge-covariant 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 quantum-limited 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 log-concave 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 quantum-limited attenuator among all the input states diagonal in the Fock basis with a given von Neumann entropy (but without the ultra log-concavity 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 quantum-limited 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 gauge-covariant 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
If is finite, we say that is a trace-class operator.
Definition II.2 (Quantum operation).
A quantum operation is a linear completely positive map on trace-class operators continuous in the trace norm.
A trace-preserving 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
has a countable orthonormal basis
called the Fock basis, on which the ladder operators act as
We can define a number operator
Definition II.4 (Hilbert-Schmidt norm).
The Hilbert-Schmidt norm of an operator is
(Hilbert-Schmidt dual) Let be a linear map acting on trace class operators and continuous in the trace norm. Its Hilbert-Schmidt dual is the map on bounded operators continuous in the operator norm defined by
for any trace-class operator and any bounded operator .
Definition II.6 (Characteristic function).
The characteristic function of a trace-class operator is
where denotes the complex conjugate.
It is possible to prove that any trace-class operator is completely determined by its characteristic function.
Theorem II.7 (Noncommutative Parceval relation).
The characteristic function provides an isometry between the Hilbert-Schmidt product and the scalar product in , i.e. for any two trace-class operators and ,
See e.g. Theorem 5.3.3 of [holevo2011probabilistic]. ∎
Definition II.8 (Gaussian gauge-covariant quantum channel).
A gauge-covariant quantum Gaussian channel with parameters and can be defined by its action on the characteristic function: for any trace-class operator ,
The channel is called quantum-limited if . If , it is a quantum-limited attenuator, while if it is a quantum-limited amplifier.
Any gauge-covariant quantum Gaussian channel is continuous also in the Hilbert-Schmidt norm.
Any gauge-covariant quantum Gaussian channel can be written as a quantum-limited amplifier composed with a quantum-limited attenuator.
See [garcia2012majorization, giovannetti2015solution, mari2014quantum, holevo2015gaussian]. ∎
The Hilbert-Schmidt dual of the quantum-limited attenuator of parameter is times the quantum-limited amplifier of parameter , hence its restriction to trace-class operators is continuous in the trace-norm.
The quantum-limited attenuator of parameter admits the explicit representation
for any trace-class operator . Then, if is diagonal in the Fock basis, is diagonal in the same basis for any also.
The channel admits the Kraus decomposition (see Eq. (4.5) of [ivan2011operator])
Using (II.4a), we have
and the claim easily follows. ∎
The quantum-limited attenuator of parameter with can be written as the exponential of a Lindbladian , i.e. , where
for any trace-class operator .
be a self-adjoint Hilbert-Schmidt operator. Then, the projectors
Easily follows from an explicit computation. ∎
Lemma II.15 (Ky Fan’s Maximum Principle).
Let be a positive Hilbert-Schmidt operator with eigenvalues in decreasing order, i.e. , and let be a projector of rank . Then
(See also [bhatia2013matrix, fan1951maximum]). Let us diagonalize as in (II.18). The proof proceeds by induction on . Let have rank one. Since
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
By the induction hypothesis on ,
Since is in the support of , and
and this concludes the proof. ∎
Let and be positive Hilbert-Schmidt operators with eigenvalues in decreasing order and , respectively. Then,
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 finite-dimensional case.
Definition III.1 (Majorization).
Let and be decreasing summable sequences of positive numbers. We say that weakly sub-majorizes , or , iff
If they have also the same sum, we say that majorizes , or .
Let and be positive trace-class operators with eigenvalues in decreasing order and , respectively. We say that weakly sub-majorizes , 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:
Given two positive operators and with the same finite trace, the following conditions are equivalent:
For any continuous nonnegative convex function with ,
For any continuous nonnegative concave function with ,
can be obtained applying to a convex combination of unitary operators, i.e. there exists a probability measure on unitary operators such that
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 trace-class operator with eigenvalues in decreasing order. We define its Fock rearrangement as
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.
The Fock rearrangement of any projector of rank is the projector onto the first Fock states:
We define the notion of passive-preserving quantum operation, that will be useful in the following.
Definition IV.3 (Passive-preserving quantum operation).
We say that a quantum operation is passive-preserving if is passive for any passive positive trace-class operator .
We will also need these lemmata:
For any self-adjoint trace-class operator ,
where the are the projectors onto the first Fock states defined in (IV.2).
where we have used that
Since is trace-class, it is also Hilbert-Schmidt, the sum in (IV.4) converges, and its tail tends to zero for . ∎
A positive trace-class operator is passive iff for any finite-rank projector
First, suppose that is passive with eigenvalues in decreasing order, and let have rank . Then, by Lemma II.15
where we have again used Lemma II.15. It follows that for any
It is then easy to prove by induction on that
i.e. is passive. ∎
Let be a sequence of positive trace-class operators with passive for any . Then also is passive, provided that its trace is finite.
Follows easily from the definition of Fock rearrangement. ∎
Let be a quantum operation. Suppose that is passive for any passive finite-rank projector . Then, is passive-preserving.
Let and be positive trace-class operators.
Suppose that for any finite-rank projector
Let be passive, and suppose that . Then (IV.16) holds for any finite-rank projector .
Let and be the eigenvalues in decreasing order of and , respectively, and let us diagonalize as in (II.18).
Suppose now that and . Then, for any and any projector of rank ,
where we have used Lemma II.15 again.
Let and be positive trace-class operators with . Then, for any positive trace-class operator ,
Let and be two sequences of positive trace-class operators, with and for any . Then
provided that both sides have finite traces.
The Fock rearrangement is continuous in the Hilbert-Schmidt norm.
Let and be trace-class operators, with eigenvalues in decreasing order and , respectively. We then have
where we have used Lemma II.16. ∎
V Fock-optimal quantum operations
We will prove that any gauge-covariant Gaussian quantum channel satisfies this property:
Definition V.1 (Fock-optimal quantum operation).
We say that a quantum operation is Fock-optimal if for any positive trace-class operator
i.e. Fock-rearranging 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.
If is trace-preserving, weak sub-majorization in (V.1) can be equivalently replaced by majorization.
We can now state the main result of the paper:
Any one-mode gauge-covariant Gaussian quantum channel is passive-preserving and Fock-optimal.
See Section VI. ∎
Any linear combination with positive coefficients of gauge-covariant quantum Gaussian channels is Fock-optimal.
In the remainder of this section, we prove some general properties of Fock-optimality that will be needed in the main proof.
Let be a passive-preserving quantum operation. If for any finite-rank projector
then is Fock-optimal.
A quantum operation is passive-preserving and Fock-optimal iff
for any two finite-rank projectors and .
Suppose first that is passive-preserving and Fock-optimal, and let and be finite-rank projectors. Then
We can now prove the two fundamental properties of Fock-optimality:
Let be a quantum operation with the restriction of its Hilbert-Schmidt dual to trace-class operators continuous in the trace norm. Then, is passive-preserving and Fock-optimal iff is passive-preserving and Fock-optimal.
Condition (V.5) can be rewritten as
and is therefore symmetric for and . ∎
Let and be passive-preserving and Fock-optimal quantum operations with the restriction of to trace-class operators continuous in the trace norm. Then, their composition is also passive-preserving and Fock-optimal.
Let and be finite-rank projectors. Since is Fock-optimal and passive-preserving,
and by Lemma IV.8
Since is Fock-optimal and passive-preserving,
Let be a quantum operation continuous in the Hilbert-Schmidt norm. Suppose that for any its restriction to the span of the first Fock states is passive-preserving and Fock-optimal, i.e. for any positive operator supported on the span of the first Fock states
Then, is passive-preserving and Fock-optimal.
Let and be two generic finite-rank projectors. Since the restriction of to the support of is Fock-optimal and passive-preserving,
Then, from Lemma IV.8
From Lemma IV.4,
The claim now follows from Lemma V.6. ∎
Let and be Fock-optimal and passive-preserving quantum operations. Then, also is Fock-optimal and passive-preserving.
Easily follows from Lemma V.6. ∎
Vi Proof of the main theorem
First, we can reduce the problem to the quantum-limited attenuator:
If the quantum-limited attenuator is passive-preserving and Fock-optimal, the property extends to any gauge-covariant quantum Gaussian channel.
From Lemma II.10, any quantum gauge-covariant Gaussian channel can be obtained composing a quantum-limited attenuator with a quantum-limited amplifier. From Lemma II.11, the Hilbert-Schmidt dual of a quantum-limited amplifier is proportional to a quantum-limited attenuator, and from Lemma V.7 also the amplifier is passive-preserving and Fock-optimal. 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
where is the generator of the quantum-limited 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 non-degenerate 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 non-degenerate spectrum. Let
be the vectors of the eigenvalues of in decreasing order, and let
their partial sums, that we similarly collect into the vector . Let instead
be the eigenvalues of (recall that it is diagonal in the Fock basis for any ), and
for their partial sums. The proof of Theorem V.3 is a consequence of:
The spectrum of can be degenerate at most in isolated points.
is continuous in , and for any such that has non-degenerate spectrum it satisfies
If is continuous in and satisfies (VI.8), then
for any and .
Lemma VI.4 implies that the quantum-limited attenuator is passive-preserving. 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 passive-preserving.
Then from the definition of majorization and Lemma VI.4 again,
for any , and the quantum-limited attenuator is also Fock-optimal.
Vi-a Proof of Lemma vi.2
The matrix elements of the operator are analytic functions of . The spectrum of is degenerate iff the function
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 non-degenerate, cannot be identically zero, and its zeroes are isolated points.
Vi-B 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 non-degenerate spectrum. Then, has non-degenerate spectrum for any in a suitable neighbourhood of . In this neighbourhood, we can diagonalize with
where the eigenvalues in decreasing order are differentiable functions of (see Theorem 6.3.12 of [horn2012matrix]), and