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Thermodynamically free quantum measurements

Thermal channels -- the free processes allowed in the resource theory of quantum thermodynamics -- are generalised to thermal instruments, which we interpret as implementing thermodynamically free quantum measurements; a Maxwellian demon using such measurements never violates the second law of thermodynamics. The properties of thermal instruments are investigated and, in particular, it is shown that they only measure observables commuting with the Hamiltonian, and they thermalise the measured system when performing a complete measurement, the latter of which indicates a thermodynamically induced information-disturbance trade-off. The demarcation of measurements that are not thermodynamically free paves the way for a resource-theoretic quantification for the thermodynamic cost of quantum measurements.


INTRODUCTION
Quantifying the thermodynamic cost of quantum processes is one of the central goals of quantum thermodynamics [1][2][3][4][5][6]. However, no consensus has yet been reached as to how such quantification should be achieved, and in particular a universally agreed upon definition for work remains elusive [7][8][9][10][11][12][13]. The resource theory of quantum thermodynamics [14][15][16] attempts to circumvent this issue by addressing what can and cannot be done when we restrict ourselves to processes that are "thermodynamically free". Such processes are referred to as (trace preserving) thermal operations, or thermal channels [17][18][19][20][21][22], which are realised by an energy conserving unitary interaction with an auxiliary system initially prepared in thermal equilibrium with an external bath. Energy conservation of the interaction implies no net-consumption of energy. Indeed, in such a case the distribution of work given by the celebrated two-point energy measurement protocol will vanish for all input states, and hence the average work will always agree with the unmeasured work-the difference in expected energy evaluated before and after the unitary evolution-as both quantities always vanish. On the other hand, thermality of the auxiliary system implies that it is freely available and hence its preparation will not accrue any costs. Within this framework, instead of directly quantifying the work cost for a given process we may instead ask what resource states-which are athermal and hence not thermodynamically free-are required to augment thermal channels so that the desired process may, at least approximately, be achieved.
In this paper, we shall generalise the notion of a thermal channel to a thermal instrument-a collection of thermal operations that sum to a thermal channel-which measures a thermal observable. As with ordinary thermal channels, a thermal instrument is implemented by a measuring apparatus with a probe that is initially prepared in thermal equilibrium with an external bath. However, both the "premeasurement" interaction between system and probe, as well as the ensuing "pointer objectification" that completes the measurement process, are energy conserving. Additionally, we do not restrict ourselves to unitary couplings and instead consider the more general class of bistochastic energy con-serving interactions. We interpret thermal instruments as implementing a thermodynamically free measurement. The justification for this interpretation follows analogous lines of reasoning as that for ordinary thermal channels cited above and, a fortiori, by the fact that a Maxwellian demon utilising such measurements never violates the second law of thermodynamics.
We investigate other properties of thermal instruments, showing that thermodynamic constraints vastly limit the types of measurements that can be performed. For example, a thermal instrument is shown to only measure observables that commute with the Hamiltonian, disturbs all observables not commuting with the measured observable, can never act as a non-trivial state preparation device, and always thermalises the measured system when performing a complete measurement, thereby destroying all the information contained therein. The demarcation of measurements that are not thermodynamically free is a first step towards a resource-theoretic quantification of the thermodynamic cost of measurements: for example, if a measurement is not thermodynamically free then its cost can be quantified by the free energy or coherence/asymmetry that must be initially present in the probe so as to approximately implement the desired measurement while still maintaining energy conservation for the measurement process [23][24][25][26]. This approach will also allow for such cost-quantification to be independent of the measured system's initial state, instead depending only on the properties of the desired measurement. This is in contrast to other methods that rely on the Landauer erasure of the probe, a quantity that is intimately statedependent [27,28].

Preliminaries:
Here, we shall cover briefly some basic elements of quantum measurement theory [29][30][31][32], which allows for the most general characterisation of the measurement process within purely operation terms. For simplicity, we shall only consider systems with a complex Hilbert space of finite dimension throughout.
An observable of a quantum system S with Hilbert space i.e., if E x are mutually orthogonal projection operators. An observable that is not sharp will be called unsharp.
Every observable E is compatible with infinitely many instruments [34], which describe how the measured system is transformed. An instrument acting in H S is a collection of operations (completely positive trace non-increasing linear maps) I := {I x : x ∈ X } such that I X (·) := x∈X I x (·) is a channel (a trace preserving operation). An instrument I is identified with a unique observable E via the relation tr[I x (ρ)] = tr[E x ρ] for all outcomes x and states ρ, and we shall refer to such I as an E-compatible instrument, or an E-instrument for short, and to I X as the corresponding Echannel. Moreover, for any ρ and x such that tr[I x (ρ)] > 0, we define ρ x := I x (ρ)/tr[I x (ρ)] as the conditional state of the system after having measured observable E with the instrument I, and having observed outcome x.
Every instrument may be implemented by infinitely many measurement schemes [35]. A measurement scheme is given by the tuple M := (H A , ξ, E, Z), where H A is the Hilbert space for the probe A of the apparatus and ξ is a fixed state of A, E is a channel acting in H S ⊗ H A which serves to correlate S with A, and Z := {Z x : x ∈ X } is a pointer observable of A. For all outcomes x, the operations of the instrument I implemented by M can be written as where tr A [·] is the partial trace over A. We may thus write the E-channel as I X (·) = tr A [E(· ⊗ ξ)]. Thermal instruments: A measurement scheme M can be understood in two ways. If we wish to measure some desired observable, with some specific choice of instrument, then we may specify the elements of M so as to achieve this. However, one may also consider the elements of M as given, and then ask what observable and instrument such a scheme implements. If we impose any constraints on M, it naturally follows that the class of implementable observables and instruments will be limited. We now define thermodynamically free measurement schemes, and subsequently determine the class of observables and instruments such schemes may or may not implement.  See Fig. 1 for a schematic representation of a thermal instrument. Condition (i) is justified by the fact that a Gibbs state describes a system when it is in thermal equilibrium with a large thermal bath. Provided that the bath is given, for example if it is the ambient environment that an experimental situation happens to find itself in, then a Gibbs state is thermodynamically free as it does not require any effort to prepare: one need only place A in thermal contact with the bath, and wait a sufficiently long time for it to reach thermal equilibrium. Indeed, in such a case there is no need to "erase" the information stored in the probe between successive measurements, and hence no Landauer erasure cost will ensue [36,37]. Condition (ii) ensures that the entropy of the compound system cannot decrease due to the measurement interaction, so that the second law is satisfied and no hidden "entropy sinks" are being used [38]. On the other hand, conditions (iii) and (iv) are both justified by energy conservation, so that the compound of system to be measured and the probe of the apparatus are energetically isolated during the entire measurement process, i.e., during both "premeasurement" and "pointer objectification".
We say that an arbitrary channel Φ acting in H (fully) conserves the Hamiltonian H if all moments of energy are invariant under its action, i.e., tr[H k Φ( )] = tr[H k ] for all states and k ∈ N [26]. But in finite dimensions, and for bistochastic channels, if the first moment is conserved (k = 1), then all moments will be conserved (k > 1). See

FIG. 1:
An instrument I measures an observable in the input state ρ and, conditional on observing outcome x, transforms the system to the (non-normalised) state Ix(ρ). I is thermal when it is implemented by a bistochastic energy conserving interaction between the measured system and a probe of the measuring apparatus, initially prepared in thermal equilibrium with a bath of inverse temperature β, followed by read-out of a pointer observable commuting with the probe's Hamiltonian.
Appendix (A) for a proof. Moreover, for unitary channels Φ(·) = U · U † which are clearly bistochastic, conservation is equivalent to the commutation relation [U, H] = O. Now note that in order for the measurement process to be complete, the pointer observable must be objectified, or "measured". We model the objectification process in the language of instruments, and so introduce some Z-compatible instrument J acting in H A that "measures" the pointer observable. While we do not consider the exact form such an instrument takes-by Eq. (1) we see that it is only the pointer observable, and not the instrument that measures it, which uniquely determines the instrument acting in the system-we do demand that the Z-channel J X conserves energy. As shown in [26], this constraint demands that Z commutes with H A . Such commutation is known as the the Yanase condition [39,40] which was first introduced in the context of the Wigner-Araki-Yanase theorem [41][42][43].
Extractable work and the second law: Given a single thermal bath of inverse temperature β, the extractable work of a system S with Hamiltonian H S , initially prepared in state ρ, is defined as where S(· ·) is the quantum relative entropy and π β = e −βH S /tr[e −βH S ] is the Gibbs state of the system. The extractable work is identified with the non-equilibrium free energy of ρ relative to π β , and has an operational meaning as the maximum amount of work that can be extracted by an isothermal process, achieved in the quasistatic limit as ρ is transformed to π β [44]. If the system is initially prepared in thermal equilibrium with the bath then no work can be extracted, since S(ρ π β ) = 0 whenever ρ = π β . This is the second law of thermodynamics in effect. But what if we are able to measure the system? We define the average extractable work of a system initialised in state ρ, given measurement by an E-compatible instrument I followed by feedback, as Here, W I ext is the maximum (average) amount of work that can be extracted if, conditional on observing outcome x, we choose a specific isothermal process so as to transform the post-measurement state ρ x to the Gibbs state π β . The average extractable work therefore quantifies the merit of an information heat engine that utilises measurement and feedback with a single thermal bath, such as the Szilard engine and its descendants [45][46][47][48][49]. Now let us consider the measurements that are usually considered in the literature. Let E be a sharp observable with effects E x = |ψ x ψ x |, where {|ψ x } is an eigenbasis of H S , and assume that E is measured by the von Neumann-Lüders instrument I L (·) = ψ x |·|ψ x |ψ x ψ x |. By a simple calculation, we can see that when the input state is thermal, i.e., ρ = π β , then it holds that ρ x = |ψ x ψ x | for all x, and Eq. (3) reduces to β −1 H , where H is the Shannon entropy of the probability distribution { ψ x |π β |ψ x }; using measurement and feedback, we have completely converted heat into work. Previous attempts to "save" the second law in such a case rely on Landauer erasure of the probe, which given certain assumptions about the measurement process can be shown to have a minimum work cost of β −1 H . But as shown by the following proposition, maintaining the second law will not need such arguments when the measurement itself is thermodynamically free: Proposition 1. Consider a system S with Hamiltonian H S . Assume that the system is measured by a thermal instrument I, implemented at inverse temperature β. Then for all states ρ, it holds that with W ext and W I ext as defined by Eq.
The proof is provided in Appendix (C), and is a consequence of the following observation: for any thermodynamically free measurement scheme, if the system is prepared in a Gibbs state π β with the same temperature as the probe, then it holds that I x (π β ) = tr[E x π β ]π β for all x. That is, not only does the thermal channel I X preserve the Gibbs state, but so do all operations of a thermal instrument. The above proposition states that when both the measuring apparatus and the measured system are in contact with a single thermal bath, then the average extractable work given thermodynamically free measurements and feedback can never exceed the extractable work without measurement; indeed, if the system is initially prepared in thermal equilibrium with the bath, then W I ext = W ext = 0 and so no work can be extracted at all. Couched in more poetic terms, a thermodynamically impotent Maxwellian demon needs no exorcism, for while it may gain information about a system in thermal equilibrium, it is unable to convert such information into useful work [50].
Time-translation covariance: An operation Φ is timetranslation covariant if it satisfies Φ(e −itH S · e itH S ) = e −itH S Φ(·)e itH S for all t. As is well-known, thermal channels are covariant, and for any covariant channel Φ and an asymmetry monotone I (H S , ·), it holds that I (H S , ρ) I (H S , Φ(ρ)) for all states ρ; covariant channels always reduce the asymmetry of a state with respect to the Hamiltonian [51]. But if I is a thermal instrument, then not only is the thermal channel I X covariant, but all the individual operations I x are covariant as well. See Appendix (D) for a proof. It follows that for any asymmetry monotone that obeys selective monotonicity [52,53]-such as the family of Wigner-Yanase-Dyson skew informations [54,55], or the quantum Fisher information [56]-and for all thermal instruments I and states ρ, it holds that I (H S , ρ) x∈X I (H S , I x (ρ)) I (H S , I X (ρ)).
In other words, a thermal instrument will always decrease the asymmetry of a state with respect to the Hamiltonian "on average". Note that even if we abandon the Yanase condition in Definition 1, a thermal channel I X is always covariant, in which case the relation I (H S , ρ) I (H S , I X (ρ)) will continue to hold. However, in such a case covariance for the individual operations of I will be broken, and so it may be the case that for some ρ we have I (H S , ρ) < x∈X I (H S , I x (ρ)).

Energy-compatibility:
As a consequence of timetranslation covariance of thermal instruments, all thermal observables are time-translation invariant, i.e., they commute with the Hamiltonian. In fact, an observable E is thermal if and only if [E, H S ] = O. See Appendix (D) for a proof. Such commutativity admits an elegant interpretation in terms of compatibility [57,58]. Two observables E := {E x : x ∈ X } and F := {F y : y ∈ Y} are compatible, or jointly measurable, if they admit a joint observable G := {G x,y : (x, y) ∈ X × Y} so that E x = y G x,y and F y = x G x,y . If E and F do not admit a joint observable, then they are incompatible. Note that while commutation of E with F is sufficient for compatibility, it is in general not necessary: if both E and F are sufficiently unsharp, they may be compatible even if they do not commute. Now let the Hamiltonian have the spectral decomposition H S = m ω m P m , with ω m the distinct energy eigenvalues and P m the corresponding spectral projections. The sharp observable P := {P m } is the spectral measure of H S , and we will refer to measurement of P and of the Hamiltonian interchangeably. Given that [E, and that commutativity is a sufficient condition for compatibility, then E and P are jointly measureable. Indeed, since P is sharp then the effects of the joint observable are uniquely given as G x,m = E x P m . In other words, for any thermal observable E we may always construct a single measurement device that jointly gives both the statistics of E and the statistics of the Hamiltonian.
Of course, while thermal observables are always jointly measureable with the Hamiltonian, this does not generally imply that two thermal observables are themselves compatible; while any pair of thermal observables E and F must both commute with H S , it may be possible for them to not commute with each other. However, there is one limiting situation where compatibility is guaranteed: when the Hamiltonian has a non-degenerate energy spectrum, for example if S is a spin-1/2 system embedded in a non-vanishing magnetic field. In such a case, we may write P m = |ψ m ψ m |, where {|ψ m } is an orthonormal basis that spans H S . If E is a thermal observable, it follows that all of its effects are simultaneously diagonalisable as E x = m p(x|m)P m , where {p(x|m)} is a family of non-negative numbers that satisfy x p(x|m) = 1 for all m [59]. It is clear that any pair of thermal observables will commute, and are therefore compatible. In such a case, we may infer that measurement of incompatible observables will always have some thermodynamic cost: even if E is a thermal observable, F will be incompatible with E only if it is non-thermal. We note that the ability to measure incompatible observables is a crucial ingredient in many quantum phenomena such as vi-olation of Bell inequalities [60] and quantum steering [61], and incompatible observables have been shown to outperform compatible ones for quantum state discrimination [62]. Indeed, measurement of incompatible observables has also been suggested as a method of efficiently fueling quantum heat engines [63].
Disturbance and commutativity: In many situations, we may want to ascertain two different properties of a system, that is, measure two observables E and F. As discussed above, if the observables are compatible then we may simultaneously determine both of their statistics by measuring their joint observable G. However, another method is with sequential measurements, where at first E is measured by an instrument I, and subsequently F is measured. The statistics of both observables can be faithfully obtained with such an approach only if I does not "disturb" F, i.e., if it holds that tr[F y I X (ρ)] = tr[F y ρ] for all states ρ and outcomes y of F. While [E, F] = O is a necessary condition for nondisturbance when either of these observables is sharp, nondisturbance may be possible for non-commuting pairs of observables if they are both sufficiently unsharp. However, recall that a thermal channel always leaves the Gibbs state invariant, i.e., I X (π β ) = π β . Since the Gibbs state is full-rank (or faithful), then as shown in [64] a thermal E-instrument will disturb all observables not commuting with E. Additionally, if F is not thermal, i.e., if F does not commute with H S , then as shown in [26] non-disturbance will be possible only if F also commutes with ∆H S := I * X (H S ) − H S , where I * X is the "Heisenberg picture" dual of the E-channel I X .
Conditional state preparation and complete measurements: An E-compatible instrument I is said to be nuclear if its operations satisfy for all ρ and x, where {σ x } is a family of states that depend only on the measurement outcome, and not the input state ρ. Nuclear instruments have utility as conditional state preparation devices, since for any input state, conditional on observing outcome x we know that the system has been prepared in the state σ x . An example of a nuclear instrument is the well-known von Neumann-Lüders measurement, which "collapses" the measured system into the eigenstates of the measured observable. The following proposition imposes constraints on nuclear thermal instruments: Proposition 2. Consider a system S with Hamiltonian H S , and assume that I is a thermal E-instrument acting in H S . If I is nuclear, then I is a trivial, thermalising instrument, with operations satisfying for all ρ and x, where π β is the Gibbs state of the system for some β > 0.
The proof is provided in Appendix (E). The above proposition implies that an instrument will be a non-trivial state preparation device only if it is not thermodynamically free.
Let us now highlight an important consequence of the above result for the class of rank-1 observables. An observable E is called rank-1 if all the effects are of the form E x = λ x P x , where λ x ∈ (0, 1] and P x is a rank-1 projection operator, i.e., P x projects onto a 1-dimensional subspace of H S . Rank-1 observables can be seen as a complete measurement, since any observable can be maximally "refined" into a rank-1 observable [65]; if the effects of some observable F can be diagonalised as is a maximal refinement of F: E provides more information than F in the sense that one may always recover the statistics of F from the statistics obtained by measuring E but not vice versa, and E itself cannot be further refined. As shown in Corollary 1 of Ref. [64] (also see Theorem 2 of Ref. [66]), all instruments compatible with a rank-1 observable are nuclear. In conjunction with Proposition 2, it follows that a thermodynamically free complete measurement necessarily thermalises the measured system. Indeed, this implies that such a measurement will disturb all non-trivial observables, even those that commute with the measured observable.
As an interesting consequence, let us consider again the situation where the system's Hamiltonian has a nondegenerate spectrum, so that the effects of any thermal observable E may be written as E x = m p(x|m)P m , where P m are the rank-1 spectral projections of the Hamiltonian. It clearly follows that measuring any thermal observable other than the Hamiltonian is superfluous: one may always reconstruct the statistics of any thermal observable by post-processing the statistics obtained by measuring the Hamiltonian with the non-negative numbers p(x|m). But the spectral measure of the Hamiltonian is a rank-1 observable, and so if we wish to measure the Hamiltonian by a thermal instrument, then the system will necessarily be thermalised. We see that there is a thermodynamically induced information-disturbance trade-off, where by obtaining all the information that is available without expending any thermodynamic resources, we must thermalise the system so as to destroy all the information contained therein. This is analogous to the case where, in the absence of any thermodynamic constraints, measurement of an informationally complete observable completely destroys all the information in the measured system [67].

DISCUSSION
Quantifying the thermodynamic cost of quantum measurements is in general a challenging task. Even if we take into account the erasure cost of the probe between successive measurements, we must also quantify the cost of preparing the probe in its initial (usually pure) state, the cost of implementing the measurement interaction between system and probe, and the cost associated with readout of the pointer observable. By taking inspiration from the resource theoretic approach to quantum thermodynamics, we have defined a thermodynamically free measurement where each step of the measurement process has zero associated costs. Indeed, such measurements never lead to an advantage in work extraction from a single thermal bath, and so the Maxwell demon paradox is resolved from the outset without need for any post-hoc exorcisms. Having provided a preliminary demarcation between free and non-free measurements, it is now possible to quantify the fundamental cost of quantum measurements that are not thermodynamically free. For example, by maintaining energy conservation of the measurement process-thereby continuing to avoid the conceptual difficulties that arise when trying to directly quantify the work cost of channels-we may obtain bounds for the free energy or coherence that must be initially present in the probe so as to approximately achieve the desired measurement. This task is left for future work.

Appendix
Before presenting the detailed proofs for the claims made in the main text, let us first introduce some notation and basic definitions. We denote by L(H) the algebra of linear operators on a finite-dimensional complex Hilbert space H, with O and 1 the zero and identity operators of L(H), respectively. A "Schrödinger picture" operation is defined as a completely positive, trace non-increasing linear map Φ : L(H) → L(K), where H is the input space and K is a potentially different output space. When both input and output spaces are the same, i.e., H = K, we say that Φ acts in H. The associated "Heisenberg picture" dual operation is a completely positive linear map Φ * : L(K) → L(H), defined by the trace duality tr[AΦ(B)] = tr[Φ * (A)B] for all A ∈ L(K), B ∈ L(H). Φ * is sub-unital, i.e., Φ * (1 K ) 1 H , and is unital when the equality holds, which is the case exactly when Φ is a channel, i.e., when Φ preserves the trace. Φ is a bistochastic channel if both Φ and Φ * are trace preserving and unital.

Appendix A: Bistochastic channels and energy conservation
Recall that energy conservation by an arbitrary channel Φ is defined as preservation of all moments of energy, i.e., tr[H k Φ( )] = tr[H k ] for all k ∈ N and . This condition can be equivalently stated in the Heisenberg picture as Φ * (H k ) = H k for all k ∈ N. While a general channel may conserve the first moment while not conserving the higher moments, we shall now show that in the special case of bistochastic channels acting in a finite-dimensional Hilbert space, a channel conserving the first moment is guaranteed to conserve all higher moments. Lemma 1. Let Φ be a bistochastic channel acting in a finite-dimensional Hilbert space H. Assume that Φ * (H) = H for some self-adjoint operator with spectral decomposition H = n λ n P n . The following hold: (iii) Φ( ) = for all states with spectral decomposition = n p n P n .
Proof. Let {K i } be any Kraus representation for Φ, so that we may write Φ(·) = i K i · K † i and Φ * (·) = i K † i · K i . If Φ is bistochastic, then it holds that both Φ and Φ * are trace preserving and unital, which implies that i if and only if all spectral projections P n commute with K i , K † i . Since e −itH = n e −itλn P n , (ii) follows trivially from above. Similarly for (iii).

Appendix B: Thermal instruments preserve the Gibbs state
It is well-known that a thermal channel preserves the Gibbs state. Here, we shall show that all operations of a thermal instrument also preserve the Gibbs state. Lemma 2. Consider a system S with Hamiltonian H S , and let I be a thermal E-instrument acting in H S . For some β > 0 there exists a Gibbs state of the system π β = e −βH S /tr[e −βH S ] such that for all x ∈ X the following holds: (B1) Proof. Assume that I is a thermal instrument, so that by Definition 1 it admits a thermodynamically free measurement scheme M β := (H A , H A , β, E, Z) where the probe is prepared in the Gibbs state ξ β = e −βH A /tr[e −βH A ]. For the Gibbs state of the system with the same temperature as the probe, π β , additivity of the total Hamiltonian H = H S ⊗1 A +1 S ⊗H A implies that π β ⊗ξ β = e −βH /tr[e −βH ] is the Gibbs state for the total system. Since E is bistochastic and energy conserving, it follows from item (iii) of Lemma 1 that E(π β ⊗ ξ β ) = π β ⊗ ξ β , and so by Eq. (1) it holds that for all x. But since I is compatible with E, then it must hold that tr[I x (π β )] = tr[E x π β ]. This implies that tr[Z x ξ β ] = tr[E x π β ]. As such, if I is a thermal instrument, then its operations must satisfy Eq. (B1).

Appendix C: Proof of Proposition 1
Let M β := (H A , H A , β, E, Z) be a thermodynamically free measurement scheme for the observable E with instrument I acting in H S . Now, let us define the instrument Φ with operations Φ x : L(H S ) → L(H S ⊗ K), ρ → I x (ρ) ⊗ |x x| where {|x } is an orthonormal basis that spans K. The action of the channel Φ X (·) := x∈X Φ x (·) on the states ρ and π β can be written as where the second line follows from Lemma 2, and where we define ρ x := I x (ρ)/tr[E x ρ] for any ρ and x such that tr[E x ρ] > 0, and ρ x := O otherwise. Now define by p := {p(x) = tr[E x ρ] : x ∈ X } and q := {q(x) = tr[E x π β ] : x ∈ X } the probability vectors arising from a measurement of E in the states ρ and π β , respectively. By the data processing inequality, and the "direct sum" property of the relative entropy [69], it holds that S(ρ π β ) S(Φ X (ρ) Φ X (π β )) = D(p q) + x∈X tr[E x ρ]S(ρ x π β ), where S(ρ σ) := tr[ρ(ln (ρ) − ln (σ))] is the quantum relative entropy between states ρ and σ whenever supp(ρ) ⊆ supp(σ), and D(p q) := x p(x) ln (p(x)/q(x)) is the classical relative entropy between probability vectors p and q whenever p(x) > 0 =⇒ q(x) > 0. Since a Gibbs state is faithful, then supp(ρ) ⊆ supp(π β ) holds for all ρ, while tr[E x π β ] > 0 holds for all x for which E x = O. As such, the quantities on both sides of the above equation are always finite and non-negative. Since D(p q) 0 always holds, then we have the inequality x∈X tr[E x ρ]S(ρ x π β ) S(ρ π β ). By Eq. (2) and Eq. (3), we obtain the bound in Proposition 1.