Device-independent quantum reading and noise-assisted quantum transmitters

In quantum reading, a quantum state of light (transmitter) is applied to read classical information. In the presence of noise or for sufficiently weak signals, quantum reading can outperform classical reading by enhanced state distinguishability. Here we show that the enhanced quantum efficiency depends on the presence in the transmitter of a particular type of quantum correlations, the discord of response. Different encodings and transmitters give rise to different levels of efficiency. Considering noisy quantum probes we show that squeezed thermal transmitters with non-symmetrically distributed noise among the field modes yield a higher quantum efficiency compared to coherent thermal quantum states. The noise-enhanced quantum advantage is a consequence of the discord of response being a non-decreasing function of increasing thermal noise under constant squeezing, a behavior that leads to an increased state distinguishability. We finally show that, for non-symmetric squeezed thermal states, the probability of error, as measured by the quantum Chernoff bound, vanishes asymptotically with increasing local thermal noise at finite global squeezing. Therefore, at fixed finite squeezing, noisy but strongly discordant quantum states with large noise imbalance between the field modes can outperform noisy classical resources as well as pure entangled transmitters with the same finite level of squeezing.

In quantum reading, a quantum state of light (transmitter) is applied to read classical information.
In the presence of noise or for sufficiently weak signals, quantum reading can outperform classical reading by enhanced state distinguishability. Here we show that the enhanced quantum efficiency depends on the presence in the transmitter of a particular type of quantum correlations, the discord of response. Different encodings and transmitters give rise to different levels of efficiency. Considering worst-case encodings with noisy quantum probes we show that symmetric squeezed thermal transmitters yield a higher quantum efficiency compared to classical and thermal quantum states. The noise-enhanced quantum advantage is a consequence of the discord of response being a nondecreasing function of increasing thermal noise under constant squeezing, a behavior that leads to an increased state distinguishability. We finally show that for non-symmetric squeezed thermal states the probability of error, as measured by the quantum Chernoff bound, vanishes asymptotically with increasing local thermal noise at finite global squeezing. Therefore noisy but maximally discordant quantum states outperform both classical resources as well as pure entangled ones.

I. INTRODUCTION
In the context of quantum information and quantum technology the idea of reading classical data by means of quantum states arises quite naturally [1,2]. In general, the standard implementations of reading are based on optical technologies: the task is the readout of a digital optical memory, where information is stored by means of the optical properties of the memory cells that are in turn probed by shining light, e.g. a laser beam, on them. The probing light is usually denoted as the transmitter. Interesting features arise in the regime in which the transmitter has to be treated quantum mechanically. The maximum rate of reliable readout defines the quantum reading capacity. The latter can overcome the classical reading capacity, obtained by probing with classical light, in several relevant settings. The (possibly quantum) transmitter that is needed to extract the encoded information is prepared in some initial state. By scanning a particular cell the transmitter changes its properties in a way depending on the cell. The task is to recognize which cell occurs based on the output state of the transmitter after it has been detected and measured. Therefore the problem of reading reduces to the problem of distinguishing the output states of the transmitter.
In such optical settings one needs to consider two main coding protocols depending on the trade off between energy and coherence of the transmitters and the channels that are being used. The first protocol is the so-called amplitude shift keying (ASK) in which the changes in the state of the transmitter are caused by the cell-dependent * Corresponding author: illuminati@sa.infn.it losses in the intensity of the transmitted signal [1][2][3]. The second main protocol is the so-called phase shift keying (PSK) [4,5]. This is a type of coding which does not produce energy dissipation. On the other hand, it requires a very high coherence of the transmitter, a feature that might be realized in realistic implementations [5].
If the transmitter is quantum, the cells play the role of effective quantum channels. The ASK protocol then corresponds to a dissipative channel coding, while in the PSK protocol the coding is realized by means of unitary operation. Within the ASK protocol, it can be shown that in the low-energy regime there is an energy threshold above which the maximally entangled transmitter, i.e. a two-mode squeezed state, yields a better reading efficiency (when properly amplified by means of many copies) than any of the classical states with the same energy [1,2]. The general result is still valid in the presence of some noise-induced decoherence. Within the ASK protocol, coding is then realized by local channels corresponding to cells with different reflectivities.
In the PSK protocol, the coding is realized by means of local unitary operations, specifically local phase shifts [4,5]. In the ideal, noise-free protocol the transmitter is taken to be in a pure Gaussian quasi-Bell state, i.e. a Bell-type superposition of quasi-orthogonal coherent states. In this scheme, the resulting quantum advantage is absolute, in the sense that quantum reading of the classical information encoded via a phase shift of π is achieved with vanishing error, while any classical state of the transmitter always yields a finite error probability.
In both the ASK and PSK protocols the transmitter is assumed to be a bipartite system such that only one part of it scans the memory cell. This choice is motivated by the fact that it maximizes distinguishability at the output when the state of the transmitter is quantum. As already mentioned, the reading efficiency is characterized by the error probability. Information is encoded in binary memory cells with indices 0 and 1. It is thus written using only two local channels that are assumed to occur with equal a priori probabilities. Given the bi-partite input transmitter AB , the two possible output states will be denoted by AB . The probability of error in distinguishing the two output states when reading a memory cell by means of the same input AB is given by the well-known Helstrom formula: where AB T r is the trace distance, with X T r = Tr √ XX † . With our normalization convention, the trace distance ranges from 0 to a maximum of 2 for orthogonal pure states.
In the original reading protocols the goal is to minimize P err over the set of possible transmitter states AB at fixed encoding in the memory cells [2,5]. The problem is thus dependent on the type of memory device being used.
Here instead we wish to provide a device-independent characterization of a given transmitter by considering the worst-case scenario that maximizes the probability of error P err over all possible codings. Once the worstcase scenario is identified, one can then compare different classes of transmitter states to identify the ones that minimize the maximum probability of error P (max) err . We will show that the maximum probability of error P (max) err is a monotonically decreasing function of the amount of quantum correlations present in the transmitter state AB , as quantified by a recently introduced measure of quantum correlations, the so-called discord of response [6] in its Gaussian version [7]. As a consequence, every state with non-vanishing discord of response is able to read any type of memory device with maximal P err < 1/2. On the other hand, for each classical transmitter state, i.e. for transmitters with vanishing discord of response, there will always exist at least one memory device which is completely invisible, i.e. for which P err = 1/2.
In Sec. II we derive the exact analytical relation between the maximum probability of error and the Gaussian discord of response. In Sec. III we discuss the properties of classical and quantum Gaussian transmitters, comparing squeezed-thermal, thermal-squeezed and coherent-thermal states; in subsection III A we derive upper and lower bounds on the maximum probability of error, and in subsection III B we identify the PSK coding that maximizes the quantum Chernoff bound and minimizes the maximum probability of error for the classes of quantum Gaussian transmitters considered.
In Sec. IV we compare the performance of Gaussian quantum states of light with classical states. We show that strongly discordant squeezed thermal states possess a higher reading efficiency than the corresponding classical states of light, that is non-discordant Gaussian thermal transmitters with the same total number of photons. This realizes an instance of noise-enhanced quantum advantage over the corresponding noisy classical resources.
In Sec. V we compare different families of discordant Gaussian states, the squeezed thermal and the thermal squeezed states at equal squeezing. While for both classes of states the entanglement obviously decreases with increasing thermal noise, we show that for squeezed thermal states the discord is an increasing function of the number of thermal photons, while the opposite holds for thermal squeezed states. Moreover, for squeezed thermal transmitters, the quantum Chernoff bound is independent of thermal noise. As a consequence, this type of transmitter plays a privileged role in the considered class of quantum Gaussian resources because the associated quantum efficiency is either enhanced or unaffected by increasing the thermal noise. Thus squeezed thermal transmitters realize an instance of noise-enhanced or noise-independent quantum resources at fixed squeezing.
The main results are summarized and some outlook perspectives for future work are discussed in Sec. VI. Detailed calculations and auxiliary reasonings are reported in four Appendixes.

II. PROBABILITY OF ERROR, BOUNDS, AND DISCORD OF RESPONSE
In the PSK protocol the two local channels acting on the input probe state AB are unitary, and denoted as U A , so that the probability of error reads Since the trace norm is invariant under local unitary transformations, one has equivalently A is still a local unitary transformation acting on the transmitted subsystem A. The absolute upper bound for the error probability is thus 1/2, corresponding to a situation in which there is no way to distinguish the two output states and therefore the memory device becomes completely invisible to the transmitter.
In general, computing the trace distance proves to be extremely challenging [8], even more so for Gaussian states of infinite-dimensional continuous-variable systems [9]. Therefore one has to look for analytically computable a priori upper and lower bounds. A natural upper bound on the probability of error, Eq. (1), in distinguishing two states 1 and 2 occurring with the same probability is provided by the quantum Chernoff bound QCB: If the states 1 and 2 are not arbitrary, but they are related by a unitary transformation (either global or local), as in the case of the PSK quantum reading protocol, for which 1 = AB and 2 = AB , then the quantum Chernoff bound QCB is achieved for t = 1/2 in Eq. (4) as it is discussed in Appendix D: and its expression coincides with the quantum Bhattacharyya coefficient. Next, considering the Uhlmann fidelity yields a complete hierarchy of lower and upper bounds [9]: where the lower bound on the probability of error LBP err ≡ (1 − √ 1 − F)/2, and the Uhlmann fidelity F between two quantum states 1 , 2 is defined as For the PSK quantum reading protocol, let us consider the maximum probability of error in distinguishing the output of a binary memory cell encoded using one identity and one arbitrary unitary channel W A chosen in the set of local unitary operations with non-degenerate harmonic spectrum. The latter is the spectrum of the complex roots of the unity and its choice is motivated by observing that it excludes unambiguously the identity from the set of possible operations: indeed, unitary operations with harmonic spectrum are orthogonal (in the Hilbert-Schmidt sense) to the identity. We further assume that the coding is unbiased, that is the two channels are equiprobable.
The worst-case scenario is defined by the probability of error Eq. (3) being the largest possible: Let us now consider a recently introduced measure of quantum correlations, the so-called discord of response [6]: where the index x denotes the possible different types of well behaved, contractive metrics under completely positive and trace-preserving (CPTP) maps. The normalization factor N x depends on the given metrics and is chosen in such a way to assure that D x R varies in the interval [0, 1]. Finally, the set of local unitary operations {W A } includes all and only those local unitaries with harmonic spectrum.
In the following, we will need to consider both the probability of error and different types of upper and lower bounds on it. Therefore we will be concerned with three different discords of response corresponding to three types of contractive distances: trace, Hellinger, and Bures.
The trace distance d T r between any two quantum states 1 and 2 is defined as: The Bures distance, directly related to the fidelity F, is defined as: Finally, the Hellinger distance is defined as: For each discord of response, trace, Hellinger, and Bures, the normalization factor in Eq. (8) is, respectively: N −1 T r = 1/4, N −1 Hell = N −1 Bu = 1/2. If the two states 1 and 2 are bipartite and related by a local unitary operation, that is 1 = AB and 2 = AB = W A AB W † A , by exploiting Eq. (5) it is straightforward to show that the quantum Chernoff bound is directly related to the Hellinger distance: It is then immediate to show that the maximum of QCB over the set of local unitary operations {W A } with completely non-degenerate harmonic spectrum is a simple linear function of the Hellinger discord of response: The discord of response quantifies the response of a quantum state to least-disturbing local unitary perturbations and satisfies all the basic axioms that must be obeyed by a bona fide measure of quantum correlations [6]: it vanishes if and only if AB is a classical-quantum state; it is invariant under local unitary operations; by fixing a well-behaved metrics such as trace, Bures, or Hellinger, it is contractive under CPTP maps on subsystem B, i.e. the subsystem that is not perturbed by the local unitary operation W A ; and reduces to an entanglement monotone for pure states, for one of which it also assumes the maximum possible value (1). By comparing Eqs. (7) and (8) with x = T r, it is immediate to relate the maximum probability of error P (max) err to the trace discord of response D T r R : From Eq. (14) it follows that half of the square root of the trace discord of response yields the difference between the absolute maximum of the error probability (i.e. 1/2) and the maximum probability of error at fixed transmitter state AB . A vanishing trace discord of response implies that there exists at least one memory that cannot be read by classical transmitters. A maximum trace discord of response (D T r R = 1) implies that, irrespective of the coding, the maximally entangled transmitter will read any memory without errors: indeed, any local unitary operation with harmonic spectrum transforms a maximally entangled state into another maximally entangled state orthogonal to it, and therefore yields perfect distinguishability at the output.

III. GAUSSIAN QUANTUM READING
In the following, in order to compare the efficiency of classical and quantum noisy sources of light in reading protocols, we will consider symmetric two-mode Gaussian states of the electromagnetic field. These states are fully described by their covariance matrix σ [10,11]: The range of values of a and c for which the corresponding states are physical (i.e. correspond to positive density matrices) is determined by the Heisenberg uncertainty relation stated in symplectic form: In the following we will focus on two rather general classes of symmetric (undisplaced) Gaussian states, the squeezed thermal states (STS) and the thermal squeezed states (TSS). The former are defined by two-mode squeezing STSs describe a physically common situation in which the thermal noise acts symmetrically on the two modes, so that the number of thermal photons in each mode is the same: N th 1 = N th 2 = N th and thus the total number of thermal photons is 2N th . Here r is the two-mode squeezing parameter and a i are the annihilation operators in each of the two modes (i = 1, 2). The diagonal and off-diagonal covariance matrix elements for these states, respectively a = a sq−th and c = c sq−th , read: where N s = sinh 2 (r) is the number of squeezed photons.
Thermal squeezed states (TSSs) describe the reverse physical situation: an initially two-mode squeezed vacuum is allowed to evolve at later times in a noisy channel and eventually thermalizes with an external environment characterized by a total number of thermal photons 2N th . The covariance matrix elements of TSSs, respectively a = a th−sq and c = c th−sq , are: The same covariance matrix, Eq. (15), describes classical uncorrelated tensor product states as well. Thermal states are obtained letting c = c cl = 0 and a = a cl = 1 + 2N th . These states are classical in the sense that they can be written as convex combinations of coherent states and, moreover, they are the only Gaussian states with vanishing discord [12,13]. In the following, without loss of generality, we will identify party A with mode a 1 and party B with mode a 2 .
A. Probability of error: upper and lower bounds, and Gaussian discords of response For PSK protocols with Gaussian transmitters, Gaussian local (single-mode) unitary operations acting on an infinite-dimensional Hilbert space are implemented by traceless local (single-mode) symplectic transformations acting on the covariance matrix σ of two-mode Gaussian input states In order to assess the performance of quantum and classical Gaussian resources in the PSK quantum reading protocol we need to evaluate the upper and lower bounds, Eqs. (4) and (6), on the maximum probability of error P (max) err , Eqs. (7) and (14), for Gaussian two-mode transmitters (σ) AB . To this end, we introduce first the Gaussian discord of response [7], i.e the discord of response obtained by minimizing over local unitaries restricted only to the subset of local symplectic, traceless, transformations F A : where the index d x stands for trace, Hellinger, or Bures distance with the same normalization factors N −1 x as before, and F T A is the transpose of the symplectic matrix F A and (σ) . The Gaussian discord of response provides an upper bound to the true discord of response of Gaussian states and vanishes on and only on Gaussian classical states (subset of separable states that are in product form). The main properties of the Gaussian discord of response are reported in Appendix A.
In complete analogy with Eq. (14) the maximum probability of error in discriminating two Gaussian transmitters related by a local symplectic transformation can be expressed as a simple function of the Gaussian trace discord of response: Specializing the bounds given by Eq. (6) to the maximum probability of error in distinguishing Gaussian states, one has: where The explicit expressions of the QCB, the Gaussian Hellinger discord of response, the Uhlmann fidelity, and the Gaussian Bures discord of response are given in Appendices B and C.
B. Maximum probability of error: π/2 phase shift The probability of error in distinguishing is given by Eq. (3) with the local symplectic transformations F A replacing W A . Among the local unitary operations F A which can implement the PSK reading protocol, an important subset includes the singlemode phase shifts P φ acting on mode a 1 , parameterized by the angle parameter φ: P φ = exp (−iφa † 1 a 1 ). Under a local phase shift the local mode a 1 is transformed as follows: a 1 = P φ a 1 P † φ = exp (−iφ)a 1 , while the two-mode covariance matrix σ transforms according For the maximum probability of error, Eq. (22), the tightest upper bound is achieved, from Eqs. (23) and (25), in terms of a simple linear function of the Hellinger Gaussian discord of response. The latter, in turn, is obtained by minimizing the Hellinger distance over the entire set of local unitary operations implemented on the covariance matrix by local symplectic, traceless, transformations. For squeezed thermal and thermal squeezed states one finds that this minimum is realized by the π/2 phase shift F π/2 , that is the only possible traceless phase shift. The details of the proof are reported in the Appendix D.
On the other hand the quantity LBP (max) err , Eq. (24), evaluated at π/2, may not be optimal but certainly still provides a lower bound on the maximum probability of error.
Since for a π/2 phase shift the corresponding transformation is implemented by the traceless symplectic matrix , the expectation values of the canonical quadrature operators x and p transform as follows: x → − p and p → x . Therefore, undisplaced classical Gaussian states ( x = p = 0) are left invariant, and the worst-case PSK coding (1, F π/2 ) is completely invisible to classical transmitters since the π/2 shift does not change their covariance matrix. The probability of error P err for every such classical transmitter always achieves the absolute maximum 1/2. Viceversa, the very same coding can always be read by any quantum Gaussian transmitter with nonvanishing Gaussian discord of response. As a consequence, quantum transmitters always outperform undisplaced classical transmitters in deviceindependent, worst-case scenario quantum reading.

IV. COMPARING CLASSICAL AND QUANTUM RESOURCES: NOISE-ENHANCED QUANTUM TRANSMITTERS
We have seen that without displacement classical transmitters are completely blind to reading. Introducing displacement enhances the distinguishability of output states and therefore allows also classical states as useful transmitters.
Let us then consider again a worst-case scenario in which we compare undisplaced but discordant quantum transmitters with displaced classical ones. We will show that even in this case, that is comparing worst-case quantum resources with distinguishability-enhanced classical ones, discordant transmitters can still outperform classical ones, and that the quantum advantage increases with increasing (thermal) noise.
Stated precisely, given the same PSK coding, we want to identify the regimes in which the probability of error associated to an undisplaced quantum transmitter is smaller than the probability of error associated to any displaced classical transmitter. Obviously, only a constrained comparison at given fixed physical quantities is meaningful. We will thus compare undisplaced squeezed thermal states and displaced thermal states (thermal coherent states) at fixed purity and fixed total number of photons.
From Eqs. (4) and (6) this is equivalent to identifying the regimes in which the upper bound on the probability of error QCB for undisplaced squeezed thermal transmitters, denoted by QCB sq−th , is smaller than the lower bound on the probability of error LBP err for thermal coherent states, that will be denoted by LBP coh−th err . With these notations, the requirement for a bona fide quantum advantage reads then as follows: The total number of photons in an undisplaced symmetric two-mode squeezed thermal states is given by the sum of the thermal and squeezed contributions, N sq−th T = 2N th + N s , where N s = sinh 2 (r), while the total number of photons in a symmetric two-mode thermal coherent state is N coh−th T = 2N th + |α| 2 , where N th is the number of thermal photons in each mode and α is the complex amplitude of displacement on, say, mode a 1 . Here we consider, again, the worst-case scenario in favor of the classical resources: indeed, it is straightforward to show that distinguishability is increased by implementing a single-mode displacement rather than a two-mode one with equal amplitudes.
In the following we will assume the same number of thermal photons 2N th for both the classical and the quantum states, and we will fix the total photon number at the same constant value N T : N sq−th This constraint then implies the relation |α| 2 = sinh 2 (r). Moreover, the state purity P = 1/(16 det σ) 1/2 depends only on the number of thermal photons and is therefore the same for both states: In order to evaluate the Uhlmann fidelity and the QCB in Eq. (27) we need to know how the phase shift F π/2 transforms the transmitters that we wish to compare: the undisplaced squeezed thermal state and the thermal coherent state. The dependence of the Uhlmann fidelity and of the QCB on the displacement vector and on the covariance matrix of general Gaussian states are reported in Appendices B and C. The Uhlmann fidelity providing the lower bound on P err for thermal coherent states depends only on the displacement vector, since the covariance matrix of classical states is unaffected by the action of the symplectic transforma- Without loss of generality, the displacement vector of a thermal coherent state can be written as u coh−th = [|α|, 0, 0, 0] T . Under a π/2 phase shift the difference δ between the final and the initial displacement vectors reads as follows: The Uhlmann fidelity of the thermal coherent state is then: where A = (1 + 2N th ). The QCB of symmetric, undisplaced squeezed thermal states depends only on the covariance matrix, Eq. (15), with entries Eqs. (17) and (18), and its explicit expression is reported in Appendix C. In Fig. 1, upper panel, we report the exact values of P sq−vac err and P coh−vac err for pure squeezed vacuum and pure displaced vacuum, N th = 0, at fixed total photon number N T = |α| 2 = sinh 2 (r). At low energies the quantum resource always outperforms the classical resource. The latter converges to the quantum efficiency only in the limit of high energies. For completeness, we also report the QCB sq−vac for the squeezed-vacuum transmitters.
Introducing thermal noise, N th , exact expressions of P err are no longer available. Therefore, in the central and lower panels of Fig. 1 we report the exact lower and upper bounds on P err based on the Uhlmann fidelity and on the QCB. We observe that at intermediate values of the total number of photons N T the quantum upper bound QCB sq−th on P err is strictly lower than both the classical lower bound LBP coh−th err and the exact error probability P coh−vac err of pure classical transmitters. The classical transmitters recover the quantum efficiency for large values of the total photon number.
Moreover, comparing the central and the lower panels in Fig. 1, we observe that as the number of thermal photons N th is increased, the range of values of the total photon number N T for which one has a quantum advantage increases (see the caption for further details).
In Fig. 2 we provide a plot of the contour lines for the differences QCB sq−th − LBP coh−th err (upper panel) and QCB sq−th − P coh−vac err (lower panel) as functions of the total photon number N T and of the purity (or, equivalently of the number of thermal photons N th ). When these differences become negative, Ineq. (27) is satisfied and the quantum resources certainly outperform the classical ones.
From the upper panel of Fig. 2, comparing noisy quantum transmitters with noisy classical ones, one observes that QCB sq−th − LBP coh−th err < 0 in a large region of parameters. Fixing the squeezing, so that the change in the total photon number N T is due only to the change in the number of thermal photons N th , corresponds to a straight line in the plane (in the figure, drawn at r = 1). Remarkably, for these iso-squeezed states the quantum advantage increases with increasing number of thermal photons. This is an instance of noise-enhanced quantum efficiency that will be discussed further in Sec. V.
In the lower panel of Fig. 2 we have also compared the noisy quantum transmitters with the noiseless classical ones. Even in this most unfavorable comparison, we observe that QCB sq−th − LBP coh−vac err < 0 in a significant range of intermediate values for both the total photon number N T and the total number of thermal photons N th . Again, fixing the squeezing, e.g. at r = 1, we notice that the quantum advantage increases with thermal noise.
We remark again that these results are obtained in a worst-case scenario, by comparing the minimum quan- tum efficiency (upper bound on the error probability using quantum transmitters) with the maximum classical efficiency (lower bound on the error probability using classical transmitters). Therefore the actual quantum advantage will be even larger.

V. COMPARING QUANTUM RESOURCES
In the previous section we compared classical and quantum transmitters and for worst-case scenarios we identified the regimes in which quantum resources anyway outperform classical ones. We also observed that the quantum advantage can increase, at fixed squeezing, with increasing thermal noise. We will now compare the behavior of squeezed-thermal and thermal-squeezed states in order to investigate how thermal noise affects the quantum efficiency of different classes of quantum transmitters. Furthermore, keeping the thermal noise fixed, we will investigate how the quantum efficiency of different quantum transmitters improves when multiple reading operations are implemented.
A. Comparing symmetric thermal squeezed and squeezed thermal transmitters: noise-enhanced vs. noise-suppressed bounds on the probability of error Let us compare quantum reading with squeezed thermal and with thermal squeezed transmitters and its performance as a function of thermal noise. This comparison is motivated by the fact that the interplay between quantum and thermal fluctuations is very different for these two classes of quantum states. Squeezed thermal states are obtained by applying a purely quantum operation (squeezing) on TS (states that have already thermalized, e.g. at the output of a noisy channel). Viceversa, TSS are realized by letting pure squeezed states evolve and eventually thermalize in a noisy channel.
When comparing the behavior of the Uhlmann fidelity or quantum Chernoff bound, and the Gaussian discords of response, under variations of the classical noise at a fixed level of quantum fluctuations (squeezing), we expect an opposite behavior for the two classes of states. On intuitive grounds, since fidelity, Chernoff bound, and discord are measures of distinguishability between an input state and the corresponding output after a local disturbance, if we compare STS and TSS we notice from the structure of their covariance matrices, see Eqs. (17)- (20), that as N th increases the correlation part of the squeezed thermal states is increased while it remains constant in a thermal squeezed state.
The origin of these different behaviors are determined quantitatively by looking at the structure of the quantum Chernoff bound and the Uhlmann fidelity for any two Gaussian states, of the form Eq. (15), related by a π/2 The region in which these quantities assume negative values corresponds to quantum transmitters certainly outperforming classical ones. The red dashed curve identifies its boundary. The straight solid yellow lines in both panels corresponds to a fixed degree of squeezing r = 1. Moving along this lines in the direction of increasing number of thermal photons N th one observes that as noise grows there is a growing advantage in using quantum transmitters over classical ones. This behavior provides an instance of noise-enhanced quantum performance. phase shift: In Fig. 3 we report the behavior of the upper bound on the probability of error QCB, Eq. (31), and of the lower bound LBP err (which is a monotonic increasing function of the Uhlmann fidelity F, Eq. (32)) for the squeezed thermal and thermal squeezed states as functions of the number of thermal photons at fixed squeezing. We observe that for thermal squeezed states they both increase with increasing thermal noise, converging asymptotically to the absolute maximum (1/2) of the probability of error. Therefore, the quantum efficiency of thermal squeezed states is suppressed by increasing the thermal noise. On the contrary, for squeezed thermal states QCB remains constant and LBP err decreases. This behavior guarantees that the probability of error, at fixed squeezing, is bound to vary in a restricted interval below 0.1. In the given example, the squeezing amplitude r has been fixed at a relatively low value r 0.88. Increasing the level of squeezing will further reduce the maximum value achievable by the probability of error. In conclusion, the quantum efficiency of squeezed thermal states is enhanced by increasing the thermal noise. A more detailed understanding of these opposite behaviors can be gained by looking at the variation of the measures of distinguishability with respect to the variations of the thermal noise and of the parameters of the covariance matrix.
Consider a generic measure of distinguishability denoted by f ( 1 , 2 ) where f can stand either for the Uhlmann fidelity, F, or the quantum Chernoff bound, QCB. Consider then the total derivative of f with respect to N th , keeping r constant: Specializing to either F or QCB we obtain the explicit expressions of their derivatives, as reported in Appendix E. From these explicit expressions it follows that it is always ∂QCB ∂a c ≥ 0, ∂QCB ∂c a ≤ 0, ∂F ∂a c ≥ 0, and ∂F ∂c a ≤ 0 irrespective of the type of quantum transmitter considered.
Hence, if f represents either the Uhlmann fidelity or the quantum Chernoff bound, the derivative ∂f ∂a c ≥ 0. This behavior agrees with the intuition that the operation of increasing the diagonal entries of the covariance matrix and keeping the off-diagonal entries constant acts like a thermal channel which makes the initial state and the final state after the phase shift less distinguishable. On the other hand, the behavior ∂f ∂c a ≤ 0 for both F and QCB is also intuitively clear, since changes in σ under F π/2 are the greater the larger the off-diagonal entries when keeping the diagonal a constant. Let us now discuss the state-dependent derivatives: for STS and TSS states, the partial derivatives ∂a ∂N th and ∂c ∂N th are non-negative, therefore they cannot oppose the behavior of the state-independent part. For STS they are given by 2 cosh (2r) and 2 sinh (2r) respectively, while for TSS ∂a ∂N th = 2 and ∂c ∂N th = 0. The behavior of F or QCB with increasing N th depends then on the ratio of the positive and negative parts on the left hand side of Eq. (33).
For TSS there is only a positive contribution in Eq. (33) and both, F and QCB, increase with increasing number of thermal photons. As a consequence, both the lower and the upper bounds on the probability of error must increase, as observed in Fig. 3. On the other hand, for STS the negative contribution always prevails when considering the Uhlmann fidelity, while the positive and negative contributions always cancel exactly when considering the quantum Chernoff bound, leading to a constant upper bound on the probability of error, as observed in Fig. 3.
The constant behavior of QCB for STS can be also seen directly from Eq. (31). This equation can be rewritten straightforwardly only in terms of a/c. Indeed, this ratio for STS does not depend on N th .
We have considered so far reading protocols with binary coding given by the identity and the phase shift π/2, and transmitters implemented by symmetric STSs. This is actually a worst-case scenario in two respects. On the one hand, the phase shift π/2 provides the worst possible coding among all traceless local symplectic operations (maximum probability of error, device-independent reading). On the other hand, the symmetric STSs provide the worst possible transmitters among general STSs.
Indeed, in the next subsection we will show that nonsymmetric STSs can provide much larger quantum efficiencies and even effectively suppress the probability of error. However, in conclusion, we have showed that even this twofold worst-case scenario considered so far guarantees the advantage of using quantum transmitters over classical ones.
B. Non-symmetric squeezed thermal states: noise-suppressed upper bound on the probability of error One might speculate that the increment of the Bures discord of response under increasing thermal noise and the corresponding decrement of the lower bound on the probability of error are related to the particular relation with the Bures metrics induced by the Uhlmann fidelity.
However, this is not the case. We will now show that if one considers non-symmetric two-mode STSs then also the Hellinger discord of response increases under increasing local thermal noise and therefore the corresponding upper bound on the probability of error decreases as well.
This is a strong indication that the true probability of error decreases as well with increasing thermal noise and thus that the use of discordant, non-symmetric STSs yields an absolute advantage over the use of the corresponding entangled pure states (two-mode squeezed vacua).
In the above expressions N th1 and N th2 denote the number of thermal photons respectively in mode A and mode B, while the number of squeezed photons is N s = sinh 2 (r). One can prove that QCB achieves its maximum for the π/2 phase shift also when using non-symmetric STSs as quantum transmitters. The proof is given in Appendix D. The exact expression of QCB for non-symmetric STSs related by a π/2 phase shift is: Let us consider the variation dQCB dN th 1 of the quantum Chernoff bound, at constant squeezing r and constant number of thermal photons N th2 in the second mode, whose analytical expression is provided in Appendix E. From this expression it is clear that there is a range of values of N th1 and N th2 , namely N th1 > N th2 , for which dQCB dN th 1 < 0. Therefore, in this regime QCB decreases with increasing N th1 . On the other hand, QCB increases with increasing N th1 if N th1 < N th2 . Henceforth, in the symmetric situation N th1 = N th2 = N th the quantum Chernoff bound is maximum and constant, independent of N th , as discussed in the previous section.
We report in Fig. 4 the behavior of QCB on N th1 for different fixed values of N th2 and fixed squeezing r. In this physical situation the quantum Chernoff bound decreases with increasing local thermal noise and vanishes asymptotically for N th1 → ∞. Therefore the probability of error in the PSK quantum reading protocol can be made essentially negligible by using non-symmetric STSs transmitters with very large local thermal noise.
This very remarkable result may look at first quite counter-intuitive. In fact, the crucial point is that this feature is obtained by the global quantum operation of two-mode squeezing applied to a two-mode thermal state with very strong asymmetry in the local thermal noises affecting the modes. It is therefore not entirely unexpected that the consequences can be dramatic. While entanglement certainly decreases, the operation of squeezing a larger amount of noise can increase quantum state distinguishability by "orthogonalizing" with respect to the thermal states.
In conclusion, Gaussian quantum reading is a remarkable protocol of quantum technology for which the best resources are neither classical, nor quantum entangled, but noise-enhanced quantum discordant, and the increased quantum efficiency is realized by a fine trade-off between local thermal noise and global two-mode squeezing that yields noise-enhanced quantum correlations and state distinguishability.

C. Many copies
Let us now analyze the case in which the total number of photons can vary by considering many copies of the transmitter, that is repeating the reading protocol many times independently. Using n copies of the system the Uhlmann fidelity and the quantum Chernoff bound decrease as powers of n. Therefore, the probability of error can decrease both in the case of squeezed thermal and thermal squeezed states.
The interesting question which arises here is how many copies we need in both cases to achieve a given level of probability of error. The number of copies defines for instance the time needed for reading one bit of information in the given coding. Therefore this process is interesting from the point of view of assessing the reading time and the strength of the sources of squeezed light that one needs. Let us for instance assume that we want to achieve the value of the probability of error 1/8, having for each copy of the squeezed thermal transmitter the thermal noise fixed at N th = 1 and the weak squeezing fixed at N s = 0.1, see Fig. 5. Looking at the upper bound (worstcase scenario), the number of copies which are needed, in order to achieve the desired level of probability of error, is at most n = 7. Taking instead the thermal squeezed transmitter with the same squeezing and the same thermal noise in each copy, we see from Fig. 5 that one needs, considering the lower bound (best-case scenario), at least n = 20 copies. These behaviors illustrate very clearly the advantage of using noise-enhanced quantum correlations. Indeed, comparing Figs. 5 and 3, we see that by keeping a fixed level of squeezing and increasing the thermal noise, the number of copies of squeezed thermal transmitters needed to achieve a given level of precision stays constant, while the number of copies of thermal squeezed transmitters must increase.

VI. CONCLUSIONS AND OUTLOOK
We have investigated quantum reading protocols realized by weak optical sources in the worst-case scenario for quantum transmitters with respect to classical ones. For protocols that involve local unitary operations in the process of reading by continuous-variable Gaussian optical fields, we have showed that the maximum probability of error in reading binary memory cells is directly related to the amount of quantum correlations in a given transmitter, as quantified by the Gaussian trace discord of response. This relation allows to quantify the reading efficiency in terms of quantum correlations, providing a natural operational interpretation to the Gaussian discord of response.
Indeed, the latter is a well-defined measure of quantum state distinguishability under the action of local unitary operations. Therefore, the more discordant is the transmitter, the smaller is the maximum probability of error when using quantum resources. This relation then allows to determine the physical regimes of state purity and signal strength for which one has a net advantage in using quantum resources over classical ones.
Since the trace distance is in general uncomputable for Gaussian states of light associated to infinite-dimensional Hilbert spaces, we have introduced exact upper and lower bounds on the maximum probability of error. We have showed that these bounds are expressed in terms of other type of quantum discords. In particular, the lower bound is expressed in terms of the Gaussian Bures discord of response, while the upper bound, provided by the quantum Chernoff bound maximized over the set of possible local unitary operations, is expressed in term of the Gaussian Hellinger discord of response.
Both bounds decrease with an increasing amount of quantum correlations, providing a precise quantitative estimate of the quantum advantage obtained by us-ing discordant resources over the corresponding classical ones. Moreover, the Bures and Hellinger discords of response are of further independent interest, as they play a central role in other quantum protocols studied recently, ranging from the assessment and use of local quantum uncertainty in optimal phase estimation [14], the efficiency of black-box quantum metrology [15], and the quantum advantage of discordant resources in the protocol of quantum illumination [16].
After comparison between quantum and classical resources, we have compared two fundamental classes of Gaussian quantum transmitters: symmetric squeezed thermal states (STSs) and thermal squeezed states (TSSs). We have showed that environmental noise can actually be of great help. Indeed, we have proven that by using TSSs transmitters both the upper and the lower bounds on the maximum probability of error increase and therefore the quantum reading efficiency decreases with increasing thermal noise.
On the other hand, using symmetric STSs as quantum transmitters, we have shown that the quantum reading efficiency must certainly be non-decreasing. Therefore, there is no experimental advantage in using pure-state squeezed transmitters or low-noise ones over large-noise STSs. Hence, a limited number of experimental resources associated with highly noisy STSs transmitters provides a better quantum efficiency. This behavior is essentially related to the fact that the Gaussian Bures discord of response increases with increasing thermal noise, yielding a decreasing lower bound on the probability of error, while the Hellinger one remains constant, yielding a constant upper bound on the maximum probability of error.
We went a step further, and investigated the use of non-symmetric STSs. For such transmitters, the quantum Chernoff bound decreases with increasing local thermal noise in only one of the two modes. Indeed, the quantum Chernoff bound vanishes asymptotically with very large local thermal noise and therefore the probability of error must also vanish. Therefore, the most efficient quantum transmitters are not only the ideal, maximally entangled, symmetric pure states at infinite two-mode squeezing (Einstein-Podolsky-Rosen states), but rather the noisy, realistic non-symmetric two-mode STSs with a very large imbalance of thermal noise between the two modes.
For these states, the Hellinger and Bures Gaussian discords of response attain their maximum value, and the quantum Chernoff bound and Uhlmann fidelity vanish. As a consequence, all upper and lower bounds on the probability of error vanish, and therefore the probability of error itself vanishes, and perfect reading is achieved.
Very recently, it has been shown that Gaussian quantum metrology, with the precision in parameter estimation measured by the Gaussian interferometric power, is a further protocol of quantum technology allowing for noise-enhanced quantum efficiency, in the sense that noisy discordant resources can outperform entangled pure ones [17] This remarkable phenomenon of noise-assisted quantum correlations and quantum efficiency is eventually due to the fact that quantum state distinguishability is intimately related to the concept of geometric quantum correlations, as measured by the discords of response, and the observation that the former can increase under increasing thermal noise.
In forthcoming studies we will provide a general characterization and quantification of noise-suppressed vs. noise-enhanced quantum correlations for different classes of quantum states [18], and we will investigate the relations between different types of quantum correlations according to states, metrics, and operations [19]. 0 1 −1 0 and S A is a symplectic matrix which diagonalize σ A , i.e. σ A = νS A 1S T A . Here ν are two equal symplectic eigenvalues of σ A . The transformation given by F π/2 is symmetry-preserving and therefore F A = S A F π/2 S −1 A leaves σ A invariant. This shows that if the state is classically correlated there exists at least one local traceless transformation F A that leaves the state invariant.
We now prove the reverse statement that only in the case of classically correlated states there exists such a symplectic traceless transformation that leaves the state invariant. We prove it by contradiction. Assume that the covariance matrix left invariant by a traceless transformation F A has the form σ AB = L 11 L 12 L 21 L 22 . Local symplectic transformation can bring the covariance matrix in the so called normal form in which L 12 = c 0 0 −c . The transformation to the normal form does not reduce the generality of the reasoning. If the state is not changed by the local transformation we have F A L 12 = L 12 . Since L 12 is reversible we obtain that F A = 1 which contradicts the assumption on the spectrum of F A . This shows that condition iii) is satisfied.
Condition iv) is guaranteed by the fact that for pure states the Gaussian discord of response reduces to the Gaussian entanglement of response [20], that is the Gaussian counterpart of the entanglement of response [21]) which is a bona fide measure of entanglement.

Appendix B: Uhlmann fidelity of Gaussian states
The Uhlmann fidelity for two-mode Gaussian states can be computed as follows [22]. Let us define the matrix of the symplectic form The displacement vector is the vector of the averages of the amplitude and phase field quadratures x and p i.e. u = ( x 1 , p 1 , x 2 , p 2 ) T , where T stands for transposition. Denote the difference of the displacement vectors of two Gaussian states 1 and 2 by δ = u 1 − u 2 . We need also the auxiliary formulas defined using the covariance matrices σ 1 and σ 2 of the respective Gaussian states: where g = 8(N th1 + N th2 + 1)(2N th2 + 1) sinh 2 (2r) N 2 th1 − 2(7N th2 + 3)N th1 + (N th2 − 6)N th2 − (N th1 + N th2 + 1) 2 cosh(4r) − 3

. (E6)
Since g ≥ 0 the derivative, Eq. (E5), is positive only if N th1 < N th2 , negative only if N th1 > N th2 , and vanishes identically for symmetric STSs, namely for N th1 = N th2 , yielding a noise-independent QCB. From the above results it follows that in the range N th1 > N th2 the QCB is a monotonically decreasing function of the number of thermal photons in the first mode (increasing local thermal noise) and vanishes asymptotically, together with the probability of error, as N th1 → ∞.