Quantum reading of quantum information

We extend the notion of quantum reading to the case where the information to be retrieved, which is encoded into a set of quantum channels, is of quantum nature. We use two-qubit unitaries describing the system-environment interaction, with the initial environment state determining the system’s input-output channel and hence the encoded information. The performance of the most relevant two-qubit unitaries is determined with two different approaches: (i) one-shot quantum capacity of the channel arising between environment and system’s output; (ii) estimation of parameters characterizing the initial quantum state of the environment. The obtained results are mostly in (qualitative) agreement, with some distinguishing features that include the CNOT unitary.


Introduction
Quantum reading is the process of retrieving classical information from a memory by using a quantum probe [1] (for a survey on the subject see [2]).It is customary to see such information encoded into a finite set of quantum channels, each one labeled by the value that a random discrete variable can take.As a such, the process involves quantum channel discrimination [3][4][5][6].Quantum reading has been applied in various contexts, ranging from physical imaging [7], to radar [8], to biology [9], to cryptography [10], and showed advantages over classical reading.
A prototypical model for a memory cell in quantum reading is the environment parametrized quantum channel [11][12][13][14][15].This is a unitary acting on two systems: the main system and the environment.Depending on the state of the latter, a channel will be realized on the former by tracing out the environment at the end.Thus, the initial state of the environment can be considered as the encoded information that has to be retrieved (while the main system plays the role of the probe).This model was employed with environment input states forming an orthonormal basis for the associated Hilbert space.In such a way it realized an incoherent model of memory cell (investigated also for private reading [16]), as well as a coherent model of memory cell (allowing entanglement generation between encoder and reader [17]).
Here, we go beyond the assumption made on the initial quantum state of the environment and consider a generic one to be determined.Thus, as figure of merit it is natural to resort to the one-shot quantum capacity [18] of the channel connecting the environment with the output of the main system (probe).In fact quantum information has to be extracted with a finite number of usage of the channel realized on the main system (probe) and a nonzero probability of error.Since explicit computation of one-shot capacity is challenging, we shall compute a lower bound on it following [19].
Additionally, we shall consider the information to be retrieved as residing on the parameters characterizing the encoded quantum state, thus tracing back the problem to continuous multi-parameter estimation.In this case as figure of merit we shall consider a Bayesian version of the quantum Cramer-Rao bound derived from the classical bound [20].
We shall confine our attention to two-qubit unitaries.In particular those that are entangling.These can be represented by points lying in a tetrahedron in R 3 (see e.g.[21]).Specifically, we characterize the quantum reading of quantum information on the edges of this tetrahedron starting from its vertices.The found results show a large qualitative agreement between the approach based on the one-shot quantum capacity bound and the approach based on Bayesian quantum Cramer-Rao bound, with some distinguishing features that include the CNOT unitary.

The model
Consider a unitary U AE→BF with systems A B and E F. By referring to Fig. 1, the environment parametrization of quantum channel [11][12][13][14][15] consists in characterizing a channel N A→B θ between A and B in terms of the E state θ .
For the purpose of quantum reading systems A and B, i.e. input and output of N A→B θ , are both held by the reader, which wants to retrieve the state θ of the system E. It is customary to consider x ∈ X (discrete and finite alphabet) encoded by E as orthogonal states |x with probability p x .Then, the objective for the reader, given N A→B x , is to find x among all possible values in X .This task can give rise to classical communication [22] as well as to quantum communication [17] between E and B. R Figure 1: General model for quantum reading based on environment parametrization of quantum channels.|ψ φ is the purification of the input φ to the A system.
In fact, for classical information transmission, an incoherent picture is used leading to the final state of the whole system as Instead, for quantum information transmission, a coherent picture is used leading to the final state of the whole system as where V x ≡ U|x is an isometry from A to BF. 1Here we change the paradigm and according to Fig. 1 we consider a generic state θ encoded into E. Then the objective is to recover such a state.This of course implies the necessity of transmitting quantum information from E to B or alternatively the necessity of estimating the environment state.
We will focus on two-qubit unitaries U AE→BF , with states φ φ φ , θ θ θ on A and E systems respectively.The former is the input of the probe system and the latter the encoded state in the memory.The bold symbols emphasize that the qubit states are characterized by vectors of R 3 .Two-qubit entangling unitaries that can be written as [24] where |Λ k are the so called magic basis states: and the eigenvalues λ k are In the canonical basis ) All unitaries of this kind form, according to Eq.( 6), a tetrahedron in the parameter space α x , α y , α z (see Fig. 2).The vertices of such a tetrahedron represent Identity, SWAP, CNOT and DCNOT unitaries.

One-shot quantum capacity
By referring to Fig. 1, for a fixed probe state φ φ φ , we can consider the channel and evaluate its capability in transmitting quantum information.In particular, we are interested to the one-shot quantum capacity for the channel N E→B φ φ φ (θ θ θ ) ⊗n .This will tell us how much quantum information we can extract from system E n by accessing system B n in one shot and with a finite probability of error.

Lower bound
Notice preliminarily that the complementary channel of Let us also set where Φ AE is maximally entangled state across the systems A and E. Analogously, it will be with Φ A n E n a maximally entangled state across the systems A n and E n .
For a given ε > 0, the one-shot quantum capacity of where Here D is a decoding map from It is where with Furthermore with ρ B n F n given by (11).We also have the following inequality for α Here H q is the conditional Renyi entropy, which is defined as with q .

Evaluation on the vertices
We now work out the calculation of the r.h.s. of (24) for the four cases of unitaries in the vertices of tetrahedron (see Fig. 2).
ii) α x = π 2 , α y = α z = 0 (U = CNOT ).In this case and assume that λ 1 and λ 2 are eigenvalues of σ F with We have ≤ − min For the last inequality we used the fact that 1 Again the lower bound (24) trivially becomes zero. iii) 1 4 e iφ 2 sin(2φ 1 ) 0 0 1 4 e −iφ 2 sin(2φ 1 ) This can also be written as ρ BF = I 2 ⊗ |φ φ | where |φ = cos φ 1 |0 + e iφ 2 sin φ 1 |1 .Then, we have As a result, the lower bound (24) reads (41) This can also be written as ρ BF = I 2 ⊗ |φ φ | where |φ = cos φ 1 |0 + ie iφ 2 sin φ 1 |1 .Then we can repeat the reasoning of case (iii) and arrive to the lower bound (24) as Summarizing, for I and CNOT no quantum information can be retrieved according to the used figure of merit.Instead for SWAP and DCNOT maximal quantum information retrieval (1 qubit) can be approached by increasing the number of shots n even with a fixed value of error ε.

Evaluation on the edges
We now extend, with the help of numerics, the analysis of the figure of merit along the edges of the tetrahedron.
In the cases analyzed in SubSec.3.2 the relevant part is given by max σ F H 2 (B|F) ρ .The continuity of H 2 (B|F) ρ (see Appendix B), hence its uniform continuity, allows us to reliable sampling discrete points for plotting its behavior along edges of the tetrahedron of Fig. 2. The results are summarized in Fig. 3.It is worth noticing that, while along the edge IC the quantity max σ F H 2 (B|F) ρ remains always zero, along the edges IS, ID it becomes nonzero only after a certain (threshold) value of |α α α| (|α α α| ≈ π/3.4).Instead, along the edges CD, CS it increases with a smooth derivative since the CNOT.Lastly, for the edge DS it is constant and equal to 1.
The results of this SubSection indicate that there should be a nonzero volume of unitaries around identity for which quantum information retrieval would not be possible.

Quantum state estimation
By referring to Fig. 1, we now consider the goal of estimating parameters θ θ θ = (r, Here σ x , σ y , σ z are the Pauli matrices.The estimation should be done by means of the channel output N θ θ θ (φ φ φ ).As a figure of merit we will derive a Bayesian version of the quantum Cramer-Rao bound following the classical bound [20].
Then the quantity of interest for us becomes where the subscript emphasizes its dependence from the input state φ φ φ of the A system.Its maximum overall states φ φ φ will be denoted by F. Due to the convexity of the Fisher information, the optimization can be restricted to pure states of the form where φ 1 ∈ [0, π] and φ 2 ∈ [0, 2π].From Ref. [27], we know that the basis-independent expression of quantum Fisher information for a single-qubit mixed state ρ(θ θ θ ) reads where ∂ a ρ = ∂ ρ/∂ θ a .For a pure qubit state Eq.( 49) reduces to

Evaluation on the vertices
We now work out the calculation of F for the four cases of unitaries in the vertices of the tetrahedron (see Fig. 2).
i) α x = α y = α z = 0 (U = I).In this case Eq.( 7) reduces to the identity and hence F = 0 for all input |φ φ φ .As a consequence F = 0. ii) Then The maximum of is achieved for In this case it is F = 1.76108.
iii) α x = α y = α z = π 2 (U = SWAP).In this case Then diverges, meaning that for any state φ φ φ we will have a good estimation on average. Then Notice that these three terms are non negative, hence maximizing (47) is equivalent to maximize them.However, their derivative with respect to cos 2 φ 1 never nullify.Hence, the maximum is at the extreme points.Actually it is easy tho see that it occurs for cos 2 φ 1 = 1.
As a consequence we will have which diverges, likewise the case of SWAP.
The fact that F diverges for DCNOT and SWAP is in line with the results in Sec. 3 where the bound for one-shot capacity in these cases turns out to be (close to) 1.In other words for DCNOT and SWAP the environment state can be estimated with perfect accuracy or analogously it can be transmitted with maximum reliability to the B system.
Also for identity we have concordance between the two approaches.In fact the environment state can be estimated with total inaccuracy or analogously it can be transmitted to the B system in a total unreliable way.
The situation is slightly different for CNOT, since according to the one-shot capacity approach it behaves like identity, while state environment estimation can be done with a finite average error.

Evaluation on the edges
We now extend, with the help of numerics, the analysis of the figure of merit along the edges of the tetrahedron.
The continuity of the average quantum Fisher information (see Appendix C), hence its uniform continuity, allows us to reliable sampling discrete points for plotting its behavior along edges of the tetrahedron of Fig. 2. The results are summarized in Fig. 4. From Fig. 4 it is left out the edge (SD).On that edge, we have (7) as Therefore and F φ φ φ does not have extrema with respect to α z inside the interval 0, π 2 , and we know that on the two ends of the interval it has the same maximum value.Thus, we can conclude, that on the edge SD we have the same value of F, i.e. infinity.
The results of this SubSection indicate that only for identity quantum information retrieval would not be possible.For all other unitaries it would be possible, although in some cases with a finite average error.

Conclusion
In conclusion, we have addressed the problem of reading quantum information by a quantum probe, thus going beyond the standard paradigm that confines quantum reading to the retrieval of classical information.As a model of quantum memory we used environment parametrized quantum channels arising from two-qubit unitaries.Since these unitaries lie in a tetrahedron in R 3 , we characterized those on the edges to have general insights.To this end, we used a lower bound to the oneshot quantum capacity of the channel connecting the environment with the output of the main system (probe) as well as a Bayesian version of the quantum Cramer-Rao bound for the initial environment state.We remark that while the first also showed the behavior vs the number n of shots, the second just refers to one-shot, Notwithstanding, the results of the first are more restrictive.In fact, according to the first figure of merit, there should be a nonzero volume of unitaries around identity for which quantum information retrieval would not be possible.Instead, the second shows that only for identity quantum information retrieval would not be possible.For all other unitaries it would be possible, although in some cases with a finite average error.
This difference of behavior between the two figures of merit (to most striking of which occurs for CNOT) should be ascribed to the non tightness of the used lower bound for to the one-shot quantum capacity.
In future, besides investigating also the unitaries inside the tetrahedron, we plan to characterize spatial arrays of the presented model to figure out the performance of realistic quantum memories.A fact that can be useful for applications in quantum cryptography as well as in quantum computation.

C Continuity of average quantum Fisher information
Let us denote by α α α the parameters vector characterizing the unitary.where C i < +∞ are positive constants.In going from (81) to (82) we used the continuity of the Fisher information [30].In going from (83) to (84) we used the the property that discarding a system cannot increase the norm [31].

Figure 2 :
Figure 2: Tetrahedron representing the parameters space of two-qubit unitaries, where I, C, S, D stand for Identity, CNOT, SWAP and DCNOT unitaries respectively.

Figure 3 :
Figure 3: Quantity H 2 (B|F) ρ evaluated along the various edges of the tetrahedron as function of the parameter |α α α|.Dashed curve corresponds to the edge IC; dotted curve to the edges IS and ID; dashed-dotted curve to the edges CD and CS; solid curve to the edge DS.

FFigure 4 :
Figure 4: Quantity F evaluated along the various edges of the tetrahedron as function of the parameter |α α α|.Curves from bottom to top refer respectively to the edges (IC), (ID), (IS), (CS and CD).The edge (DS) is not represented because along it, the value of F is infinity.