Asymmetric quantum hypothesis testing with Gaussian states
We consider the asymmetric formulation of quantum hypothesis testing, where two quantum hypotheses have different associated costs. In this problem, the aim is to minimize the probability of false negatives and the optimal performance is provided by the quantum Hoeffding bound. After a brief review of these notions, we show how this bound can be simplified for pure states. We then provide a general recipe for its computation in the case of multimode Gaussian states, also showing its connection with other easier-to-compute lower bounds. In particular, we provide analytical formulae and numerical results for important classes of one- and two-mode Gaussian states.
pacs:03.67.-a, 89.70.Cf, 03.67.Hk, 03.65.Ta, 02.10.Ud
Quantum hypothesis testing (QHT) is a fundamental topic in quantum information theory Wilde (); Nielsen (), playing a non-trivial role in protocols of quantum communication and quantum cryptography QCprot (); QCprot2 (). The typical formulation of QHT is given in terms of quantum state discrimination QHT (); QHT2 (); RMP (); Helstrom (), where a certain number of generally non-orthogonal quantum states (the quantum hypotheses) have to be discriminated by means of a quantum measurement. In particular, the simplest scenario regards the statistical discrimination between two non-orthogonal quantum states, corresponding to the ‘null’ and the ‘alternative’ quantum hypotheses, occurring with some a priori probabilities. In symmetric testing, these hypotheses have the same cost QHT (); QHT2 (); Helstrom () and the goal is to minimize the mean error probability of confusing them by suitably optimizing the quantum measurement.
For such a basic problem, we know closed analytical formulae identifying both the minimum error probability, given by the Helstrom bound Helstrom (), and the optimal quantum detection, expressed in terms of the Helstrom matrix Helstrom (). Furthermore, we can also use an easier-to-compute bound which becomes tight in asymptotic conditions. This is the recently-introduced quantum Chernoff bound QCB (), for which we know simple formulae in the case of multi-mode Gaussian states QCB2 (), (i.e., those states with Gaussian Wigner function RMP ()).
In this paper, we consider asymmetric QHT, where two quantum hypotheses have different associated costs QHT (); QHT2 (); Helstrom (). In this approach, we aim to minimize the probability that the alternative hypothesis is confused for the null hypothesis, an error which is known as ‘false negative’. This minimization has to be done by suitably constraining the probability of another possible error, known as a ‘false positive’, where the null hypothesis is confused for the alternative hypothesis. This is clearly the best approach for instance in medical-type testing, where the null hypothesis typically represents absence of a disease, while the alternative corresponds to the presence of a disease.
Asymmetric QHT is typically formulated as a multi-copy discrimination problem, where a large number of copies of the two possible states are prepared and subjected to a collective quantum measurement. From this point of view, the aim is to maximize the error-exponent describing the exponential decay of the false negatives, while placing a reasonable constraint on the false positives. For this calculation, we can rely on two mathematical tools. The first is the quantum relative entropy RMP () between the two states, while the other is the recently-introduced quantum Hoeffding bound (QHB) QHB (), which performs the optimization of the error-exponent while providing a better control on the false positives.
In this work, we start by giving some basic notions on asymmetric QHT and briefly reviewing the QHB, also showing how its computation simply reduces to the quantum fidelity Fidelity () in the presence of pure states. Then, we provide a general recipe for computing this bound in the case of multimode Gaussian states, for which it can be expressed in terms of their first- and second-order statistical moments. In the general multimode case, we derive a relation between the QHB and other easier-to-compute bounds, which are based on well-known mathematical inequalities. Finally, we derive analytical formulas and numerical results for the most important classes of one-mode and two-mode Gaussian states.
By developing the theory of asymmetric QHT for Gaussian states, our work could be useful in tasks and protocols involving Gaussian quantum information RMP (), including technological applications of quantum channel discrimination (e.g., quantum illumination Qill (); Qill2 () or quantum reading Qread (); Qread2 (); Qread3 (); Qread4 ()) where we are interested in increasing our ability to accept one specific quantum hypothesis.
Ii Brief review of asymmetric testing
ii.1 Basic formulation
In binary QHT we consider a quantum system which is prepared in some unknown quantum state , which can be or . For instance we can imagine one party, say Alice, who prepares such a system. This system is then passed to Bob, who does not know which choice Alice has made. Thus, Bob must decide between the following two hypotheses
In order to discriminate between these two hypotheses, i.e., distinguish between the two states, Bob applies a quantum measurement, generally described by a positive operator valued measure (POVM). Without loss of generality, Bob can always reduce his measurement to be a dichotomic POVM with Helstrom (). The outcome , with POVM operator , is associated to the null hypotheses , while the other outcome , with POVM operator , is associated with the alternative hypothesis .
Since the two quantum states and are generally non-orthogonal, there is a non-zero error probability to confuse the two hypotheses. We can identify two different types of error: Type-I and type-II errors, with associated conditional error probabilities. By definition, the type-I error, also known as a ‘false-positive’, is where Bob accepts the alternative hypothesis when the null hypothesis holds. We have a corresponding error probability expressed by
Then, the type-II error or ‘false-negative’ is where Bob accepts the null hypothesis when the true hypothesis is the alternative. This error occurs with conditional probability
Note that we can introduce other probabilities, but they are fully determined by and . For instance, we may also consider the ‘specificity’ or ‘true-negativity’ of the test which is the success probability of identifying the null hypothesis, i.e., which is simply given by . Similarly, we may also consider the ‘sensitivity’ or ‘true-positivity’ of the test which is the success probability of identifying the alternative hypothesis, i.e., .
The costs associated with the two types of error can be very different especially in the medical and histological settings. For instance, in a medical test, is typically associated with no illness, while with the presence of the disease. It is therefore clear that we would like to have tests where the false-negative probability (or rate) is the lowest possible, so that ill patients are not diagnosed as healthy. For this reason, in a medical setting, hypothesis testing is almost always asymmetric, meaning that we aim to minimize one of the two conditional error probabilities.
ii.2 Multi-copy formulation
These systems are passed to Bob who performs a collective measurement on them. As before, this general POVM can be chosen to be dichotomic with .
The error probabilities now depend on the number of copies . In particular, the probability of false positives is given by
and the probability of false negatives is
In the limit of a large number of copies , these probabilities go to zero exponentially, i.e., we have
where the coefficients
are called the ‘error-exponents’ or ‘rate limits’ QHB ().
Bob’s aim is to maximize the error exponent , so that the error probability of false negatives has the fastest exponential decay to zero. This must be done while controlling the rate of false positives. Here a well known result is the ‘quantum Stein lemma’ QHB () which connects with the quantum relative entropy between the single-copy states and . For a large number of copies , there is a dichotomic POVM such that the error probability of the false positives is bounded
and the error probability of false negatives goes to zero with error-exponent
More powerfully, we may use the notion of the QHB QHB (). For , there is a dichotomic POVM such that the error-exponent of false positives is lower-bounded by a positive parameter
and the error-exponent of false negatives satisfies
where is the QHB defined by
is the ‘s-overlap’ between the single-copy states and . Note that the quantum Hoeffding bound enforces a stronger constraint on false-positives, since these are bounded at the level of the error-exponent and not at the level of the error probability as happens for the quantum relative entropy bound.
Iii Asymmetric testing with pure states
Asymmetric testing becomes very simple when one of the states (or both) is pure. In this case, we can in fact relate the QHB to the quantum fidelity between the two states.
Let us start by considering the case where only one of the states is pure, e.g., . We can write JPAgae ()
This bound can be further simplified by explicitly performing the maximization with regard to the parameter . After a simple calculation we find
which depends on the comparison between the parameter and the fidelity of the two states.
Iv Asymmetric testing with Gaussian states
iv.1 Basics of bosonic systems and Gaussian states
These operators satisfy the vectorial commutation relations SPEDcomm ()
where is the symplectic form, defined as
Correspondingly, a real matrix is called ‘symplectic’ when it preserves by congruence, i.e., .
By definition, we say that a bosonic state is ‘Gaussian’ when its phase-space Wigner representation is Gaussian RMP (). In such a case, we can completely describe the state by means of its first- and second-order statistical moments. These are the mean value or displacement vector , and the covariance matrix (CM) with generic element
where denotes the anticommutator. The CM is a real symmetric matrix, which must satisfy the uncertainty principle RMP ()
An important tool in the manipulation of Gaussian states is Williamson’s theorem RMP (): For any CM , there is a symplectic matrix such that
The matrix is the ‘Williamson form’ of , and the set is the ‘symplectic spectrum’ of . According to the uncertainty principle, each symplectic eigenvalue must satisfy the condition , with for all if and only if the Gaussian state is pure.
iv.2 Computation of the quantum Hoeffding bound
Our goal is to find a general recipe for the calculation of the QHB for Gaussian states. We start from the general formula in Eq. (15) involving the logarithm of the -overlap defined in Eq. (16). Given two -mode Gaussian states, and , we can write an explicit Gaussian formula for the -overlap in terms of their statistical moments (, ) and (, ). This is given by QCB2 (); JPAgae ()
where is the difference between the mean values, while and depends on the CMs and . In particular, introducing the two real functions
we can write the formulas
where and are the symplectic spectra of the two states, with and being the symplectic matrices which diagonalize the two CMs according to Williamson’s theorem, i.e.,
iv.3 Other computable bounds
Note that computing the -overlap and its logarithmic form could be difficult due to the presence of the symplectic matrices, and , in the term in Eq. (33). A possible solution is to compute an upper bound, known as the ‘Minkowski bound’, which is based on the Minkowski determinant inequality Bathia () and depends only on the two symplectic spectra QCB2 (). Specifically, we have , where
Another easy-to-compute upper bound is the ‘Young bound’ , which is based on Young’s inequality Young () and satisfies
where QCB2 ()
Taking the negative logarithm of Eq. (39), we can write the following inequality for the QHB
In the specific case where one of the two Gaussian states is pure, we can compute their fidelity and apply the upper bound given in Eqs. (18) and (19), which becomes tight when both states are pure [see Eq. (20)]. In particular, for two multimode Gaussian states and , we can easily write their fidelity in terms of the statistical moments JPAgae ()
where . As a result, we can use Eq. (19) with
V Discrimination of one-mode Gaussian states
In this section, we examine the case of one-mode Gaussian states. This means we fix in the previous formulas of Sec. IV, with matrices becoming , vectors becoming 2-dimensional, and symplectic spectra reducing to a single eigenvalue. For instance, the -overlap can be more simply computed using the expressions
v.1 Asymmetric testing of coherent amplitudes
The expression of the QHB is greatly simplified in the case of one-mode coherent states and . Since both states are pure, the QHB is equal to the fidelity bound in Eq. (19), i.e., . Therefore, it is sufficient to compute the fidelity between the two coherent states, which is given by
so that , and we can write
Assuming that we impose a good control on the rate of false positives (so that ), then the error-exponent for the false negatives is simply given by . More explicitly, this corresponds to an asymptotic error rate
Note that, if we have poor control on the rate of false positives, i.e., , then the QHB is infinite. This means that the probability of false negatives goes to zero super-exponentially, i.e., more quickly than any decreasing exponential function.
v.2 Asymmetric testing of thermal noise
In this section we derive the QHB for one-mode thermal states and , with variances equal to and , respectively (in our notation, , where is the mean number of thermal photons). These Gaussian states have zero mean () and CMs in the Williamson form and (so that ). Thus, we can write
This is the -overlap to be used in the QHB of Eq. (15).
Given two arbitrary and , the maximization in Eq. (15) can be done numerically. The results are shown in Fig. 1 for thermal states with variances up to vacuum units (equivalent to mean thermal photon). From the figure we can see an asymmetry with respect to the bisector which is a consequence of the asymmetric nature of the hypothesis test. The bottom-right part of the figure is related to the minimum probability of confusing a nearly-vacuum state () with a thermal state having one average photon (). By contrast, the top-left part of the figure is related to the probability of confusing a thermal state having one average photon () with a nearly-vacuum state (). These probabilities are clearly different.
We are able to derive a simple analytical result when we compare a thermal state with the vacuum state. Let us start by considering the vacuum state to be the null hypothesis () while the thermal state is the alternative hypothesis (). In this specific case, we find
and we get
Since is a constant, the maximization of over corresponds to minimizing the function , whose minimum occurs at . As a result, we have
Since , we can write the QHB in terms of the mean number of thermal photons, i.e.,
This is the optimal error exponent for the asymptotic probability of false negatives, i.e., of confusing a thermal state with the vacuum state.
Let us now consider the thermal state to be the null hypothesis () while the vacuum state is the alternative hypothesis (). In this case, we derive
which leads to the following expression for the QHB
This is related to the minimum probability of confusing the vacuum state with a thermal state. Note that this is very different from Eq. (56).
Vi Discrimination of two-mode Gaussian states
In this section we consider two important classes of two-mode Gaussian states. The first is the class of Einstein-Podolsky-Rosen (EPR) states, also known as two-mode squeezed vacuum states. The second (broader) class is that of two-mode squeezed thermal (ST) states, for which the computation of the QHB is numerical.
vi.1 Asymmetric testing of EPR correlations
The expression of the QHB in the case of EPR states is easy to derive. Since EPR states are pure, the QHB is given by of Eq. (19). As a result, we need only to compute the fidelity between the two states.
vi.2 Squeezed thermal states
In this section we consider symmetric ST states , which are Gaussian states with zero mean and CM
Note that, for , we have no correlations, and the ST state is a tensor-product of thermal states, i.e., . For the correlations are maximal, and the ST state becomes an EPR state, i.e., . Finally, for , we have maximal separable correlations. In other words, is the separable ST state with the strongest correlations (e.g., highest discord).
The symplectic decomposition of a symmetric ST state is known. From the CM of Eq. (63), one can check that the symplectic spectrum is degenerate and given by the single eigenvalue
The symplectic matrix which diagonalizes in Williamson form is given by
As a result, the s-overlap between two symmetric ST states, and , can be computed using the simplified formulas
Let us start with simple cases involving the asymmetric testing of correlations with specific ST states. First we consider the asymmetric discrimination between the uncorrelated thermal state as null hypothesis and the correlated (but separable) ST state as alternative hypothesis. A false negative corresponds to concluding that there are no correlations where they are actually present Other (). It is straightforward to derive their degenerate symplectic eigenvalues which are simply and . Then, we have , while can be easily computed from Eqs. (65) and (66). By substituting these into Eqs. (67) and (68), we can compute the s-overlap and therefore the QHB via Eq. (15). The results are plotted in Fig. 2, for values of thermal variance up to (i.e., from zero to mean photon) and small values of the parameter , bounding the rate of false-positives. As expected, the QHB improves for decreasing and increasing .
Now let us consider the asymmetric discrimination between and the EPR state , i.e., the most correlated and entangled ST state Other (). Thanks to the simple symplectic decomposition of the EPR state (), we can further simplify the previous Eqs. (67)-(68) and write
with being given by Eq. (59). As before, we compute the QHB which is plotted in Fig. 3, for and . As expected the QHB improves for decreasing and increasing . Note a discontinuity identifying two regions, one where the QHB is finite, and the other where it is infinite (white region in the figure).
In fact, by expanding the term in Eq. (15) for , that we find
For values of and such that , we find that the term diverges at the border, making the QHB infinite. For a given , this happens when
Finally, we consider the most general scenario in the asymmetric testing of correlations with ST states. In fact, we consider two generic ST states, and , with the same thermal noise but differing amounts of correlation. For this computation, we use Eqs. (64)-(66) with or , to be replaced in Eqs. (67)-(68), therefore deriving the s-overlap and the QHB. At small thermal variance () and for the numerical value , we plot the QHB as a function of the correlation parameters and . As we can see from Fig. 4, the QHB is not symmetric with respect to the bisector (where it is zero) and increases away from this line.
In this work we have considered the problem of asymmetric quantum hypothesis testing by adopting the recently-developed tool of the quantum Hoeffding bound (QHB). After a brief review of these notions, we have shown how the QHB can be simplified in some cases (pure states) and estimated using other easier-to-compute bounds based on simple algebraic inequalities.
In particular, we have applied the theory of asymmetric testing to multimode Gaussian states, providing a general recipe for the computation of the QHB in the Gaussian setting. Using this recipe, we have found analytic formulas and shown numerical results for important classes of one-mode and two-mode Gaussian states. In particular, we have studied the behavior of the QHB in the low energy regime, i.e., considering Gaussian states with a small average number of photons.
Our results could be exploited in protocols of quantum information with continuous variables. In particular, they could be useful for reformulating Gaussian schemes of quantum state discrimination and quantum channel discrimination in such a way as to give more importance to one of the quantum hypotheses. This asymmetric approach could be the most suitable in the development of quantum technology for medical applications.
This work has been supported by EPSRC (EP/J00796X/1). G.S. has been supported by an EPSRC DTA grant. The authors thank C. Ottaviani and S. Pirandola for enlightening discussions.
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