The ultimate precision of quantum illumination

The ultimate precision of quantum illumination

Giacomo De Palma QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark    Johannes Borregaard QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
Abstract

Quantum illumination is a technique for detecting the presence of a target in a noisy environment by means of a quantum probe. We prove that the two-mode squeezed vacuum state is the optimal probe for quantum illumination in the scenario of asymmetric discrimination, where the goal is minimizing the probability of a false positive with a given probability of a false negative. Quantum illumination with two-mode squeezed vacuum states offers a 6 dB advantage in the error probability exponent compared to illumination with coherent states. Whether more advanced quantum illumination strategies may offer further improvements had been a longstanding open question. Our fundamental result proves that nothing can be gained by considering more exotic quantum states, such as e.g. multi-mode entangled states. Our proof is based on a new fundamental entropic inequality for the noisy quantum Gaussian attenuators. We also prove that without access to a quantum memory, the optimal probes for quantum illumination are the coherent states.

Quantum entanglement can increase both resolution and sensitivity in a number of metrological tasks Giovannetti et al. (2004, 2006, 2011); Degen et al. (2017). By using highly entangled states such as NOON states Demkowicz-Dobrzański et al. (2015), the sensitivity can in principle reach the Heisenberg limit yielding an improvement scaling with the square root of the number of probes compared to the standard quantum limit Mitchell et al. (2004); Afek et al. (2010); Facon et al. (2016). However, in most of these metrological tasks the improvement quickly vanishes in the presence of noise. One remarkable exception is quantum illumination.

Quantum illumination was introduced in the setting of detecting the presence of a low-reflective target in a noisy environment using single photons as probes Lloyd (2008). By entangling the signal photons with a quantum memory kept at the measurement station, it was shown that the initially strong quantum correlations between the signal photons and the memory resulted in a suppressed error probability compared to non-entangled signals. However, entangled single photon states still perform worse than coherent states Tan et al. (2008); Shapiro and Lloyd (2009). Nonetheless, quantum illumination with a two-mode squeezed vacuum state exhibits a 6 dB decrease in the error probability exponent compared to the strategy with coherent states Tan et al. (2008); Guha (2009); Guha and Erkmen (2009); Zhuang et al. (2017a). The advantage of the two-mode squeezed vacuum state was experimentally verified in the optical regime Lopaeva et al. (2013, 2014); Zhang et al. (2015) and methods for extending this to the microwave regime have been proposed Barzanjeh et al. (2015). However, despite numerous theoretical studies Wilde et al. (2017); Zhang et al. (2014); Ragy et al. (2014); Weedbrook et al. (2016); Cooney et al. (2016); Zhuang et al. (2017b), the ultimate limit of quantum illumination beyond the two-mode squeezed vacuum still remains a fundamental open question.

In this Letter, we answer this question by proving that the two-mode squeezed vacuum state constitutes the optimal probe in the scenario of asymmetric discrimination, where the goal is minimizing the probability of a false positive with a given probability of a false negative (Theorem 1). We also prove that coherent states constitute the optimal probe when a quantum memory is not available. These two fundamental results completely solve the problem of the optimal probe for quantum illumination for asymmetric discrimination. The striking and surprising consequence is that nothing can be gained with arbitrary many-mode entangled states compared to the two-mode squeezed vacuum state.

The proof of Theorem 1 is based on a new fundamental entropic inequality (Theorem 2), stating that the tensor powers of the two-mode squeezed vacuum states minimize the quantum conditional entropy of the output of the noisy quantum Gaussian attenuators among all the input states with a bounded average energy.

Quantum illumination.—

In the standard setup of quantum illumination (see Figure 1), Alice wants to determine the presence of a low-reflective target in a noisy environment by sending an electromagnetic signal towards it. If the target is present, Alice receives the reflected fraction of the signal plus the environmental noise. If the target is not present, Alice receives just the noise. Alice has then to distinguish between two quantum states: the received state with the reflected signal plus the noise and the received state with just the noise 111We mention that also another scenario has been considered in the literature Cooney et al. (2016). In this scenario, the receiver is located on the other side of the target with respect to Alice, and he receives just the noise if the target is present, or the noise plus the transmitted signal if the target is not present..

Let us suppose that Alice wants to distinguish between the quantum state associated to the null hypothesis and the quantum state associated to the alternative hypothesis. Let the POVM used for the discrimination have elements , where . A type-I error or “false positive” occurs when Alice concludes that the state is when it actually is , and has probability . Conversely, a type-II error or “false negative” occurs when Alice concludes that the state is when it actually is , and has probability . From the quantum Stein’s lemma Hiai and Petz (1991); Ogawa and Nagaoka (2005), the quantum relative entropy

 S(ρ1∥ρ0)=Tr[ρ1(lnρ1−lnρ0)] (1)

governs the exponent of the asymptotic optimal decay of the probability of a false positive with a given probability of a false negative: for any ,

 limn→∞1nlninf0≤Mn≤In{p(n)1|0:p(n)0|1<ϵ}=−S(ρ1∥ρ0). (2)

In the quantum illumination scenario, the natural choice of null hypothesis is “target not present”, and the consequent choice of alternative hypothesis is “target present”. This setup has been analyzed in Wilde et al. (2017) in the particular case of Gaussian probes. An -mode Gaussian quantum system is the quantum system of harmonic oscillators, or modes of the electromagnetic radiation. Let Alice hold the quantum state , where is the -mode Gaussian quantum system of the signal sent towards the target, and is the memory quantum system that Alice keeps. We stress that our analysis is not restricted to Gaussian quantum states, and can be any state of the joint quantum system . If the target is present, Alice gets back an attenuated signal with some thermal noise, and her final state is , where is an -mode noisy quantum Gaussian attenuator with thermal noise. Let and , , be the ladder operators of the system and the noise, respectively. Alice then receives the system with ladder operators

 bi=√ηai+√1−ηei,i=1,…,n, (3)

where is the fraction of the signal that is reflected (see Figure 1). Alice’s goal is performing a measurement on her final state to detect whether the target is there. If the target is not present, Alice gets back only the noise, and her final state is , where is an -mode thermal quantum Gaussian state, is the -mode vacuum and is the marginal on of . This enforces that Alice cannot get any information on the presence of the target if she does not send any signal, i.e., if .

If Alice could send an arbitrarily bright signal, she could always detect whether the target is present. Therefore, we impose that the marginal state of the signal must have average energy at most , i.e.

 TrA[HAρA]≤E, (4)

where is the standard Hamiltonian of that counts the total number of photons. We do not consider the energy of the memory system.

The optimal probes.—

We consider the asymmetric discrimination in quantum illumination with and . The optimal asymptotic error exponent is governed by the following quantum relative entropy:

 S(ρBM∥ωB⊗ρM)=−S(B|M)ρBM−TrB[ρBlnωB], (5)

where

 S(X|Y)=S(XY)−S(Y) (6)

is the quantum conditional entropy Holevo (2013); Wilde (2017) (see Kuznetsova (2010); Wilde and Qi (2016) for the definition in the case where ). If is not there, the relevant quantum relative entropy is

 S(ρB∥ωB)=−S(Φ(ρA))−TrB[ρBlnωB]. (7)

Our main result states that Alice’s best probes for the asymmetric discrimination problem are the tensor powers of the two-mode squeezed vacuum states if she has a quantum memory, and the coherent states otherwise. The coherent states of a one-mode Gaussian quantum system are the states of the form

 |α⟩=e−|α|22∞∑k=0αk√k!|k⟩,α∈C, (8)

where is the Fock basis. The coherent states of an -mode Gaussian quantum system are the tensor products of the coherent states of each mode. The two-mode squeezed vacuum states of a two-mode Gaussian quantum system are the states of the form

 |ψ(z)⟩=√1−z2∞∑k=0zk|k⟩⊗|k⟩,0≤z<1. (9)

They can be obtained applying a two-mode squeezing to the two-mode vacuum state.

Theorem 1.

The -th tensor power of the two-mode squeezed vacuum state (9) that saturates the energy constraint (4) maximizes the quantum relative entropy (5) among all the joint states of the signal-memory system satisfying (4), and hence constitutes Alice’s optimal strategy in the asymmetric discrimination scenario. Moreover, the -mode coherent states that saturate the energy constraint (4) maximize the quantum relative entropy (7) among all the states of the signal satisfying (4), and hence constitute Alice’s optimal strategy in the asymmetric discrimination scenario when a quantum memory is not available.

In other words, let be a generic quantum system, and let be a joint quantum state of such that . Then,

 S(ρBM∥ωB⊗ρM)≤S(σBA′∥ωB⊗σA′), (10)

where is an -mode Gaussian quantum system, is the -th tensor power of the two-mode squeezed vacuum state such that its marginal on has average energy , and . Moreover, let be a quantum state of such that . Then,

 S(Φ(ρA)∥ωB)≤S(Φ(|α⟩⟨α|)∥ωB), (11)

where , is any -mode coherent state of with average energy (i.e., with ).

First, the term in (5) and (7) is a function of alone. Indeed, since is a thermal Gaussian state, with . Hence, , and the claim follows since is a linear increasing function of Holevo (2013).

Since coherent states minimize the output entropy of the noisy quantum attenuators Giovannetti et al. (2015); Mari et al. (2014), Alice’s best choice in (7) is choosing to be a -mode coherent state with the maximum allowed average energy. The best choice is not unique, since the average energy can be distributed in an arbitrary way among the modes.

The optimality of the two-mode squeezed vacuum states in (5) follows from the following new fundamental entropic inequality for the noisy quantum Gaussian attenuators.

Theorem 2.

Let and be -mode Gaussian quantum systems and a generic quantum system, and let be a noisy quantum Gaussian attenuator with thermal noise. Let , and let be a joint quantum state of such that and , where is the Hamiltonian of , and and are the marginals of on and , respectively. Then,

 S(B|M)ρBM≥S(B|A′)σBA′, (12)

where is an -mode Gaussian quantum system, is the -th tensor power of the two-mode squeezed vacuum state such that its marginal on has average energy , and .

Proof.

We can assume pure. Indeed, let be a purification of . Since the marginals on of and coincide, they have the same average energy. The claim then follows from the strong subadditivity Holevo (2013); Wilde (2017):

 S(B|M)ρBM≥S(B|MR)ρBMR, (13)

where .

Let then be a pure state. We have

 S(B|M)ρBM =S(ρBM)−S(ρM)=S(~Φ(ρA))−S(ρA), S(B|A′)σBA′ =S(~Φ(σA))−S(σA), (14)

where is the complementary channel of Holevo (2013); Wilde (2017), and since and has finite average energy Holevo (2013). Let be the quantum additive noise channel such that , and let be a purification of . Since is a quantum Gaussian channel, we have from De Palma and Trevisan (2018), Lemma 10

 S(~Φ(ρA))−S(ρA)≥S(~Φ(ρ′A))−S(ρ′A), (15)

hence

 S(B|M)ρBM≥S(B|M′)ρ′BM′, (16)

where . We can then assume .

Now is the thermal Gaussian state with the same average energy as , and Holevo (2013)

 S(ρA∥σA) =S(σA)−S(ρA), S(~Φ(ρA)∥~Φ(σA)) =S(~Φ(σA))−S(~Φ(ρA)), (17)

where the second line follows from Lemma 1. The claim then follows from the data processing inequality for the relative entropy Holevo (2013); Wilde (2017):

 S(B|M)ρBM−S(B|A′)σBA′ =S(ρA∥σA)−S(~Φ(ρA)∥~Φ(σA))≥0. (18)

Lemma 1.

Let be the complementary channel of the -mode noisy quantum Gaussian attenuator with thermal noise. Then, for any -mode quantum state

 S(~Φ(ρA)∥~Φ(σA))=S(~Φ(σA))−S(~Φ(ρA)), (19)

where is the -mode thermal Gaussian state with the same average energy as .

Proof.

We will prove that

 ~Φ†(ln~Φ(σA))=xHA+yIA, (20)

where , is the dual channel of Holevo (2013) defined by

 Tr[X~Φ(Y)]=Tr[~Φ†(X)Y] (21)

for any bounded operator and any trace class operator , and is the Hamiltonian on , where the are the ladder operators associated to the modes of . The claim (19) then follows since

 S(~Φ(ρA)∥~Φ(σA)) =S(~Φ(σA))−S(~Φ(ρA)) =+TrA[(σA−ρA)~Φ†(ln~Φ(σA))], (22)

and and have the same average energy.

Since both and are the -th tensor power of the corresponding one-mode versions, the left-hand side of (20) is the sum of one term for each of the modes. Then, it is sufficient to prove the claim (20) for , when . Since is a Gaussian channel, is still a Gaussian state, is a quadratic polynomial in the quadratures, and is a quadratic polynomial in and Holevo (2013). The claim (20) follows if we prove that

 eiHAt~Φ†(ln~Φ(σA))e−iHAt=~Φ†(ln~Φ(σA))∀t∈R. (23)

Indeed, and are the only invariant quadratic polynomials since .

Let us now prove (23). The complementary channel of the one-mode noisy attenuator is Holevo (2013)

 ~Φ(ρA)=TrA[(UAE⊗IE′)(ρA⊗γEE′)(U†AE⊗IE′)], (24)

where is a two-mode squeezed vacuum state of the one-mode Gaussian quantum systems and , and implements a beam-splitter on . We have for any

 e−i(HE−HE′)tγEE′ei(HE−HE′)t =γEE′, e−i(HA+HE)tUAEei(HA+HE)t =UAE, (25)

where and are the Hamiltonians of and , respectively. Then, for any quantum state , any bounded operator and any

 ~Φ(e−iHAtρAeiHAt)=e−i(HE−HE′)t~Φ(ρA)ei(HE−HE′)t, ~Φ†(ei(HE−HE′)tXe−i(HE−HE′)t)=eiHAt~Φ†(X)e−iHAt, (26)

and the claim (20) follows. ∎

Discussion.—

We have proven that the tensor powers of the two-mode squeezed vacuum states are Alice’s best probe in the asymmetric discrimination problem of quantum illumination. This fundamental striking result implies that nor correlations nor entanglement among the modes of the signal that Alice sends can decrease the error probability. Alice’s best strategy is then the simplest one, and nothing can be gained by using any more exotic probe, such as states with multi-mode entanglement.

Putting together our results with Cooney et al. (2016); Berta et al. (2016), the optimality of the two-mode squeezed vacuum states may be extended to the adaptive scenario with feedback.

The main open question left is whether the optimality of the two-mode squeezed vacuum states is limited to the asymmetric discrimination. We conjecture that this is not the case, and the two-mode squeezed vacuum states are optimal also in the symmetric discrimination problem. Proving this conjecture will be the subject of future work.

Acknowledgements.—

We thank Matthias Christandl for very helpful discussions and Mark Wilde for a careful reading of the paper and useful comments.

GdP acknowledges financial support from the European Research Council (ERC Grant Agreements no 337603 and 321029), the Danish Council for Independent Research (Sapere Aude), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and the Marie Skłodowska-Curie Action GENIUS (grant no. 792557). JB acknowledges financial support from the European Research Council (ERC Grant Agreements no 337603), the Danish Council for Independent Research (Sapere Aude), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059), and Qubiz - Quantum Innovation Center.

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