Abstract
A fundamental problem in quantum information is to explore what kind of quantum correlations is responsible for successful completion of a quantum information procedure. Here we study the roles of entanglement, discord and dissonance needed for optimal quantum state discrimination when the latter is assisted with an auxiliary system. In such process, we present a more general joint unitary transformation than the existing results. The quantum entanglement between a principal qubit and an ancilla is found to be completely unnecessary, as it can be set to zero in the arbitrary case by adjusting the parameters in the general unitary without affecting the success probability. This result also shows that it is quantum dissonance that plays as a key role in assisted optimal state discrimination and not quantum entanglement. A necessary criterion for the necessity of quantum dissonance based on the linear entropy is also presented. PACS numbers: 03.65.Ta, 03.67.Mn, 42.50.Dv.
Introduction
An important distinctive feature of quantum mechanics is that quantum coherent superposition can lead to quantum correlations in composite quantum systems like quantum entanglement^{1}, Bell nonlocality^{2} and quantum discord^{3,4}. Quantum entanglement has been extensively studied from various perspectives and it has served as a useful resource for demonstrating the superiority of quantum information processing. For instance, entangled quantum states are regarded as key resources for some quantum information tasks, such as teleportation, superdense coding and quantum cryptography^{5}.
In contrast to quantum entanglement, quantum discord measures the amount of nonclassical correlations between two subsystems of a bipartite quantum system. A recent report regarding the deterministic quantum computation with one qubit (DQC1)^{6,7} demonstrates that a quantum algorithm to determine the trace of a unitary matrix can surpass the performance of the corresponding classical algorithm in terms of computational speedup even in the absence of quantum entanglement between the the control qubit and a completely mixed state. However, the quantum discord is never zero. This result is somewhat surprising and it has engendered much interest in quantum discord in recent years. In particular, it has led to further studies on the relation of quantum discord with other measures of correlations. Moreover, it has been shown that it is possible to formulate an operational interpretation in the context of a quantum state merging protocol^{8,9} where it can be regarded as the amount of entanglement generated in an activation protocol^{10} or in a measurement process^{11}. Also, a unified view of quantum correlations based on the relative entropy^{12} introduces a new measure called quantum dissonance which can be regarded as the nonclassical correlations in which quantum entanglement has been totally excluded. For a separable state (with zero entanglement), its quantum dissonance is exactly equal to its discord.
It is always interesting to uncover nontrivial roles of nonclassical correlations in quantum information processing. The quantum algorithm in DQC1 has been widely regarded as the first example for which quantum discord, rather than quantum entanglement, plays a key role in the computational process. Moreover, a careful consideration of the natural bipartite split between the control qubit and the input state reveals that the quantum discord is nothing but the quantum dissonance of the system. This simple observation naturally leads to an interesting question: Can quantum dissonance serve as a similar key resource in some quantum information tasks? The affirmative answer was shown in an interesting piece of work by Roa, Retamal and AlidVaccarezza^{13} where the roles of entanglement, discord and dissonance needed for performing unambiguous quantum state discrimination assisted by an auxiliary qubit^{14,15} was studied. This protocol for assisted optimal state discrimination (AOSD) in general requires both quantum entanglement and discord. However, for the case in which there exist equal a priori probabilities, the entanglement of the state of systemancilla qubits is absent even though its discord is nonzero and hence the unambiguous state discrimination protocol is implemented successfully only with quantum dissonance. This protocol therefore provides an example for which dissonance and not entanglement, plays as a key role in a quantum information processing task.
In this work, we show more generally that quantum entanglement is not even necessary for AOSD. Moreover, we look at the roles of correlations in the AOSD under the most general settings by considering a generic AOSD protocol. We also show that only dissonance in general is required for AOSD and quantum entanglement is never needed.
Results
The general AOSD protocol
Suppose Alice and Bob share an entangled twoqubit state (see Fig. 1), where p_{±} ∈ [0, 1] and p_{+} + p_{−} = 1, ψ_{±}〉 are two nonorthogonal states of the qubit of Alice (system qubit S) and {0〉_{c}, 1〉_{c}} are the orthonormal bases for the one of Bob (qubit C). The reduced state of system qubit ρ = p_{+}ψ_{+}〉 〈ψ_{+} + p_{−}ψ_{−}〉〈ψ_{−} is a realization of the model in^{13} in which a qubit is prepared in the two nonorthogonal states ψ_{±}〉 with a priori probabilities p_{±}. To discriminate the two states ψ_{+}〉 or ψ_{−}〉 unambiguously, the system is coupled to an auxiliary qubit A, prepared in a known initial pure state k〉_{a}. Under a joint unitary transformation between the system and the ancilla, one obtains
where Φ〉 = cos β0〉 + sin βe^{iδ}1〉, {0〉, 1〉} and {0〉_{a}, 1〉_{a}} are the bases for the system and the ancilla, respectively. The probability amplitudes α_{+} and α_{−} satisfy , where α = 〈ψ_{+}ψ_{−}〉 = αe^{iθ} is the priori overlap between the two nonorthogonal states. The unitary transformation can be constructed by performing an operation on the original one in Ref. 13, where and . It has the form as
where are the components of ψ_{±}〉 orthogonal to and , with being two arbitrary states orthogonal to the right hands of Eq. (1) and and . Obviously, only the terms with have effect on the initial state ψ_{±}〉k〉_{a}.
The state of the systemancilla qubits is given by
which depends on β and δ and it is generally not equivalent to the corresponding one in^{13} under local unitary transformations unless Φ〉 = +〉. The state discrimination is successful if the ancilla collapses to 0〉_{a}. This occurs with success probability given by
where is the unit matrix for the system qubit. Without loss of generality, let us assume that p_{+} ≤ p_{−} and denote . The analysis of the optimal success probability can be divided into two cases: (i) , P_{suc} is attained for ; (ii) , P_{suc} is attained for α_{+} = 1 (or equivalently α_{−} = α). One has
Before proceeding further to explore the roles of correlations in the AOSD, we make the following remarks.
Remark 1
State discrimination of a subsystem in a reduced mixed state has practical interest in conclusive quantum teleportation where the resource is not prepared in a maximally entangled state (see Refs. 16,17,18). In the conclusive teleportation protocol, the sender Alice possesses an arbitrary onequbit state φ〉_{Alice} = a0〉 + b1〉 and she shares a nonmaximally entangled state Ψ_{+}(θ)〉 = cos θ00〉 + sin θ11〉 with the receiver Bob. Under the protocol, one has
where Ψ_{±}(θ)〉 = cos θ00〉 ± sin θ11〉, Φ_{±}(θ)〉 = sin θ01〉 ± cos θ10〉 and σ_{x}, σ_{y}, σ_{z} are Pauli matrices. The concurrences^{19} of the states Ψ_{±}(θ)〉 and Φ_{±}(θ)〉 are all equal to . The states Ψ_{±}(θ)〉 are orthogonal to the states Φ_{±}(θ)〉, but {Ψ_{+}(θ)〉, Ψ_{−}(θ)〉} (or {Φ_{+}(θ)〉, Φ_{−}(θ)〉}) are not mutually orthogonal. To teleport the unknown state φ〉_{Alice} from Alice to Bob with perfect fidelity (equals to 1), state discrimination^{16,17,18} is generally required. It should also be noted that only the maximally entangled states (with θ = π/4) can realize the perfect teleportation with unit success probability.
Remark 2
Through quantum teleportation, we see that our model recover the scheme in^{13}, in which the principal qubit is randomly prepared in one of the two pure states ψ_{+}〉 or ψ_{−}〉. Let us conisder replacing the entangled resource Ψ_{+}(θ)〉 by maximally entangled states randomly prepared with a probabilities as {, , , }. Although they are all maximally entangled states and each of them is a resource for perfect teleportation, perfectly faithful teleportation cannot be realized in this case. It can be shown that the fidelity of teleportation is the one corresponding to the average state^{16}
Consequently, the amount of entanglement contributing to teleportation is not just the average value of the entanglement which is , but the entanglement of the average state as . Therefore the amount of entanglement available depends crucially on the knowledge of the entangled state. The amount of quantum entanglement that is needed for the AOSD scheme considered here, as well as the one in Ref. 13, refers to the entanglement of the average state, and not to the average value of the entanglement as .
We are now ready to investigate the roles of correlations in the AOSD. To this end, let us first calculate the concurrence of ρ_{SA}:
with . When β = π/4 and δ = 0, Eq. (5) reverts to the result in^{13}.
Let us impose the constraint for any α, α_{+} and p_{+}. It is then easy to see that
Based on Eq. (6), state (2) is a separable state as
where η_{1}〉 and η_{2}〉_{a} are two unnormalized states as
where .
Note that the state (2) has rank two and it is really the reduced state of the following tripartite pure state
Its discord can be derived analytically as D(ρ_{SA}) = S(ρ_{A}) − S(ρ_{SA}) + E(ρ_{SC}) = S(ρ_{A}) − S(ρ_{C}) + E (ρ_{SC}) using the KoashiWinter identity^{20}, where S(ρ) is the von Neumann entropy, E(ρ_{SC}) is the entanglement of formation^{19} between the principal system and the qubit C. The explicit expression for the discord is
where , τ_{A} is the tangle between A and SC, τ_{C} is the tangle between C and SA and is the concurrence between S and C in the state ρ_{SC}. One can obtains
with τ_{S} the tangle between S and AC, and τ(Ψ〉) the threetangle^{21}. The tangle between S and AC is given by
and the threetangle is
Dissonance for cases (i) and (ii)
For case (i), upon the substitution , p_{−} = 1 − p_{+} and Eqs. (6)(11)(12)(13) into Eq. (10), one has the analytical expression for the dissonance, which depends only on α and p_{+}. In Fig. 2, we plot the curves of the dissonance versus α for p_{+} = 1/2, 1/4, 1/8, respectively (see the curves with D(ρ_{SA}) > 0). For case (ii), because α_{+} = 1, one has β = π/2 and the state ρ_{SA} is
with ρ_{a} = p_{+}1〉_{a}〈1 + p_{−}μ〉_{a}〈μ, . The state (14) is clearly a directproduct state hence its dissonance is zero. In Fig. 2, for case (ii), we also plot the curves of dissonance versus α for the same p_{+}'s (see the curves with D(ρ_{SA}) = 0). Fig. 2 shows that dissonance is a key ingredient for AOSD other than entanglement for case (i) and that the classical state can accomplish the task of AOSD for case (ii).
Geometric picture
It can be observed that the optimal success probability P_{suc,max} in Eq. (4) can be analyzed in two different regions: and . Here based on the positiveoperatorvalued measure (POVM) strategy^{15}, we provide a geometric picture of P_{suc,max}. Since the success probability, the concurrence and the discord of state ρ_{SA} under the constraints in Eq. (6) are all independent of the phase θ of α, one can simply set θ = 0 and regard the states ψ_{±}〉 as two unit vectors in with the angle γ = arccos α between them. The square roots of the a priori probabilities, i.e., and , behave like wave amplitudes and the effects of the coherence can be seen from the states ζ〉 and Ψ〉. In Fig. 3, we plot two vectors and with to denote and , respectively. The two POVM elements that identify the states can be implemented as , with r_{±} ≥ 0. The vectors and correspond to the unnormalized states with the coefficients . The third POVM element giving the inconclusive result is . The elements Π_{±,0} are required to be positive  this is a constraint on the POVM strategy. Finally, the probability of successful discrimination is P_{POVM} = (r_{+}p_{+} + r_{−}p_{−})(1 − α^{2}), which is
When , the optimal P_{POVM} is attained at . The vectors and , where E is the intersection point of AE and BE (see Fig. 3). The maximum value of P_{POVM} is the square of AB, nanmely , which recovers Eq. (4a). When , the point E coincides with B for p_{+} < p_{−} (or A for p_{+} > p_{−}), for the optimal P_{POVM} one has r_{−} = 1 (or r_{+} = 1) and P_{POVM} = p_{−}(1 − α^{2}). For and p_{+} < p_{−}, E lies outside of the angle and is opposite to . Consequently, we do not get a physically realizable value of r_{+}. The optimal P_{POVM} strategy then occurs at r_{−} = 1 and r_{+} = 0 (i.e., E coincides with B), one has , which is Eq. (4b).
Discussion
In summary, based on a sufficiently general AOSD protocol, we found that the entanglement between the principal qubit and the ancilla is completely unnecessary. Moreover, this quantum entanglement can be arbitrarily zero by adjusting the parameters in the joint unitary transformation without affecting the success probability. Theoretically, this fact clearly indicates that dissonance plays a key role in assisted optimal state discrimination other than entanglement. Experimentally, the absence of entanglement can be more easily observed because there is no restriction on the a priori probabilities. In Fig. 4, we present a realization of the unitary transformation in Eq. (1) for the initial states ψ_{+}〉 = 0〉, and k〉_{a} = 0〉_{a} by using singlequbit gates and twoqubit controlledunitary gates. These gates can be demonstrated experimentally in many systems^{23,24} in recent years. The success probability of state discrimination is determined by steps (i) to (iii), which transform the systemancilla state into
It is not affected by the controlledU_{Φ} in step (iv), which can adjust the correlations in state (2).
Let us also reiterate a necessary criterion for the requirement of dissonance in AOSD based on linear entropy. Under the general protocol, Alice and Bob share the entangled state ζ〉, encoded in the basis of the polarization of the qubit, Bob can acquire knowledge of the linear entropy of Alice's qubit. If , he can be sure that Alice needs dissonance for her AOSD (see Fig. 5). Finally, we would like to mention that local distinguishability of multipartite orthogonal quantum states was studied in Ref. 22 where again the local discrimination of entangled states does not require any entanglement.
References
Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009).
Bell, J. S. On the Einstein Podolsky Rosen paradox. Physics (Long Island City, N.Y.) 1, 195–200 (1964).
Ollivier, H. & Zurek, W. H. Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001).
Henderson, L. & Vedral, V. Classical, quantum and total correlations. J. Phys. A 34, 6899–6905 (2001).
Ekert, A. K. Quantum cryptography based on Bell's theorem. Phys. Rev. Lett. 67, 661–663 (1991).
Lanyon, B. P., Barbieri, M., Almeida, M. P. & White, A. G. Experimental quantum computing without entanglement. Phys. Rev. Lett. 101, 200501 (2008).
Datta, A., Shaji, A. & Caves, C. M. Quantum discord and the power of one qubit. Phys. Rev. Lett. 100, 050502 (2008).
Cavalcanti, D. et al. Operational interpretations of quantum discord. Phys. Rev. A 83, 032324 (2011).
Madhok, V. & Datta, A. Interpreting quantum discord through quantum state merging. Phys. Rev. A 83, 032323 (2011).
Piani, M. et al. All nonclassical correlations can be activated into distillable entanglement. Phys. Rev. Lett. 106, 220403 (2011).
Streltsov, A., Kampermann, H. & Bruß, D. Linking quantum discord to entanglement in a measurement. Phys. Rev. Lett. 106, 160401 (2011).
Modi, K., Paterek, T., Son, W., Vedral, V. & Williamson, M. Unified view of quantum and classical correlations. Phys. Rev. Lett. 104, 080501 (2010).
Roa, L., Retamal, J. C. & AlidVaccarezza, M. Dissonance is required for assisted optimal state discrimination. Phys. Rev. Lett. 107, 080401 (2011).
Neumann, J. V. Mathematical Foundations of Quantum Mechanics Vol. 2, (Princeton University Press 1996).
Jafarizadeh, M. A., Rezaei, M., Karimi, N. & Amiri, A. R. Optimal unambiguous discrimination of quantum states. Phys. Rev. A 77, 042314 (2008).
Horodecki, M., Horodecki, P. & Horodecki, R. General teleportation channel, singlet fraction and quasidistillation. Phys. Rev. A 60, 1888–1898 (1999).
Roa, L., Delgado, A. & FuentesGuridi, I. Optimal conclusive teleportation of quantum states. Phys. Rev. A 68, 022310 (2003).
Kim, H., Cheong, Y. W. & Lee, H. W. Generalized measurement and conclusive teleportation with nonmaximal entanglement. Phys. Rev. A 70, 012309 (2004).
Wootters, W. K. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998).
Koashi, M. & Winter, A. Monogamy of quantum entanglement and other correlations. Phys. Rev. A 69, 022309 (2004).
Coffman, V., Kundu, J. & Wootters, W. K. Distributed entanglement. Phys. Rev. A 61, 052306 (2000).
Walgate, J., Short, A. J., Hardy, L. & Vedral, V. Local distinguishability of multipartite orthogonal quantum states. Phys. Rev. Lett. 85, 4972–4975 (2000).
Chow, J. M. et al. Universal quantum gate set approaching faulttolerant thresholds with superconducting qubits. Phys. Rev. Lett. 109, 060501 (2012).
Brunner, R. et al. Twoqubit gate of combined singlespin rotation and interdot spin exchange in a double quantum dot. Phys. Rev. Lett. 107, 146801 (2011).
Acknowledgements
F.L.Z. is supported by NSF of China (Grant No. 11105097). J.L.C. is supported by National Basic Research Program (973 Program) of China under Grant No. 2012CB921900, NSF of China (Grant Nos. 10975075 and 11175089) and partly supported by National Research Foundation and Ministry of Education of Singapore.
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F.L.Z and J.L.C. initiated the idea. F.L.Z. derived the formulas and prepared the figures. J.L.C., F.L.Z., L.C.K. and V.V. wrote the main manuscript text. All authors contributed to the derivation and the manuscript.
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Zhang, FL., Chen, JL., Kwek, L. et al. Requirement of Dissonance in Assisted Optimal State Discrimination. Sci Rep 3, 2134 (2013). https://doi.org/10.1038/srep02134
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DOI: https://doi.org/10.1038/srep02134
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