Behavior of Quantum Correlations under Local Noise
aa r X i v : . [ qu a n t - ph ] J un Behavior of Quantum Correlations under Local Noise
Alexander Streltsov, ∗ Hermann Kampermann, and Dagmar Bruß
Heinrich-Heine-Universität Düsseldorf, Institut für Theoretische Physik III, D-40225 Düsseldorf, Germany
We characterize the behavior of quantum correlations under the influence of local noisy channels.Intuition suggests that such noise should be detrimental for quantumness. When considering qubitsystems, we show for which channel this is indeed the case: the amount of quantum correlationscan only decrease under the action of unital channels. However, non-unital channels (e.g. suchas dissipation) can create quantum correlations for some initially classical state. Furthermore, forhigher-dimensional systems even unital channels may increase the amount of quantum correlations.Thus, counterintuitively, local decoherence can generate quantum correlations.
Composite quantum states often reveal puzzling fea-tures of nature. Recently, much interest [1] has been de-voted to the study of quantum correlations that may arisewithout entanglement: here, the quantumness of a com-posite system manifests itself even in a separable state.The fact that such quantum correlations are present [2]in an algorithm for mixed state quantum computing [3]has stimulated intensive investigations into measures forquantum correlations [4–13] and their properties and in-terpretations [14–26]. Some studies of the dynamics ofquantum correlations have been presented in [27, 28].An appeal of mixed state quantum computation liesin the possibility to be run in a noisy environment: pureentangled states are typically fragile, and the resource ofentanglement is easily destroyed by noise. For an opensystem the transition from entangled to separable statesis only a matter of time - as the volume of the set ofseparable states is non-zero [29], typically it takes a finitetime for entanglement to disappear under noise such asdissipation or decoherence [30].Mixed state quantum computation as suggested in [3]already uses separable states, so it is natural to assumethat it can be run also in a noisy environment. However,in order to verify or falsify this conjecture, one has tostudy the behavior of quantum correlations under noisychannels (described by trace-preserving completely pos-itive maps). Here we only consider local noisy channels- as correlated channels may also preserve entanglement(with or even without some degradation, depending onthe amount of correlation), see e.g. [31]. The goal ofthis Letter is to answer questions such as: Which typesof noisy channels decrease the amount of quantum cor-relations? Are there any noisy channels that might even increase the amount of quantum correlations? How doesdissipation influence quantum correlations, and how arethey affected by decoherence? - We point out that ouranswers to these questions also apply to the situationwhere one actively performs local operations on a com-posite quantum system, e.g. with the aim of creating orpreserving quantum correlations.In general, a bipartite quantum state is called fully classically correlated , if it can be written in the form [6, 7] ρ cc = X i,j p ij | i A i h i A | ⊗ | j B i h j B | , (1)where {| i A i} and {| j B i} are sets of orthogonal states ofparty A and B, respectively, with nonnegative probabili-ties p ij that add up to one. If a state cannot be writtenas in Eq. (1), it is called quantum correlated. These def-initions can be extended to any number of parties [13].As a simple example consider the classically correlatedstate of two qubits ρ cc = 12 | A i h A | ⊗ | B i h B | + 12 | A i h A | ⊗ | B i h B | . (2)Using a local channel on qubit A only (namely a localmeasurement and subsequent replacement) it is possi-ble to create from the classically correlated state (2) thequantum correlated state ρ = 12 | A i h A |⊗| B i h B | + 12 | + A i h + A |⊗| B i h B | (3)with | + A i = √ ( | i + | i ). The quantum channel thatachieves this transformation can be formally written asthe completely positive trace-preserving map ρ = Λ A ( ρ cc ) = E ρE † + E ρE † (4)with local Kraus operators E = | A i h A | and E = | + A i h A | acting only on qubit A . The state in Eq. (3)is not of the form (1), i.e. it is quantum correlated.As will become clear below in this Letter, one rea-son why the local quantum channel in Eq. (4) is ableto create quantum correlations lies in its action on themaximally mixed state A . Observe that Λ A (cid:0) A (cid:1) = | A i h A | + | + A i h + A | 6 = A . This property is alsoknown as non-unitality . A single-qubit quantum channelΛ is called unital if and only if it maps the maximallymixed state onto itself: Λ (cid:0) (cid:1) =
1, see also Fig. 1.We will turn this observation into Theorem 1 by show-ing that non-unitality is one property which enables alocal channel to create quantum correlations in a multi-qubit system. In Theorem 2 we will show that on theother hand local unital quantum channels cannot increase b bb bb Λ u b b | ψ i | ψ i σσ σ Λ nu b b | ψ i | ψ i bb ρ ρ bb b Figure 1. Quantum channels on a single qubit: The up-per figure shows a unital quantum channel Λ u (green ar-row) which maps the maximally mixed state onto it-self: Λ u (cid:0) (cid:1) = . Two orthogonal states | ψ i and | ψ i with collinear Bloch vectors are mapped onto the states ρ = Λ u ( | ψ i h ψ | ) and ρ = Λ u ( | ψ i h ψ | ) with collinearBloch vectors. The lower figure shows a non-unital quantumchannel Λ nu (yellow arrow) which maps the maximally mixedstate onto the state σ = Λ nu (cid:0) (cid:1) = . The Bloch vectorsof σ = Λ nu ( | ψ i h ψ | ) and σ = Λ nu ( | ψ i h ψ | ) add up totwice the non-zero Bloch vector of σ , see main text. quantum correlations in a multi-qubit system. However,this statement does not hold for higher dimensions.Before presenting the main result of this Letter, weintroduce the semi-classical channel Λ sc . It maps all in-put states ρ onto states Λ sc ( ρ ) which are diagonal in thesame basis: Λ sc ( ρ ) = X k p k ( ρ ) | k i h k | . (5)The nonnegative probabilities p k ( ρ ) can in general de-pend on the input state ρ , while the orthogonal states | k i are independent of ρ . Such a channel is e.g. realizedby complete decoherence, after which only the diagonalelements of a density matrix may be non-zero. We arenow in the position to prove the following theorem. Theorem 1.
A local quantum channel acting on a sin-gle qubit can create quantum correlations in a multi-qubitsystem if and only if it is neither semi-classical nor uni-tal.Proof.
For simplicity we restrict ourselves to two qubitsonly. A generalization to an arbitrary number of qubitsis straightforward. The action of a local semi-classicalchannel Λ
Asc on the classically correlated state (1) is, due to linearity,Λ
Asc ( ρ cc ) = X i,j p ij Λ Asc (cid:0) | i A i h i A | (cid:1) ⊗ | j B i h j B | . (6)The definition of a semi-classical channel in Eq. (5) di-rectly implies that Λ Asc ( ρ cc ) is classically correlated.Now we will show that a local unital channel nevercreates quantum correlations in a multi-qubit system. Alocal unital channel Λ Au on the qubit A takes a classicallycorrelated state to the stateΛ Au ( ρ cc ) = X i,j p ij Λ Au (cid:0) | i A i h i A | (cid:1) ⊗ | j B i h j B | . (7)The action of the unital channel on the pure state | i A i h i A | can be studied using the Bloch representa-tion: | A i h A | = (cid:0) A + P i r i σ Ai (cid:1) , where σ Ai are thePauli operators with i ∈ { x, y, z } , and | A i h A | = (cid:0) A − P i r i σ Ai (cid:1) . Using linearity and unitality of Λ Au wesee that the state | A i h A | is mapped onto the state ρ A =Λ Au (cid:0) | A i h A | (cid:1) = (cid:0) A + P i r i Λ Au (cid:0) σ Ai (cid:1)(cid:1) . The sameprocedure for | A i h A | results in ρ A = Λ Au (cid:0) | A i h A | (cid:1) = (cid:0) A − P i r i Λ Au (cid:0) σ Ai (cid:1)(cid:1) . Note that the Bloch vectors ofthe states ρ A and ρ A point into opposite directions, seeFig. 1 for illustration. States with this property canbe diagonalized in the same basis. This implies that itis possible to write the state Λ Au ( ρ cc ) in the form (1).Thus we proved that local unital quantum channels can-not create quantum correlations in a classically correlatedmulti-qubit state.In the following we will complete the proof of Theorem1 by showing that any local quantum channel Λ Anu that isneither unital nor semi-classical can create quantum cor-relations. By definition Λ
Anu maps the maximally mixedstate A onto some state that is not maximally mixed:Λ Anu (cid:18) A (cid:19) = 12 A + X i s i σ Ai ! , (8)with P i s i = 0. Since we demand that the quantumchannel is not semi-classical, there exists a state | ψ A i such that Λ Anu (cid:0) | ψ A i h ψ A | (cid:1) is not diagonal in the eigenba-sis of Λ Anu (cid:0) A (cid:1) . Again we consider the Bloch represen-tation Λ Anu (cid:0) | ψ A i h ψ A | (cid:1) = 12 A + X j r j σ Aj (9)and note that the two Bloch vectors r and s are linearlyindependent. Otherwise the states Λ Anu (cid:0) | ψ A i h ψ A | (cid:1) andΛ Anu (cid:0) A (cid:1) could be diagonalized in the same basis, whichis in contradiction to the definition of | ψ A i . Consider nowthe classically correlated state ρ cc = 12 | ψ A i h ψ A | ⊗ | B i h B | + 12 | φ A i h φ A | ⊗ | B i h B | (10)with orthogonal states h ψ A | φ A i = 0. We can writethe states as | ψ A i h ψ A | = (cid:0) A + P i v i σ Ai (cid:1) , and | φ A i h φ A | = (cid:0) − P i v i σ Ai (cid:1) . We define the vector w such that the equality Λ nu (cid:0)P i v i σ Ai (cid:1) = P i w i σ Ai with P i w i = 0 is satisfied. This is always possible, since Λ nu is trace-preserving. The action of the channel onto thetwo states | ψ A i and | φ A i is as follows:Λ Anu (cid:0) | ψ A i h ψ A | (cid:1) = 12 A + X i ( s i + w i ) σ Ai ! , (11)Λ Anu (cid:0) | φ A i h φ A | (cid:1) = 12 A + X i ( s i − w i ) σ Ai ! . (12)As noted above, the two Bloch vectors s and r = s + w are linearly independent. The same must hold for thevectors s + w and s − w . This implies that the two statesΛ Anu (cid:0) | ψ A i h ψ A | (cid:1) and Λ Anu (cid:0) | φ A i h φ A | (cid:1) are not diagonal inthe same basis. This completes the proof.So far we saw that local unital and local semi-classicalchannels acting on a single qubit cannot create quan-tum correlations from a classically correlated multi-qubitstate. These results hold independently of the chosenmeasure for quantum correlations. In the following wewill go one step further by showing that these local chan-nels never increase a very general class of measures forquantum correlations in multi-qubit systems. We con-sider distance-based measures of quantum correlations Q D , which are defined via the minimal distance D to theset of the classically correlated states CC [8, 9], Q D = min σ ∈ CC D ( ρ, σ ) , (13)where D does not necessarily have to be a distance inthe mathematical sense. The statement mentioned abovewill be shown to hold for all distance measures D withthe property of being non-increasing under any quantumchannel Λ, i.e. D (Λ ( ρ ) , Λ ( σ )) ≤ D ( ρ, σ ) . (14)This property is also frequently used for defining entan-glement measures [32, 33]. Theorem 2.
Quantum correlations Q D in multi-qubitsystems do not increase under local unital channels Λ lu and local semi-classical channels Λ lsc : Q D (Λ lu ( ρ )) ≤ Q D ( ρ ) , (15) Q D (Λ lsc ( ρ )) ≤ Q D ( ρ ) . (16) Proof.
Let ξ be the classically correlated state which min-imizes the distance, i.e. Q D ( ρ ) = D ( ρ, ξ ). Using theproperty (14) of the distance to be nonincreasing underquantum channels we obtain Q D ( ρ ) = D ( ρ, ξ ) ≥ D (Λ lu ( ρ ) , Λ lu ( ξ )) , (17) Q D ( ρ ) = D ( ρ, ξ ) ≥ D (Λ lsc ( ρ ) , Λ lsc ( ξ )) . (18) Now we use Theorem 1 noting that local unital chan-nels Λ lu and local semi-classical channels Λ lsc map theclassically correlated state ξ onto another classically cor-related state Λ ( ξ ) which is not necessarily the one thatminimizes the distance to Λ ( ρ ). This observation finishesthe proof.One example for a measure that satisfies the proper-ties (15) and (16) - and thus Theorem 2 holds - is the geometric measure of quantumness which we define as Q G ( ρ ) = min σ ∈ CC (1 − F ( ρ, σ )) (19)with the fidelity F ( ρ, σ ) = (cid:0) Tr p √ ρσ √ ρ (cid:1) . Using thefact that the fidelity is non-decreasing on quantum chan-nels together with Theorem 2, we see that the geometricmeasure of quantumness does not increase under localunital channels and local semi-classical channels. Al-ternatively, we can use the quantum relative entropy S ( ρ || σ ) = − Tr [ ρ log σ ] + Tr [ ρ log ρ ], which also ful-fills the property (14) [32, 33]. From Theorem 2 fol-lows that the resulting measure of quantum correlations Q S = min σ ∈ CC S ( ρ || σ ) does not increase under local uni-tal and local semi-classical channels. Q S was also studiedin [13], where it was called relative entropy of quantum-ness.So far we considered states consisting of an arbitrarynumber of qubits. We have shown that local unitaland local semi-classical channels acting on a single qubitnever increase quantum correlations as defined by Q D in Eq. (13). On the other hand, any local channel which isnon-unital and not semi-classical can in principle createquantum correlations, independently of the consideredmeasure, out of a classically correlated state. An exam-ple for such a channel is the amplitude damping channelas a model for dissipation. Thus, dissipation can increasequantum correlations.At the present stage it is natural to ask the question,for what kind of input states this behavior can or cannotbe observed in general. The following theorem shows thatpure states are special.
Theorem 3.
The geometric measure of quantumness ofmultipartite systems with arbitrary dimension cannot in-crease under any local quantum channel, if the initialstate is pure: Q G (Λ l ( | ψ i h ψ | )) ≤ Q G ( | ψ i h ψ | ) , (20) where Λ l is an arbitrary local quantum channel.Proof. Let ξ ∈ CC be defined such that Q G ( | ψ i h ψ | ) =1 − F ( | ψ i h ψ | , ξ ). Using the properties of the fidelity F we see that ξ can be chosen to be a pure prod-uct state ξ = | φ i h φ | . Moreover 1 − F does not in-crease under the action of any quantum channel, i.e.1 − F ( | ψ i h ψ | , | φ i h φ | ) ≥ − F (Λ l ( | ψ i h ψ | ) , Λ l ( | φ i h φ | )).Since | φ i is a product state, Λ l ( | φ i h φ | ) is also a productstate. This observations completes the proof.Note that Theorem 3 does not follow from the factthat for pure states the amount of quantum correlationsis equal to the amount of entanglement.So far we have shown that quantum correlations inmulti-qubit systems cannot increase under local unitalquantum channels. A prominent example for a unitalchannel is the phase damping channel, which is a modelfor decoherence in a quantum system. Under decoherencethe quantum state ρ = P i,j ρ ij | i i h j | is transformed tothe stateΛ ( ρ ) = X i ρ ii | i i h i | + (1 − p ) X i = j ρ ij | i i h j | (21)with the damping parameter 0 ≤ p ≤
1. Since Λ is unital,it is not possible to create quantum correlations with lo-cal phase damping in a multi-qubit system. Surprisingly,this is not true if the local systems are not qubits: qubitsare special. This can be demonstrated via the classicallycorrelated state as input: ρ cc = 12 | ψ A i h ψ A | ⊗ | B i h B | + 12 | φ A i h φ A | ⊗ | B i h B | (22)with the orthogonal single-qutrit states | ψ A i = √ (cid:0) − | A i + | A i + | A i (cid:1) and | φ A i = √ (cid:0) | A i + | A i (cid:1) .We will show that a local phase damping channel Λ A act-ing on subsystem A generates quantum correlations. Weconsider the action of the channel (21) with the dampingparameter p = on the state ρ cc in Eq. (22):Λ A ( ρ cc ) = 12 X i =1 λ i | ψ Ai i h ψ Ai | ⊗ | B i h B | + 12 X j =1 µ j | φ Aj i h φ Aj | ⊗ | B i h B | , (23)where the states (cid:8) | ψ Ai i (cid:9) are the eigenstates ofΛ A (cid:0) | ψ A i h ψ A | (cid:1) with the corresponding eigenvalues λ i . Similarly the states (cid:8) | φ Aj i (cid:9) are eigenstates ofΛ A (cid:0) | φ A i h φ A | (cid:1) with the eigenvalues µ j . One can seeas follows that the state Λ A ( ρ cc ) is quantum correlated:The eigenvalues of Λ A (cid:0) | ψ A i h ψ A | (cid:1) are given by λ = ,and λ = λ = . The eigenstate to the largest eigen-value λ is given by | ψ A i = | ψ A i . It is easy to check that | ψ A i is not an eigenstate of Λ A (cid:0) | φ A i h φ A | (cid:1) , and thereforethe state in Eq. (23) is not classically correlated. Thuswe proved that it is possible to create quantum correla-tions with a local phase damping channel, i.e. via localdecoherence.In conclusion, we have investigated the effect of local noisy channels (i.e. trace-preserving completely positivemaps) on quantum correlations. While entanglement cannever increase under such local channels, quantum corre-lations without entanglement may or may not increase,depending on the type of channel and the type of in-put state. For multi-qubit systems we fully answer the question which local channels can increase quantum cor-relations: unital and semi-classical local channels can-not enhance quantum correlations, while non-unital andnon-semi-classical local channels (e.g. dissipation, cor-responding to amplitude damping) can increase quan-tum correlations. Surprisingly, for higher-dimensionalsystems, even unital channels such as decoherence, corre-sponding to phase-damping, can generate quantum corre-lations from an initially classically correlated state. How-ever, quantum correlations as quantified by the geometricmeasure of quantumness can become larger under localchannels only when the initial state is mixed. - Thus, wehave shed some light on the behavior of quantum corre-lated states in a noisy environment.We acknowledge partial financial support by DeutscheForschungsgemeinschaft (DFG) and by the ELES foun-dation. Note added:
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