Entanglement, discord and the power of quantum computation
aa r X i v : . [ qu a n t - ph ] J a n Entanglement, discord and the power of quantum computation
Aharon Brodutch ∗ and Daniel R. Terno † Department of Physics & Astronomy, Faculty of Science, Macquarie University, NSW 2109, Australia
We show that the ability to create entanglement is necessary for execution of bipartite quantumgates even when they are applied to unentangled states and create no entanglement. Starting witha simple example we demonstrate that to execute such a gate bi-locally the local operations andclassical communications (LOCC) should be supplemented by shared entanglement. Our resultspoint to the changes in quantum discord, which is a measure of quantumness of correlations evenin the absence of entanglement, as the indicator of failure of a LOCC implementation of the gates.
The question “What makes a quantum computertick?” goes back to the early discussions of quantum algo-rithms [1]. Two different explanations of the speed-up ofquantum algorithms are centered on the two fundamen-tal aspects of quantum theory: superposition of quantumstates and their entanglement [2, 3].The latter view is supported by the make-up of auniversal set of gates [2]. To run a quantum compu-tation it is sufficient to execute certain one-qubit gatesand one entangling gate, such as a two-qubit controlled-
NOT ( CNOT ). An entangling gate turns a generic non-entangled input into an entangled output. On the otherhand, any pure-state quantum computation that utilizesonly a restricted amount of entanglement can be effi-ciently simulated classically [4].According to the alternative view, it is a superposi-tion of all possible computational paths in a quantumcomputer that is responsible for a speed-up, while entan-glement may be just incidental. Indeed, the algorithmDQC1 demonstrates such a speed-up without entangle-ment [5, 6].We show that entanglement is required for the im-plementation of bipartite gates, even if they operate ona restricted set L of unentangled input states that aretransformed into unentangled outputs. This remains truewhen the set is chosen to contain only mixtures of somepure states, and not their coherent superpositions.A distributed implementation of a gate is a naturalsetting to study the effects of entanglement. A(lice) andB(ob) execute a bipartite gate U using local operationsand different shared resources. We show that under quitegeneral assumptions U can be implemented bi-locally on L only if Alice and Bob share some entanglement. Thebuild-up of quantum correlations other than entangle-ment as ρ in is transformed into ρ out indicates the demandfor shared entanglement. The correlations are quantifiedby quantum discord [7].We first introduce quantum discord and review some ofits properties, then present a simple example and followwith general results. Discord is defined through the difference in the gen- ∗ Electronic address: [email protected] † Electronic address: [email protected] eralizations of two expressions for the classical mutualinformation, I ( A : B ) = H ( A ) + H ( B ) − H ( AB ) , (1)and J ( A : B ) = H ( A ) − H ( A | B ) = H ( B ) − H ( B | A ) , (2)where H ( X ) is the Shannon entropy of the probabilitydistribution X , H ( Y | X ) the conditional entropy of Y given X , and H ( XY ) is the entropy of a joint proba-bility distribution [8]. The two classical expressions areequivalent. The quantum measurement procedure Λ ona state ρ leads to a probability distribution X Λ ρ . The vonNeumann entropy S ( ρ X ) = − tr ρ X log ρ X replaces theShannon entropy [9], but the conditional entropy nowexplicitly depends on the measurement procedure [7, 10]and the optimization goal it tries to achieve. For our pur-poses it is enough to assume that the measurement Π A on Alice’s subsystem is represented by a complete set oforthogonal projections, and the optimization is chosen tolead to the discord measure D [11, 12]. Then J Π A ( ρ ) := S ( ρ B ) − S ( ρ B | Π A ) + S ( ρ A ) − S ( ρ Π A A ) , (3)where the averaged post-measurement state of A is ρ Π A A = X a p a Π aA , p a = tr ρ A Π aA , (4)(classically the last two terms in (3) cancel out), and theconditional entropy of the post-measurement state of B , S ( ρ B | Π A ) := X a p a S ( ρ B | Π aA ) . (5)is the weighted average of the entropies of the states ρ B | Π aA = tr A (Π aA ⊗ B ρ Π aA ⊗ B ) /p a (6)that correspond to the individual outcomes. Finally, thediscord is D A ( ρ ) := min Π A [ H ( A Π ρ ) + S ( ρ B | Π A )] − S ( ρ AB ) > . (7)Discord has a number of interesting properties and ap-plications [11–14]. We use the property [12]: D Π A ( ρ ) = S ( ρ Π A ) − S ( ρ ) > D A ( ρ ) , (8)which holds for any set Π A that induces the averagedpost-measurement state ρ Π A . Gate implementation . We investigate a bi-local imple-mentation of the gate U on a restricted set L . Alice andBob can perform arbitrary local operations and measure-ments on their respective qubits, are allowed to exchangeunlimited classical messages, but have no shared entan-glement. While it is just a standard LOCC paradigm, wepoint out one important feature of the reduced dynamicsof the system.The measurements are represented by arbitrary localpositive operator-valued measures (POVM), so Alice’smeasurement is given by a family of positive operatorsof the form E µA = Λ µA ⊗ B , Λ µA > P Λ µA = A . Ateach stage the operations and measurements are inte-grated together with the help of an ancilla, which canbe further divided into two parts A ′ and A ′′ , as in[15]. The measurement is accomplished in two stages:first some unitary operation U AA ′ A ′′ is applied to theentire system, and then a projective measurement Π a , a = 1 , . . . dim A ′′ , Π a Π b = Π a δ ab is done on the sys-tem A ′′ . Depending on the outcome, a unitary U AA ′ ( a )is applied to the remaining part AA ′ . While the en-tire evolution of A is completely positive, that is, ρ in A ρ A | Π a ρ out A = P µ K µ ρ in A K † µ for some set of Kraus ma-trices K µ [2], the evolution of a post-measurement state ρ A | Π a ρ out A = tr A ′ U AA ′ ( a ) ρ ′ AA ′ U † AA ′ ( a ) generally de-pends on the correlations between A and A ′ and may benot completely positive [16]. Example: A CNOT gate . This gate can be performedbi-locally by Alice and Bob if they share one ebit of en-tanglement per gate use [17]. In our example Alice andBob share an unknown state from the known list L andtry to implement the CNOT gate by LOCC. It is obviousthat if the set L is locally distinguishable, then the gatecan be implemented by LOCC. It is also obvious thatif the action creates entanglement, the implementationfails. However, absence of entanglement is not sufficient.Consider the set L in Table I. TABLE I: Four inputs and outputs for the CNOT gate a | i| Y + i → i | i| Y − i c | Y + i| X − i → | Y − i| X − i b | i| Y + i → | i| Y + i d | Y + i| X + i → | Y + i| X + i Here σ y | Y ± i = ±| Y ± i , σ x | X ± i = ±| X ± i , where σ x,y,z are Pauli matrices.We demonstrate that ability to implement the CNOTgate on L without shared entanglement makes it possi-ble to unambiguously discriminate between these non-orthogonal states using just one input copy, which is im-possible [9]. Without specifying the local operations ofAlice and Bob we classify them according to their ac-tion on the sate | Y + i . An operation Φ is flipping (F)if up to a phase Φ( | Y + i ) = | Y − i , non-flipping (N) ifΦ( | Y + i ) = | Y + i , and is undetermined otherwise. Knowing the operation type allows Alice and Bob tonarrow down the list of possible inputs: For example,Bob’s F is incompatible with having the input b , whilefor Alice’s operation not to have a definite type excludesboth c and d . The list of possible inputs if both oper-ations are of a definite type is presented in Table II. Ifone of the performed operations is neither F nor N, thenthe type of other operation allows to determine the inputuniquely. TABLE II: Possible inputsAlice BobF NF ( a c ) ( cb ) N ( a d ) ( b d ) Any pair of outputs can be reset to their original inputstate by local unitaries and resent through the gate. Forexample, if the overall operation is of the FF type, the op-eration σ Az ⊗ σ Bx will transform the outputs ψ ′ a = | i| Y − i and ψ ′ c = | Y − i| X − i into the inputs ψ a and ψ c , respec-tively.The operations that implement the gate on its sec-ond run may be the same or different from the operationin the previous run. If the gate’s design allows a finiteprobability of having a different operation type, it willbe realized after a finite number of trials. This othertype (FN or NF in the preceding example) will uniquelyspecify the input. If a particular pair of inputs is alwaysprocessed by the same type of operations, then the gatecan be used to unambiguously distinguish between onestate from this pair and at least one of the two remainingstates in a single trial. (cid:4) Definition . A bi-local implementation G of a gate U on some (finite) set of unentangled states L = { ρ in i } Ni =1 (and their convex combinations) is a completely positivemap that is implemented by local operations on the sub-systems A and B , performed separately, that are assistedby unlimited classical communication such that for anystate ρ i ∈ L G ( ρ in i ) = X k K k ρ in i K † k ≡ U ρ i U † = ρ out i . (9)Successful implementation of the gate on pure inputsguaranties that it is “reversible”, with the dual map [18]playing the role of the inverse. Property 1.
The dual map G + ( ρ ) := P k K † k ρK k satis-fies ρ in i = G + ( ρ out i ) (10)for all pure input states ρ ψ ∈ L . Proof:
Since ρ out ψ = G ( ρ in ψ ) = U ρ ψ U † is pure, using theHilbert-Schmidt inner product we see that1 = h ρ out ψ , ρ out ψ i = h ρ in ψ , G + ( U ρ in ψ U † ) i , (11)hence G + acts as an inverse for all allowed pure inputsand their convex combinations. (cid:4) It is straightforward to see that if we restrict local op-erations to projective measurements and unitaries, thenthe zero discord becomes a necessary criterion for suchimplementation’s success. Namely, since entropies of ini-tial and final states are the same, but a local measure-ment on a state of non-zero discord increases it accordingto Eq. (8), we reach a contradiction.A symmetrized version of the discord is used in whatfollows: D ( ρ ) := min[ D A ( ρ ) , D B ( ρ )] = 0 . (12)Unlike the exact value of discord that can be calculatedanalytically only in special cases, it is straightforward tocheck weather the discord is zero or not [12]. Moreover,sates of zero discord (say, D A = 0) are of the form ρ = X a p a Π aA ⊗ ρ aB , p a ≥ , X a p a = 1 . (13)Now we consider different bi-local implementation of U . Assume first that the the set of possible inputs L in-cludes the maximally mixed state (i.e. the gate is unital, G ( ) = ), and at least one pure state that we write as | i . Also restrict the allowed local operations to com-pletely positive (CP) maps (this is realized, in particular,if at each stage the ancilla is entirely consumed by themeasurement, i.e., dim A ′ = 0). Lemma 1.
If a set L contains one pure product state( | i ) and the maximally mixed sate ( /
4) in L , and theaction of U is realized by local operations restricted toarbitrary POVM and CP maps and classical communi-cation, then all other allowed inputs (and their arbitraryconvex combinations) satisfy D ( ρ in ) = 0. Proof:
Assume that some states in L have D ( ρ in ) = 0.Introduce a CP map Φ( ρ ) = G + (cid:0) G ( ρ ) (cid:1) . It is a unitalmap, because G + is unital [18]. According to Property1 its application to ρ := | ih | gives Φ( ρ ) = ρ .Assume that Alice is the first party to perform a measure-ment on the inputs, and consider a state ρ := | ih | (not necessarily an allowed input). Since Φ is unital,Φ( − ρ ) = Φ( ρ + ρ + ρ ) = − ρ , (14)so the positivity of density matrices enforces h | Φ( ρ ) | i = 0, and similarly for two other statesin the preceding equation. As a result, Φ( ρ ) has adisjoint support from ρ .Separate the map Φ into Alice’s first measurement { Λ µA } and everything else. Evolution of any state ρ in can be schematically written as ρ in ρ µ ρ out ρ ′ ,with ρ out = U ρ in U † for ρ in ∈ L , and ρ ′ = ρ in for purestates in L . We write ρ µ for ρ | Λ µA to simplify the notation.Since ρ ′ = Φ µ ( ρ µ ) for some CP map Φ µ by the lemma’sassumption, and CP maps cannot improve state distin-guishability [2, 19], the post-measurement states ρ µ and ρ µ should have disjoint supports for any outcome µ . Re-call that in dealing with these two states Alice measures pure qubits while Bob’s sides are identical. Hence Alice’smeasurement reduces to the projective measurement insome basis (say 0 ′ ,1 ′ ),Λ aA = Π aA = | a ih a | A , a = 0 ′ , ′ . (15)Let Alice perform this measurement on inputs withnon-zero discord. For pure states ρ in A the average post-measurement entropy becomes non-zero [2, 9]. For mixedstates with D A = 0 Eq. (8) ensures that S ( ρ Π A in ) >S ( ρ in ) = S ( ρ out ). However, projective measurements arerepeatable, and a second measurement by Alice will cer-tainly give the same result and induce no further changein the state. Hence, if the state ρ in ∈ L , then for anyoutcome a the gate operates successfully, G ( ρ in | Π aA ) = G ( ρ in ) = U ρ in U † . Since unitary maps preserve entropyand and unital CP maps do not decrease it [2, 18], wereach a contradiction.In case the first measurement is performed by Bob weconsider the state | i and use the discord D B . (cid:4) Now we consider what happens if the operation is per-formed on d -dimensional systems, and the set L con-tains two non-orthogonal quantum states, | ψ i i = | a i i| b i i , i = 1 ,
2. Obviously as | ψ ′ i i = U | ψ i i , h a | a ih b | b i = h a ′ | a ′ ih b ′ | b ′ i . (16)This time we do not have to assume anything about thegate G apart from its being implemented by LOCC. Thefollowing lemma explains our original example. Lemma 2.
If the set L contains two pure non-orthogonal states, and the unitary operation is such that D ( ρ ) = D ( U ρU † ), where ρ = wρ ψ + (1 − w ) ρ ψ , forsome 0 < w <
1, then it cannot be implemented on L byLOCC alone. Proof:
Eq. (16) holds either through the constancy of theoverlap on both sides individually, |h a | a i| = |h a ′ | a ′ i| , |h b | b i| = |h b ′ | b ′ i| , or by increasing one overlap and de-creasing the other, as, for example, |h a | a i > |h a ′ | a ′ i , |h b | b i| < |h b ′ | b ′ i| . The latter possibility precludesLOCC gate execution, since the inequality |h a | a i| > |h a ′ | a ′ i| entails that the distinguishability of two statesimproved as a result of some CP map, which is impossible[2, 19].The product form of the final states makes it is possi-ble to find (non-unique) local unitary operations U iA , U iB such that | a ′ i i = U iA | a i i , | b ′ i i = U iB | b i i . The norm conser-vation requires that when restricted to the linear spansof the states | a i i and | b i i , respectively, these operators tosatisfy U A = e iα U A and U B = e iβ U B for some phases α and β . As a result, on the states ρ = wρ ψ + (1 − w ) ρ ψ the gate is realized by a bi-local unitary operation, ρ ′ = U ρU † = U A ⊗ U B ρU † A ⊗ U † B , (17)which implies [7, 12] that D ( ρ ) = D ( ρ ′ ), contraindicat-ing the assumption. Hence the LOCC implementation of U is impossible. (cid:4) It is possible to draw several conclusions. First the ab-sence of entanglement in both input and output does notautomatically enable a remote implementation by LOCC.Second, a discrepancy between local and global informa-tion content of non-entangled states (which is capturedby the discord D in our setting and may have to begeneralized in more sophisticated scenarios) requires en-tanglement for their processing. In the preceding casespresented above we see that entanglement is required for any gate which changes the discord of the state. Recentresults [20, 21] suggest that a change in discord ratherthen entanglement is the required resource in computa-tional speed-up. Our result shows that the two are inti-mately linked.We thank G. Brennen, A. Datta, R. Duan, F. Fanchini,K. Modi, J. Twamley, and K. ˙Zyczkowski for useful dis-cussions and helpful comments. [1] A. Ekert and R Jozsa, Phil. Trans. R. Soc. London A , 1769 (1998).[2] D. Bruß and G. Leuchs, Lectures on Quantum Informa-tion (Wiley-VCH, Weinheim, 2007); M. A. Nielsen andI. L. Chuang,
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