Comparing globalness of bipartite unitary operations: delocalization power, entanglement cost, and entangling power
aa r X i v : . [ qu a n t - ph ] M a y Comparing globalness of bipartite unitary operations:delocalization power, entanglement cost, and entangling power
Akihito Soeda and Mio Murao
1, 21
Department of Physics, Graduate School of Science,The University of Tokyo, Tokyo 113-0033, Japan. Institute for Nano Quantum Information Electronics,The University of Tokyo, Tokyo 113-0033, Japan
We compare three different characterizations of the globalness of bipartite unitaryoperations, namely, delocalization power, entanglement cost, and entangling power,to investigate global properties of unitary operations. We show that the globalness ofthe same unitary operation depends on whether input states are given by unknownstates representing pieces of quantum information, or a set of known states for thecharacterization. We extend our analysis on the delocalization power in two ways.First, we show that the delocalization power differs whether the global operation isapplied on one piece or two pieces of quantum information. Second, by introducinga new task called LOCC one-piece relocation, we prove that the controlled-unitaryoperations do not have the delocalization power strong enough to relocate one of twopieces of quantum information by adding LOCC.
I. INTRODUCTION
Understanding the source of quantum advantage in quantum computation is a long-standing issue in quantum information science. Previous researches have shown that cer-tain quantum computation is ‘classical’, for the reason that it is efficiently simulateable byclassical computers. One example is any computation performed just by local operationsand classical communication (LOCC) [Horodecki et al. globalness of quantum operations.It is also known that not all global operations result in quantum speedup for quantumcomputation. There must be a specific globalness that differentiates the quantum opera-tions leading to quantum speedup from those do not. The difference may be due to morethan one kind of globalness, but even this is not clear at this point. For this reason, hav-ing a good understanding of the globalness of quantum operations is important. In thispaper, we try to understand the simplest case of the global operations, namely, bipartiteunitary operations.To investigate globalness of unitary operations, it is important to clarify what kind ofstates is given as inputs of the unitary operations. We want to evaluate the globalnessthat does not depend on a choice of a particular input state. By introducing the concept of pieces of quantum information , we analyze characterizations of unitary operations for twopieces of quantum information represented by arbitrary unknown states, in terms of delo-calization power [Soeda and Murao 2010] and entanglement cost [Soeda et al. entangling power of globaloperations [Kraus and Cirac 2001, Wolf et al. et al.
LOCC one-piece relocalization for one piece of delocalized quantum information that corresponds to the case when apart of input state is unknown and arbitrary but the other part can be chosen from a setof known state. The other task is
LOCC one-piece relocation for two pieces of delocalizedquantum information, which evaluates the ability of the unitary operation to relocate oneof the two pieces of quantum information from one Hilbert space to another by addingLOCC to the unitary operation.The rest of the paper is organized as following. In Section II, we introduce the conceptof pieces of quantum information and present an overview on the three characterizations.We summarize the comparison of different aspects of the globalness of bipartite unitaryoperations presented in the previous works in Section III. We extend the analysis of thedelocalization power in Sections IV and V. In Section IV, we show the result on LOCCone-piece relocalization for one piece of delocalized quantum information. In Section V,we analyze LOCC one-piece relocation of two pieces of quantum information. Finally, inSection VI, we present our conclusion.
II. THREE CHARACTERIZATIONS OF THE GLOBALNESS OF QUANTUMOPERATIONSA. Delocalization power
First, we define a piece of quantum information for a d -dimensional quantum system,or qudit , whose Hilbert space is denoted by H = C d . Definition 1
If a pure quantum state of n qudits | ψ i ∈ H ⊗ n is given by | ψ i = X i α i | ϕ i i , where {| ϕ i i} d − i =0 is a fixed set of normalized and mutually orthogonal states in H ⊗ n and thecoefficients α i ∈ C are arbitrary and unknown except for the normalization P i | α i | = 1 ,the unknown state | ψ i is said to represent one piece of quantum information for a qudit. In the formalism presented above, a piece of quantum information for a single quditcan be stored in an n -qudit system using an arbitrary set of orthonormal states, {| ϕ i i} d − i =0 .Any such set of states would form a logical qudit space, but in a special case satisfying | ϕ i i = | i i ⊗ | ξ i for all i ∈ { , . . . , d − } , where the set of states {| i i} forms an orthonormal basis of H and | ξ i ∈ H ⊗ ( n − is independent of i , the piece of quantum information is stored in a physical qudit. Hence it is possible to assign one physical qudit for each piece of quantuminformation.Using this formalism, now we provide the formal definition of one piece of localized quantum information for a qudit. We label the qudits of an n -qudit system from 0 to n − k by H k . The Hilbert space of n − excluding a certain qudit k will be denoted by H ¬ k . We will also assume that two differentpieces of quantum information in the same system are assigned to different physical qudits. Definition 2
For n ≥ , a piece of quantum information represented by an unknown n -qudit state | ψ i is said to be localized at an assigned Hilbert space H k , or simply localizedwhen there is no fear of confusion, if it is represented in the form | ψ i = X i α i | i i ⊗ | ξ i , where {| i i} d − i =0 is any basis of the Hilbert space of the assigned qudit (i.e., H k ), | ξ i ∈ H ¬ k is an ( n − -qudit state determined independently of the set of coefficients { α i } , and { α i } are arbitrary coefficients satisfying the normalization condition P i | α i | = 1 . Note that the global phase factor of the coefficients is not a physical quantity, so wetake the global phase equivalence. There are d − n = 1, since H = C d is the minimal Hilbert spaceto store one piece of quantum information for a qudit, one piece of quantum informationhas to be localized in H .We define the concept of delocalized quantum information, which is the complement oflocalized quantum information, and also the concept of delocalization of quantum infor-mation. Definition 3
If a piece of quantum information is not localized, then it is said to bedelocalized. The task of delocalizing quantum information is called delocalization.
Next, we consider two-qudit states, where the state of each qudit represents one pieceof localized quantum information. We denote the two Hilbert spaces of the qudits by H A = C d and H B = C d . The two pieces of localized quantum information can berepresented by a tensor product state | ψ A i A ⊗ | ψ B i B , where the subscripts of the ketsdenote the assignment of the Hilbert spaces of the qudits, | ψ A i A ∈ H A and | ψ B i B ∈ H B .We define delocalization for two pieces of quantum information as the following. Definition 4
Two pieces of quantum information are said to be delocalized, if the staterepresenting the two pieces of quantum information cannot be written by u A | ψ A i A ⊗ u B | ψ B i B , where u A on H A and u B on H B are arbitrary local unitary operations butindependent of | ψ A i A and | ψ B i B . We again note that we have already assigned a Hilbert space for each piece of quantuminformation, so the state | ψ B i A ⊗ | ψ A i B represents delocalized two pieces of quantuminformation out from the assigned Hilbert spaces.Now we investigate the effects of a global unitary operation U applied on two pieces oflocalized quantum information | ψ A i A ⊗ | ψ B i B . If the unitary operation U is not a tensorproduct of two local unitary operations on H A and H B , U always transforms each piece oflocalized quantum information to delocalized quantum information. In this paper, we saythat the unitary operations have delocalization power , which in a sense is the ‘strength’of delocalization of quantum information due to the unitary operations.How pieces of quantum information are delocalized is determined only by the set oforthonormal states representing the quantum information, which in turn is determinedby the unitary operation used for the delocalization. Therefore, the globalness of unitaryoperations can be studied by understanding how a unitary operation delocalizes pieces ofquantum information.Later, we argue that certain pieces of quantum information are ‘more’ delocalized thanothers. The difference in the level of delocalization can only have come from the differencein the globalness of the unitary operation, namely, the delocalization power. Hence, wecan classify the delocalization power by analyzing the level of the delocalization that eachunitary operation achieves.To define and classify the level of delocalization, we introduce the following LOCCtask, LOCC one-piece relocalization , that aims to localize just one of the two pieces ofdelocalized quantum information by sacrificing the other piece of quantum information in H A ⊗ H B . We denote the set of density operators on the Hilbert space H by S ( H ). Definition 5
LOCC one-piece relocalization of qudit B for two pieces of quantuminformation delocalized by a global unitary operation U is a task to find an LOCC-implementable completely positive trace preserving (CPTP) map Λ LOCC U : S ( H A ⊗ H B ) →S ( H B ) satisfying Λ LOCC U [ U ( | ψ A i A h ψ A | ⊗ | ψ B i B h ψ B | ) U † ] = | ψ B i B h ψ B | for any | ψ A i A ∈ H A and | ψ B i B ∈ H B . We characterize the delocalization power of global unitary operations in terms of theirability to allow LOCC one-piece relocalization on two pieces of quantum informationdelocalized by the global unitary operations. We define the order of the delocalizationpower of two global unitary operations U and U ′ on two pieces of quantum informationby the following. Definition 6
If LOCC one-piece relocalization of two pieces of delocalized quantum infor-mation is possible for a unitary operation U , but not possible for another unitary operation U ′ , the order of the delocalization power of U is defined to be smaller than that of U ′ interms of LOCC one-piece relocalization. B. Entanglement cost
Another way to quantify the globalness of a unitary operation applied on quantuminformation is to evaluate how much extra global resource is required on top of LOCCoperations to implement the unitary operation on two pieces of quantum information. Theminimum amount of entanglement required to deterministically implement a given globaloperation is unique, based on the fact that entanglement cannot be generated by LOCC.We define an LOCC task, entanglement assisted deterministic LOCC implementation andthen define entanglement cost of the unitary operation on quantum information in termsof this LOCC task.
Definition 7
Entanglement assisted deterministic LOCC implementation of a global uni-tary operation U on two pieces of localized quantum information | ψ A i A ⊗ | ψ B i B ∈ H in = H in A ⊗ H in B using a fixed bipartite resource state | Φ i AB ∈ H r = H r A ⊗ H r B is a task of findingan LOCC-implementable CPTP map Γ LOCC U : S ( H in ⊗ H r ) → S ( H in ) satisfying Γ LOCC U ( | ψ A i A h ψ A | ⊗ | ψ B i B h ψ B | ⊗ | Φ i AB h Φ | ) = U ( | ψ A i A h ψ A | ⊗ | ψ B i B h ψ B | ) U † for any | ψ i A ∈ H r A and | ψ i B ∈ H r B . Definition 8
Entanglement cost of a unitary operation U on two pieces of quantum in-formation is given by the minimum amount of entanglement of the resource state | Φ i AB required to perform entanglement assisted deterministic LOCC implementation of U ontwo pieces of localized quantum information. The entanglement cost of unitary operations can be regarded as the minimum entan-glement cost for delocalizing two pieces of quantum information. Thus this is anotherway to characterize the globalness of unitary operations applied on quantum information.
C. Entangling power
We can also quantify the globalness of a unitary operation by evaluating its ability ofentanglement generation in place of entanglement cost, similarly to the pair of the defi-nitions for evaluating the globalness of quantum states, namely, distillable entanglementand entanglement cost. However, the amount of entanglement generated by a unitaryoperation strongly depends on the choice of input states, therefore it is difficult to definea quantity in terms of quantum information, namely, unknown states. Instead, entanglingpower of a global operation [Kraus and Cirac 2001, Wolf et al. et al.
Definition 9
The entangling power of a bipartite unitary operation U (denoted by E ep ( U ) ) is defined as the maximum amount of entanglement generated between the bipar-tite cut by applying U on a known state, i.e., E ep ( U ) ≡ max ρ in E ( U ρ in U † ) − E ( ρ in ) , where E ( ρ ) is an entanglement measure of choice and ρ in is chosen among the given setof states. III. COMPARISON OF GLOBALNESS BY DIFFERENTCHARACTERIZATIONS
In the previous section, we introduced three different characterizations for the global-ness of bipartite unitary operations: delocalization power, entanglement cost and entan-gling power. In this section, we summarize known results on the three characterizationsand investigate whether the globalness characterized by each method is same to or differ-ent from that of by the others.
Theorem 1
LOCC one-piece relocalization for two pieces of quantum information ofqudits delocalized by a unitary operation U is possible if and only if U is a locally unitaryequivalent to a controlled-unitary operation C { u k } = P k | k i h k | ⊗ u k , where {| k i} formsan orthonormal basis for one of the subsystems and { u k } is a set of unitary operations onthe other subsystem. [Soeda and Murao 2010] The characterization of globalness based on the delocalization power reveals that thereare two classes of globalness for bipartite unitary operations, one class is a local unitaryequivalent of a controlled-unitary operations, and the other class is all the rest of globalunitary operations.
Theorem 2
For any given two-qubit controlled-unitary operation, its entanglementcost for entanglement assisted deterministic LOCC implementation is 1 ebit when theSchmidt number, the number of non-zero Schmidt coefficients, of the resource state is 2.[Soeda et al.
For other two-qubit unitary operations, the entanglement cost of the swap operation U SWAP , of which action is given by U SWAP | ψ A i A ⊗ | ψ B i B = | ψ B i A ⊗ | ψ A i B for any | ψ A i and | ψ B i , is easily shown to be 2 ebit by considering the situation where the two inputqubits are parts of maximally entangled states. However, for more general operations, itis not easy to evaluate the minimum entanglement cost of entanglement assisted LOCCimplementation, therefore, it is still unknown.The formulation of entangling power depends on the set of allowed input states andthe measure of entanglement. Entangling power is usually difficult to calculate becauseit involves two optimizations. One is the maximization over all possible input states(usually taken to be separable or product states). The other is the calculation of theamount of generated entanglement according to the chosen entanglement measure. Evenwhen the quantum operation is restricted to bipartite unitary operations, the exact valueis obtained for only limited cases [Kraus and Cirac 2001, Chefles 2005]. For example, itis known that the CNOT operation C X ( C { u k } where u = I and u is given by the Paulimatrix X ) has the entangling power of 1 ebit and the swap operation U SWAP has theentangling power of 2 ebit when we allow to use ancilla qubits.Nevertheless, we can make a relatively generic statement about entangling power if theentanglement measure is continuous. The statement is as follows. The identity operationclearly generates no entanglement at all, hence its entangling power should be zero. In-voking a continuity argument, there should be a set of operations in the neighborhood ofthe identity operation such that their entangling power is arbitrarily small.However, when we evaluate the globalness in terms of the delocalization power andentanglement cost, a fundamental difference arrises. By using these two characterizations,all two-qubit controlled-unitary operations of the form C u = | i h | ⊗ I + | i h | ⊗ u, (1)where u is a single qubit unitary operation, belong to the same class of globalness irrelevantto their entangling power. Thus, two-qubit controlled-unitary operations and their localunitary equivalent operations belong to a distinct class from the the class of identityoperation, even when the local unitary operations are close to the identity ( u ≈ I ), incontrast to the characterization in terms of entangling power.On the other hand, for general known bipartite pure qudit states, the LOCC convert-ibility condition between two states is known. Theorem 3
A bipartite state | Ψ i AB can be transformed to another state | Ψ ′ i AB by usingonly LOCC, if and only if the Schmidt coefficients of the | Ψ i AB is majorized by those of | Ψ ′ i AB [Nielsen 1999]. Since the LOCC conversion protocol depends on the choice of states | Ψ i AB and | Ψ ′ i AB (when LOCC conversion from | Ψ i AB to | Ψ ′ i AB is possible), it is essential that thesestates are known. By taking | Ψ i AB = | ψ A i A ⊗ | ψ B i B and | Ψ ′ i AB = U | Ψ i AB and usingTheorem 3, we can see that if we have a resource state of which Schmidt coefficients areequal to the those of | Ψ ′ i AB , it is possible to obtain U | Ψ i AB by LOCC. For two-qubitunitary operations, which can only create an entangled state with Schmidt number 2, themajorization condition is equivalent to the comparison of the amount of entanglement.Thus, we have the following corollary. Corollary 1
The entanglement cost of the resource state for entanglement assisted de-terministic LOCC implementation of unitary operations on a given known state | ψ A i A ⊗| ψ B i B is given by the the amount of entanglement of U | ψ A i A ⊗ | ψ B i B . This corollary gives a justification to define the entanglement cost for an entanglementassisted deterministic LOCC implementation on a set of known states by finding thelargest minimum entanglement cost to perform U over the set of input states. In thiscase, the entanglement cost for a set of known states coincides to the entangling power of U .The results in this section indicate that there are several aspects of globalness inquantum operations. It is particularly important to clarify the types of input states,known states or unknown states representing pieces of quantum information, for analyzingglobalness of unitary operations, since they lead to a fundamental difference. IV. DELOCALIZATION POWER FOR ONE PIECE OF QUANTUMINFORMATION
In the classification of globalness of unitary operations in terms of the delocalizationpower presented in the previous sections, we analyzed global properties of two pieces ofdelocalized quantum information. That is, we analyzed the globalness of unitary oper-ations totally independent of input states. On the other hand, in Section II A, we alsodefined one piece of delocalized quantum information. This situation corresponds to thecase where one of the two input qudits is in an arbitrary and unknown state, but that ofthe other qudit is in a known state, and we can choose the most suitable state for per-forming tasks. In this section, we extend our analysis on globalness of unitary operationsin terms of delocalization power to the case for one piece of quantum information.We define the task of LOCC one-piece relocalization of one piece of delocalized quantuminformation.
Definition 10
LOCC one-piece relocalization of the qudit B for one piece of quantuminformation delocalized by a global unitary operation U is a task to find an LOCC-implementable CPTP map Λ LOCC U : S ( H A ⊗ H B ) → S ( H B ) and a state | ξ A i ∈ H A satisfying Λ LOCC U [ U ( | ξ A i A h ξ A | ⊗ | ψ B i B h ψ B | ) U † ] = | ψ B i B h ψ B | for any | ψ B i B ∈ H B . We show the following lemma.
Lemma 1
The global unitary operations that allow LOCC one-piece relocalization forone piece of delocalized quantum information is in a strictly wider class of global unitaryoperations than that allows LOCC one-piece relocalization for two pieces of delocalizedquantum information.
To prove this lemma, we present an example of two-qubit unitary operations, U ex ,where LOCC one-piece relocalization is impossible for delocalized two pieces of quantuminformation, but it becomes possible if one of the qubits is promised to be in a particularpure state. Let us take the computational basis, which is an orthonormal basis of thecomposite Hilbert space H A ⊗ H B given by {| i i A ⊗ | j i B } i,j , where {| i i A } and {| j i B } areorthonormal base for H A and H B , respectively. The matrix representation of U ex in thecomputational basis is given by U ex = − . First, we show that LOCC one-piece relocalization of the qubit B for one piece ofquantum information delocalized by U ex is possible by presenting that U ex can be simulatedby a locally unitary equivalent operation to a controlled-unitary operation if qubit A isset to a particular state. We set the state | ξ A i A ∈ H A to be | + i = ( | i + | i ) / √
2. It iseasy to check that for an arbitrary | ψ B i B , U ex ( | + i A ⊗ | ψ B i B ) = ( H ⊗ I ) · C X ( | + i A ⊗ | ψ B i B )where C X is a controlled-NOT operation and H denotes the single-qubit Hadamard op-eration represented in the computational basis by H = 1 √ (cid:18) − (cid:19) . (The same calculation can be done using the stabilizer formalism [Gottesman 1997] byexploiting the fact that U ex is a Clifford operation.) Thus the action of U ex can besimulated by ( H ⊗ I ) · C X , a locally unitary equivalent operation to the controlled-NOToperation, when we fix one of the qubits to be in the state | + i A .From Theorem 1, any operation which is locally unitary equivalent to a controlled-unitary operation is LOCC one-piece relocalizable for two pieces of delocalized quantuminformation. Note that, if an LOCC protocol relocalizes two pieces of delocalized quantuminformation, the same protocol must also relocalize one piece of delocalized quantuminformation. Therefore, U ex is LOCC one-piece relocalizable for one piece of delocalizedquantum information.Next, we show that U ex itself is not locally unitary equivalent to controlled-unitaryoperations, therefore it is not LOCC one-piece relocalizable for two pieces of delocalizedquantum information. To show this, we analyze the Cartan numbers for two-qubit unitaryoperations.It is known that any two-qubit unitary operator on H A ⊗ H B has the following Cartandecomposition [Kraus and Cirac 2001], u A ⊗ u B · exp[ i ( γ X X A ⊗ X B + γ Y Y A ⊗ Y B + γ Z Z A ⊗ Z B )] · v A ⊗ v B , by taking appropriate local unitary operations u A , v A on H A and u B , v B on H B , co-efficients 0 ≤ γ k ≤ π/ k = X, Y, Z ), where X , Y and Z denote the Pauli matricesand each subscript of the Pauli matrix indicates the corresponding Hilbert space. In thisdecomposition, the nonlocal component of the unitary operation is represented by theset of coefficients { γ k } . In this paper, we refer γ k to be a Cartan coefficient, and thenumber of non-zero Cartan coefficients to be the Cartan number. The Cartan numberof a unitary operation cannot be changed by local unitary operations, and two unitaryoperations with different Cartan numbers cannot be locally unitary equivalent to eachother [Nielsen et al. U ex is given by U ex = u A ⊗ u B · exp[ iπ/ X A ⊗ X B + Y A ⊗ Y B )] · v A ⊗ v B by using appropriate local unitary operators u A , v A , u B and v B [Anders et al. U ex is 2. On the other hand, the Cartan decomposition ofa controlled phase operation C S θ = | i h | ⊗ I + | i h | ⊗ S θ where the single-qubit phaseoperation S θ is defined by S θ = | i h | + e iθ | i h | is given by C S θ = e − iθ/ S θ/ ⊗ S θ/ · exp [ iθ/ Z A ⊗ Z B )] . Thus, the Cartan number of the controlled-phase operation is 1. It is also known thatany controlled-unitary operations C u is locally unitary equivalent to C S θ , therefore, theCartan number of the operations that are locally unitary equivalent to controlled-unitaryoperations is also 1. Therefore, U ex cannot be locally unitary equivalent to the controlled-unitary operations.Note that some unitary operations remain LOCC one-piece un relocalizable even for onepiece of delocalized quantum information. Such an example is the swap operation U SWAP .By performing U SWAP , even if one of the input qubit states is fixed to a particular knownstate, a piece of quantum information represented by the other qubit’s unknown inputstate completely moves out from the original Hilbert space and stored in the other Hilbertspace. This phenomena is an example of what we call a relocation of a piece of quantuminformation. Once this relocation happens, it is not possible to relocalize the piece ofquantum information back to the original Hilbert space by LOCC alone. It requires 1ebit of entanglement to relocalize the one piece of relocated quantum information on topof LOCC by using quantum teleportation [Bennett et al.
V. RELOCATION OF QUANTUM INFORMATIONA. LOCC one-piece relocation
In the previous section, we briefly introduced the concept of relocation of a piece ofquantum information. But actually, U SWAP provides relocation of both two pieces of quan-tum information. U SWAP is the only unitary operation that has the delocalization powerstrong enough to relocate two pieces of quantum information simultaneously without anyadditional operation or resource. A wider class of unitary operations, namely, the localunitary equivalents of U SWAP , also relocates two pieces of quantum information, if localoperations are allowed as an extra operation. To define and classify delocalization powerof unitary operations in terms of relocation, we further relax the condition of the addi-tional operations to LOCC and investigate LOCC relocatability of one of the two piecesof quantum information delocalized by unitary operations.
Definition 11
LOCC one-piece relocation from H A to H B for two pieces of quantum in-formation delocalized by a unitary operation U is the task to find an LOCC-implementableCPTP map Λ LOCC U : S ( H A ⊗ H B ) → S ( H B ) satisfying Λ LOCC U [ U ( | ψ A i A h ψ A | ⊗ | ψ B i B h ψ B | ) U † ] = | ψ A i B h ψ A | (2) for any | ψ i A ∈ H A and | ψ i B ∈ H B . This is a task similar to LOCC one-piece relocalization for two pieces of quantuminformation, in the sense by sacrificing one of two pieces of quantum information, weobtain one piece of localized quantum information. The difference between these tasksis the location of the piece of localized quantum information. We define the order of thedelocalization power of two global unitary operations U and U ′ on two pieces of quantuminformation in terms of LOCC one-piece relocation by the following. Definition 12
If LOCC one-piece relocation of two pieces of delocalized quantum infor-mation is possible for a unitary operation U , but not possible for another unitary operation U ′ , the order of the delocalization power of U is defined to be larger than that of U ′ interms of LOCC one-piece relocation. Note that for LOCC one-piece relocation, feasibility of the task implies more delocalizationpower, whereas for LOCC one-piece relocalization, feasibility of the task implies lessdelocalization power.0As the first step to classify the delocalization power of global unitary operations interms of LOCC one-piece relocation, we show that where the locally unitary equivalentclass of controlled-unitary operations lies in this classification.
Lemma 2
If two pieces of quantum information are delocalized by an operation locallyunitary equivalent to controlled-unitary operations, LOCC one-piece relocation is not pos-sible.
To prove this lemma, we employ the formulation of LOCC using accumulated oper-ators [Soeda and Murao 2010, Soeda et al.
B. Formulation of LOCC using accumulated operators
We adopt the standard formulation of LOCC [Donald et al. { M ( r ) } satisfying the completeness relation P r M ( r ) † M ( r ) = I . There exists one operator for each outcome in the measurement, which is denoted bythe superscript r .We add a subscript to the outcome index, for example r k , to specify to which measure-ment operation the index belongs. In this notation, r k belongs to the k -th measurementoperation in the sequence. We use ~R k = ( r , r , . . . , r k ) to denote the set of measurementoutcomes of the first k measurement operations in the sequence. The ( k + 1)-th mea-surement operation is a function of ~R k and we denote the set of operators describing thismeasurement operation by { M ( r k +1 | ~R k ) } r k +1 . Let us denote Alice’s measurement operations by M ( r n | ~R n − ) A and Bob’s by M ( r n | ~R n − ) B .We set M ( r | ~R ) A = M ( r ) A and M ( r | ~R ) B = M ( r ) B . Note that M ( r n | ~R n − ) A is an operator on H A and M ( r n | ~R n − ) B is on H B . When n -th turn is Alice’s turn then ( n + 1)-th turn is Bob’sturn, which implies that Alice does not perform any operation during this ( n + 1)-thturn. In this case, we set Alice’s measurement operation to the identity operation, i.e. , { M ( r n +1 | ~R n ) A } = { I } . If this ( n + 1)-th turn happens to be Bob’s, then his measurementoperation is set to the identity operation.The effect of the measurement operations accumulates as an LOCC protocol proceeds.The accumulated effect up to a particular turn is expressed by the product of all themeasurement operators corresponding to all the measurement outcomes obtained up tothat point. Given a particular sequence of measurement outcomes ~R n , we represent theaccumulated effect corresponding to this sequence by an accumulated operator A ~R n definedby A ~R n = n Y k =1 M ( r k | ~R k − ) A . B ~R n defined by a similar way of A ~R n . C. Impossibility of relocation
Let us focus on two-qubit controlled-unitary operations for simplicity. The followingargument can be extended to arbitrary two- qudit controlled-unitary operations. We proveby contradiction that LOCC one-piece relocation of two pieces of quantum informationdelocalized by any controlled-unitary operation is impossible. Now, consider the followingscenario where Alice has an extra ancilla qubit, whose Hilbert space is denoted by H a .Let | Φ i Aa denote a maximally entangled state between Alice’s input qubit and the ancillaqubit defined by | Φ i Aa = 1 √ | i A ⊗ | i a + | i A ⊗ | i a ) ∈ H A ⊗ H a . Suppose that there is an LOCC one-piece relocation protocol for the given controlled-unitary operation C u defined by Eq. (1). Let Alice set her input qubit and the ancilla inthe state of | Φ i Aa while Bob’s input remains arbitrary. Alice and Bob perform C u andthe LOCC protocol to complete the relocation of one piece of quantum information from H A to H B . Then Alice’s ancilla qubit and Bob’s input qubit are in the state of | Φ i aB = 1 √ | i a ⊗ | i B + | i a ⊗ | i B ) ∈ H a ⊗ H B , which implies that Λ LOCC C u ( C u | Φ i Aa h Φ | ⊗ | ψ B i h ψ B | C † u ) = | Φ i aB h Φ | holds for an arbitrary | ψ B i ∈ H B . Using the accumulated operator representation ofΛ LOCC C u , we haveTr A [ X ~R n ( A ~R n ⊗ B ~R n )( C u | Φ i Aa h Φ | ⊗ | ψ B i h ψ B | C † u )( A ~R n ⊗ B ~R n ) † ] = | Φ i aB h Φ | . (3)We modify the LOCC protocol Λ LOCC C u by adding an extra measurement operation byAlice described by {| i A h | , | i A h |} , just after Alice’s final measurement. We denotethis modified protocol by Λ ′ LOCC C u . Direct substitution reveals thatΛ ′ LOCC C u ( C u | Φ i Aa h Φ | ⊗ | ψ B i h ψ B | C † u )= Tr A [ X ~R n ( | i A h | A ~R n ⊗ B ~R n )( C u | Φ i Aa h Φ | ⊗ | ψ B i h ψ B | C † u )( | i A h | A ~R n ⊗ B ~R n ) † ]+ Tr A [ X ~R n ( | i A h | A ~R n ⊗ B ~R n )( C u | Φ i Aa h Φ | ⊗ | ψ B i h ψ B | C † u )( | i A h | A ~R n ⊗ B ~R n ) † ] . Since the partial trace Tr A is taken and the additional measurement introduced for theprotocol Λ ′ LOCC C u acts only on H A , we haveΛ ′ LOCC C u ( C u | Φ i Aa h Φ | ⊗ | ψ B i h ψ B | C † u ) = Λ LOCC C u ( C u | Φ i Aa h Φ | ⊗ | ψ B i h ψ B | C † u ) . Thus we obtain Λ ′ LOCC C u ( C u | Φ i Aa h Φ | ⊗ | ψ B i h ψ B | C † u ) = | Φ i aB h Φ | . A [( | i A h k | A ~R n ⊗ B ~R n )( C u | Φ i Aa h Φ | ⊗ | ψ B i h ψ B | C † u )( | i A h k | A ~R n ⊗ B ~R n ) † ]= p ~R n ,k,ψ B | Φ i aB h Φ | (4)for all ~R n and k = 0 ,
1, where p ~R n ,k,ψ B is a positive coefficient normalized by X ~R n ,k p ~R n ,k,ψ B = 1 . Since Eq. (4) holds for any | ψ B i ∈ H B , we can replace | ψ B i by a completely mixedstate I /
2, and obtainTr A [( | i A h k | A ~R n ⊗ B ~R n )( C u · | Φ i Aa h Φ | ⊗ I · C † u )( | i A h k | A ~R n ⊗ B ~R n ) † ]= 12 Tr A [( | i A h k | A ~R n ⊗ B ~R n )( C u | Φ i Aa h Φ | ⊗ | i h | C † u )( | i A h k | A ~R n ⊗ B ~R n ) † ]+ 12 Tr A [( | i A h k | A ~R n ⊗ B ~R n )( C u | Φ i Aa h Φ | ⊗ | i h | C † u )( | i A h k | A ~R n ⊗ B ~R n ) † ]= ( p ~R n ,k, + p ~R n ,k, ) | Φ i aB h Φ | . (5)Note that | i A h k | A ~R n acts only on Alice’s input qubit. Taking the partial trace overAlice’s ancilla qubit Tr a , Eq. (5) givesTr A [( | i A h k | A ~R n ⊗ B ~R n )( C u · I ⊗ I · C † u )( | i A h k | A ~R n ⊗ B ~R n ) † ] = ( p ~R n ,k, + p ~R n ,k, )2 I , where we have used the relation Tr a | Φ i aA h Φ | = Tr a | Φ i aB h Φ | = I / . Noting that theidentity operator commutes with any unitary operators, after performing the partial traceTr A , we have A h k | A ~R n A ~R n † | k i A B ~R n B ~R n † = ( p ~R n ,k, + p ~R n ,k, )2 I . This equation guarantees that Bob’s accumulated operator B ~R n for each sequence ofmeasurement outcomes ~R n is proportional to a unitary operator, i.e. , B ~R n = c ~R n u ~R n , (6)where the coefficient c ~R n is set to satisfy( c ~R n ) = ( p ~R n ,k, + p ~R n ,k, )2 A h k | A ~R n A ~R n † | k i A . For any linear operator T on H A and the maximally entangled state given by | Φ i Aa ,( T ⊗ I ) | Φ i Aa = ( I ⊗ t T ) | Φ i Aa , where t T denotes the transpose of T in the computationalbasis, holds. Let { S ( i ) x } denote a set of operators forming a basis for the operators on H x (where x = a, A, or B ). That is, for any T on H x , there exists a set of complex numbers c ( i ) such that T = P i c ( i ) S ( i ) x . (An example of such a basis is the set of Pauli operators3and the identity operator, if the Hilbert space in question has the dimension of 2.) Withthis basis, C u on H A ⊗ H B can be expressed as a linear combination of S ( i ) A ⊗ S ( j ) B , namely, C u = X i,j u ij S ( i ) A ⊗ S ( j ) B , where u ij denotes the coefficient of S ( i ) A ⊗ S ( j ) B . Let us choose S ( i ) a to satisfy S ( i ) a = t S ( i ) A in the computational basis and define ˜ C u on H a ⊗ H B by˜ C u = X i,j u ij S ( i ) a ⊗ S ( j ) B . Under these conventions, we have( | i A h k | A ~R n ⊗ B ~R n )( C u | Φ i Aa h Φ | ⊗ | ψ B i h ψ B | C † u )( | i A h k | A ~R n ⊗ B ~R n ) † = | i A h | ⊗ ˜ C ut A ~R n | k i a h k | a A ~R n ∗ ⊗ B ~R n | ψ B i h ψ B | B ~R n † ˜ C † u . Comparing this equation to Eq. (4), it must be that˜ C ut A ~R n | k i a h k | a A ~R n ∗ ⊗ B ~R n | ψ B i h ψ B | B ~R n † ˜ C † u = p ~R n ,k,ψ B | Φ i aB h Φ | , (7)which is equivalent to˜ C ut A ~R n | k i a ⊗ B ~R n | ψ B i = exp( iθ ~R n ,k,ψ B ) q p ~R n ,k,ψ B | Φ i aB . Let an ancilla state (not necessarily normalized) (cid:12)(cid:12)(cid:12) v ~R n ,k E be defined by (cid:12)(cid:12)(cid:12) v ~R n ,k E = t A ~R n | k i a . By substituting Eq. (6) into Eq. (7), we conclude that˜ C u · ( I ⊗ u ~R n ) (cid:12)(cid:12)(cid:12) v ~R n ,k E ⊗ | ψ B i = exp( iθ ~R n ,k,ψ B ) q p ~R n ,k,ψ B /c ~R n | Φ i aB holds for all | ψ B i . The right hand side is collinear to | Φ i aB for all | ψ B i . On the otherhand, because ˜ C u · ( I ⊗ u ~R n ) is invertible, the left hand side returns linearly independentvectors when {| ψ B i} are chosen linearly independently. This, however, is a contradictionproving that the assumption that LOCC one-piece relocalization is possible for two piecesof quantum information delocalized by the controlled-unitary operations C u must nothold.This proof strongly depends on the fact that Bob’s input state is kept arbitrary, namely,we considered the situation of delocalized two pieces of quantum information. Indeed,if we are allowed to choose Bob’s input state, one-piece relocation is possible for certaincontrolled-unitary operations. An example is the controlled-NOT operation on two qubits.4 VI. CONCLUSION AND DISCUSSION
In this paper, we first introduced the concept of pieces of quantum information andreviewed three different characterizations of the globalness of bipartite unitary operations,which were delocalization power, entanglement cost, and entangling power. The first twocharacterizations are on the globalness of the unitary operations on two pieces of quantuminformation represented by unknown states, and the last one is on the globalness of theunitary operations on a set of known states. We showed the fundamental differencebetween these two types of globalness of the unitary operations.Next, we extended our analysis on characterization in terms of the delocalization powerby introducing a new LOCC task, LOCC one-piece relocalization of one piece of quan-tum information delocalized by a unitary operation. We showed that there are unitaryoperations which belong to a higher globalness class in terms of the delocalization powerthan the local unitary equivalents of controlled-unitary operations, and such operationscan be further divided into two subclasses depending on the possibility of this task.We also introduced another new task, LOCC one-piece relocation of two pieces of de-localized quantum information. We proved that LOCC one-piece relocation is impossiblefor any controlled-unitary operations. This confirms that the local unitary equivalents ofcontrolled-unitary operations, which are LOCC one-piece relocalizable, belong to a classof global operations with relatively weak globalness also in terms of LOCC relocation ofquantum information.In our analysis of the LOCC tasks, we focused on the LOCC tasks that transform twopieces of quantum information within the two-qudit Hilbert space. This is because ourmain purpose is to investigate the delocalization power of two-qudit unitary operations.But in general, we can investigate more general properties of delocalized pieces of quantuminformation by considering LOCC tasks that transforms n pieces of quantum informationdelocalized in an m -qudit subspace of a totally M -qudit Hilbert space ( n ≤ m and m ≤ M ) to n ′ pieces of quantum information delocalized in an m ′ -qudit subspace ( n ′ ≤ n and n ′ ≤ m ′ ), by limiting the allowed operations to be LOCC for a certain devision of thetotal Hilbert space.Self-teleportation [Matsumoto 2007] can be interpreted as a special case of this gener-alized LOCC task for n = n ′ = 2, m = m ′ = 2 and M = 3. By denoting the total Hilbertspace by H A ⊗ H B ⊗ H C , only LOCC is allowed between the division of H A and H B ⊗ H C in this case. It is shown that asymptotically, k -copies of any delocalized two pieces ofquantum information in H A ⊗ H B can be approximately ‘relocated’ to H B ⊗ H C . Theerror probability of this relocation drops exponentially with the number of copies k aslong as the two pieces of quantum information are delocalized, namely, not in a productstate.In our analysis of the delocalization power, we characterized the order of delocalizationpower of unitary operations by their ability allowing the LOCC tasks. For more quantita-tive analysis of the unitary operations that do not allow the LOCC tasks, it is importantto analyze the entanglement cost of the corresponding entanglement assisted versions ofthe LOCC relocalization/relocation tasks.Quantum state merging [Horodecki et al. et al. n = n ′ = 3, m = m ′ = 3 and M = 4 ( H A ⊗ H B ⊗ H C ⊗H R ), where only LOCC is allowed between the division of H A and H B ⊗ H C and no oper-ation is allowed on H R . This is an entanglement assisted LOCC task to achieve relocationof three pieces of quantum information delocalized in H A ⊗H B ⊗H R to H B ⊗H C ⊗H R . Inthe asymptotic limit, it is shown that the entanglement cost coincides with the quantum5conditional entropy, which provides an operational interpretation of the quantum condi-tional entropy. To understand information theoretical meanings of our LOCC tasks, it isinteresting to analyze asymptotic settings of our LOCC and entanglement assisted LOCCtasks. We leave these investigations for future works. ACKNOWLEDGMENTS
This work is supported by the Special Coordination Funds for Promoting Science andTechnology, Institute for Nano Quantum Information Electronics, and by Global COEProgram “the Physical Sciences Frontier”, MEXT, Japan. [Anders et al.
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