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Qubit Carriers with Internal Degrees of Freedom in a Non-Factorable State Qubit carriers with internal degrees of freedom in a non-factorable state Martina Mikov´a,Helena Fikerov´a,Ivo Straka, Michal Miˇcuda, Miroslav Jeˇzek,Miloslav Duˇsek,and Radim Filip Department of Optics, Faculty of Science, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic A directly measurable parameter quantifying effective indistinguishability of particles as a resource for quantum information transfer and processing is proposed. In contrast to commonly used overlap of quantum states of particles, defined only for a factorable states, this measure can be generally applied to any joint state of the particles. The relevance of this generalized measure for photons produced in parametric down-conversion has been experimentally verified. The simplest linear- optical quantum-state-transfer protocol, for which this measure directly determines fidelity of the transferred state, was experimentally tested. It has been found that even if some degrees of freedom of two particles are entangled, the particles can still serve as good carriers of qubits. PACS numbers: 03.67.Mn, 42.50.Ex, 03.67.Lx I. INTRODUCTION uniquely on indistinguishability of particles. We consider transfer of a state of a source single-photon qubit (S) to a single-photon target qubit (T ). The transfer is performed In last few decades quantum physics has offered novel by a partial exchange of photons, optimal measurement applications in information and communication technol- on S, and conditional feed-forward correction on T . We ogy. Their performance crucially depends on the quality show that fidelity of the transferred state depends di- of elements of quantum information { qubits [1]. Nec- rectly on D even if the internal degrees of freedom of the essary conditions for high-fidelity qubits are their coher- particles are entangled. In more complex quantum proto- ence [2, 3], which appears when no information is leak- cols, the quality of information processing may depend on ing into an environment, and indistinguishability [4{6]. a nontrivial combination of the effective indistinguisha- By effective indistinguishability of two (spatially sepa- bility and the properties of other resources. rated) particles we mean that all their internal degrees of freedom not used to carry information are identical. For factorable state ρA ⊗ ρB of two particles A and B II. OPERATIONAL MEASURE OF it means that the action of a flip (exchange) operator, INDISTINGUISHABILITY F (ρA ⊗ ρB) F = ρB ⊗ ρA, does not change the state. The mean value of the flip operator for a factorable state Let us have two particles, S and T , carrying the same equals to the overlap of states of individual subsystems, qubit states, let ρE;ST denote the state, not necessarily Tr [F (ρA ⊗ ρB)] = Tr [ρAρB] [7]. This is a hint for an in- separable, of all of the other (inaccessible) degrees of free- distinguishability measure. But these considerations are dom. The internal environmental degrees of freedom can still valid only for factorable states. Direct measurement even be entangled with an external environment. Clearly, of the overlap was already suggested for qubits [8] and they are responsible for distinguishability of the particles. harmonic oscillators [9]. For the simplest case of two pho- Let us define a measure jDj describing an effective indis- tons, it was measured by Hong-Ou-Mandel (HOM) type tinguishability, where interferometry [10]. There are many other related two- photon experiments [11{17]. Indistinguishability of par- D = Tr [FA ρE;ST ] : (1) ticles is crucial in a number of quantum protocols which were intensively studied in recent years and are still of a P Operator FA = m;n j niSh mj ⊗ j miT h nj is a flip arXiv:1209.0908v2 [quant-ph] 11 Apr 2013 great interest [18{28]. operator acting on the joint environment of both par- In this paper, we propose a directly measurable param- ticles, which exchange basis states corresponding to a eter, D, quantifying effective indistinguishability of par- given observable A, where Aj ni = anj ni. Properties ticles which can be used for an arbitrary state. Effective of Tr [FA ρE;ST ] follows from the features of operator FA. y indistinguishability can be defined by means of the flip Operator FA is both hermitian FA = FA and unitary y y operator exchanging relevant degrees of freedom of the FAFA = FAFA = 1 [7]. Since FA is commuting with particles. Full flip of particles corresponds to complete any local unitary transformation UES ⊗ UET , it is invari- exchange of their quantum states. On the other hand, ant to a choice of operator A and therefore, we can con- transfer by quantum teleportation relies both on the par- sider F instead FA as a basis-independent operation. F ticle indistinguishability and entanglement. Therefore, can be also expressed as a difference, F = Πsym − Πanti, to show how distinguishability of particles used as infor- of orthogonal projectors Πsym and Πanti onto the sym- mation carriers affects quantum information processing metric and anti-symmetric subspace of the total space, without the influence of other imperfections of resources, respectively. Therefore it is directly linked to indistin- we designed a quantum-state-transfer protocol depending guishability of the environmental states. Clearly, F is a 2 dichotomic observable (with eigenvalues ±1) and we get Thus parameter jDj quantifies effective indistinguishabil- −1 ≤ Tr [F ρ] ≤ 1. Parameter D is invariant under sym- ity of resources for quantum information processing. metric local unitary transformations UES ⊗ UET (indis- tinguishability cannot change if the particles go through the same unitary channels). In contrast, entanglement of III. PHOTONIC QUBIT TRANSFER the environments have to be invariant under more gen- eral unitary transformations UES ⊗VET , where UES may To demonstrate relevance of effective indistinguishabil- differ from VET . The conceptual difference is that a lo- ity jDj, we have proposed and experimentally tested the cal unitary applied on the environment of one particle simplest example of a quantum information transfer, in can make its state distinguishable from the state of the which jDj alone directly determines quantum fidelity of other particle, although it does not change the amount the transferred states. It manifests a clear operational of correlations and entanglement between the environ- meaning of the above defined effective indistinguishabil- ments. Any symmetric state of two particles satisfying ity. ρE;ST = F ρE;ST = ρE;ST F has D = 1, irrespective of We consider only the equatorial states of qubit S repre- its entanglement. States related by the permutation op- sented by a dual-rail superposition of single photon states 0 eration, ρE;ST = F ρE;ST F , have the same values of D. P 1 For separable state ρE;ST = n pn ρn;ES ⊗ ρn;ET of two jΨiS = p [j0; 1iS + exp(iθ)j1; 0iS]; (2) particles, D is always positive semi-definite. If a twirling 2 transformation is applied to any input state ρE;ST (i.e., if identical random unitaries are applied to both qubits) where phase θ may be unknown during the transfer. This parameter D does not change and the resulting state is state should be transferred to target qubit T represented the Werner state which is fully parameterized by D. For by another single photon, which is in state factorable states D = Tr [ρ ρ ] and it reduces to the ES ET 1 overlap of S and T particles. Notice however, that in jΦiT = p (j0; 1iT + j1; 0iT ) (3) general D is not equal to the overlap. The problem of 2 the overlap lies in the assumption that quantum states of at the beginning. All other degrees of free- all, even unaccessible, degrees of freedom of two bosons dom are described by a density matrix ρ = are factorized. Such assumption cannot be operationally E;ST P c j i h j ⊗ j i h j, where i; j; k; l are certified, except one achieves an unrealistic complete to- i;j;k;l ij;kl i ES j k ET l multi-indices over many different sub-degrees of freedom mography of a joint quantum state corresponding to all of the joint \environment" of photons and j i denote degrees of freedom of both particles. x basis states of each photon. We consider that all physi- Alternatively, the flip operator can be rewritten as cal differences between the particles are contained in this TA F = jΨ~ +iST hΨ~ +j representing an entanglement wit- environmental state. So the overall initial state of the ~ P qubits reads ρini = jΨiShΨj ⊗ jΦiT hΦj ⊗ ρE;ST . ness, where jΨ+i = k j kiSj kiT is an unnormalized In general, an imperfect interaction between qubits symmetric state. Thus D < 0 is a witness of entangle- can also limit quality of the transfer. We there- ment in ρE;ST (due to the presence of an anti-symmetric fore consider implementation without any direct in- component) [7, 9, 29]. However, we do not focus on en- teraction. To transfer a quantum state, we swap tanglement. Instead, indistinguishability is in our atten- two rails between S and T (see Fig. 1). It means, tion. basis states j0; 1iSj1; 0iT j iiESj kiET are changed Measure jDj characterizes resources, i.e. quantum sys- to j0; 1iSj1; 0iT j kiESj iiET (here the environmen- tems, information is encoded in. Among other condi- tal basis states are swapped). While basis states tions, theory of quantum information processing requires j1; 0iSj0; 1iT j iiESj kiET remain unchanged. All other all the resources to be in the same states which are de- possibilities are excluded by post-selection considering coupled from each other (their total state must be fac- only single photon in S and single photon in T . This torable). In our notation ρE;ST = ρE;S ⊗ ρE;T with conditional \partial exchange" process entangles qubits ρE;S = ρE;T .
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