Separability Criterion for Quantum Effects

Separability Criterion for Quantum Eﬀects Ikko Hamamura1 ∗

1Department of Nuclear Engineering, Kyoto University, 6158540 Kyoto, Japan

Abstract. Entanglement of quantum states is one of the most important concepts in modern quantum science and technology. The existence of entanglement is shown by the violation of the famous Bell–Clauser–Horne–Shimony–Holt (Bell–CHSH) inequality. On the other hand, entanglement of the measurement of observable can also be considered. An effect, in the title, represents a binary observable. In this poster presentation, we will discuss the definition of the entanglement of observables, the detection way of the effect using a dual version of the Bell–CHSH inequality, and the application of our inequality to quantum teleportation.

Keywords: Bell–CHSH inequality, Entanglement of observables, IBM Quantum Experience

1 Introduction separable. An observable is an effect valued measure (corresponding to positive operator valued mea- Entanglement of quantum states is one of the sure). A separable observable is a separable effect most important concepts in modern quantum sci- valued measure. An entangled observable is defined ence and technology. To detect the entanglement as an observable which is not separable. of states, several ways have been proposed includ- In the poster, we discuss the entanglement of ob- ing the Bell–Clauser–Horne–Shimony–Holt (Bell– servables. The detection way of the entanglement is CHSH) inequality [1, 2]. Alice and Bob share a discussed in section 2. Here, the dual version of the quantum state ρ. After that, Alice makes a mea- Bell–CHSH inequality is proposed. This is a sep- surement of a binary observable A or A , or Bob 0 1 arability criterion for quantum effects (binary ob- makes a measurement of a binary observable B or 0 servables). In section 3, we show the experimental B . The output values of these measurements are 1 setting and the experimental result by IBM’s cloud mapped to +1 or −1. Let B be quantum computer. In section 4, we discuss the application of our inequality to quantum teleporta- tr [ρA0 ⊗ B0] + tr [ρA1 ⊗ B0] tion. Some concluding comments are addressed in + tr [ρA ⊗ B ] − tr [ρA ⊗ B ] . (1) 0 1 1 1 section 5. If the shared state ρ are not entangled (separable), 2 Dual version of the Bell–CHSH in- then equality −2 ≤ B ≤ 2 (2) holds. This inequality is called the Bell–Clauser– Now, we discuss the detection way of the entan- Horne–Shimony–Holt (Bell–CHSH) inequality. If gled effect. We proposed the separability criterion some state and observables violate the inequal- for quantum effect, called a dual version of the Bell– ity (2), then the given state is entangled. This is CHSH inequality [3]. the detection of the entanglement of states. The inequality formulates as follows: Figure 1 On the other hand, entanglement of observables shows the setting of the inequality. Alice and Bob prepare states respectively and they make a mea- can be considered. Let X be an effect. That is, X surement with output { +1 −1 }. Let and is an operator on a Hilbert space H and 0 ≤ X ≤ 1 , M1 M−1 be the corresponding POVM elements (effects). We is satisfied. An effect X is called separable if X can be decomposed to introduce the expectation operator M defined as

X X 0 0 M = 1 × M1 + (−1) × M−1. (4) X = Ai ⊗ Bi = 1 − Aj ⊗ Bj, (3) i j For simplicity, we assume that each system is where A ,B ,A0 ,B0 are positive operators. An en- qubit system in this manuscript. That is HA = i i j j 2 tangled effect is defined as an effect which is not HB = C . The general description is shown in Ref. [3] ∗[email protected] 3 Experiment We use the cloud computing, called IBM Quan- A tum Experience, to show our inequality violates. Al- ρi ice prepare an eigenstate with an eigenvalue +1 of M 0 1 1 0 B X = ,Z = . (10) ρj 1 0 0 −1 Bob prepares an eigenstate with eigenvalue +1 of 1 1 1 1 √ Z + √ X, √ Z − √ X. (11) Figure 1: The setting of our experiment. Alice pre- 2 2 2 2 pares states ρA and ρA, and Bob prepares states ρB 0 1 0 They perform a Bell measurement as a nonlocal and ρB. After that, they make a nonlocal measure- 1 measurement. The Bell measurement is defined as ment M. We assume that this nonlocal measurement following POVM with four outcomes: M has a binary output { +1, −1 }.  1 0 0 ±1 We consider the difference from ignorance defined 1  0 0 0 0  MΦ± =   (12) as 2  0 0 0 0  1 ±1 0 0 1 tr ρ − M . (5) , 2 and The difference from ignorance for two quantum sys- 0 0 0 0 tems is defined as 1 0 1 ±1 0 M ± =   . (13) 1 1 Ψ 2 0 ±1 1 0 E(ρA, ρB, M) = 2 tr ρA − ⊗ ρB − M . 2 2 0 0 0 0 (6) Our theorem requires the condition tr [Mi] ≤ 1. This coefficient 2 is not essential, but this is neces- Therefore, we prepare the completely mixed state 1 and check the value of tr 1 . At least one sary to set the maximum value to 2 of our inequality. 2 2 Mi Mi A B satisfies the condition, so we choose it as M1 and By using effect M1, this E(ρ , ρ , M) is equal to P summarise the others like M−1 = others Mi. Af- 1 1 terthat, we check the value D in the dual Bell–CHSH 4 tr ρA − ⊗ ρB − M (7) 2 2 1 inequality. In our experiment, D is greater than 2, so the Let D be violation of the dual Bell–CHSH inequality is con- A B A B firmed. A detailed setting of our experiment can be E(ρ0 , ρ0 , M) + E(ρ0 , ρ1 , M) A B A B seen in the jupyter notebook in the Qiskit Tutori- + E(ρ1 , ρ0 , M) − E(ρ1 , ρ1 , M). (8) als [4]. In the notebook, an experiment with a sim- ulator and a cloud quantum computer is presented. Theorem 1 We assume that a binary observable M with two outcomes { +1 −1 } satisfies the condition , 4 Application to quantum teleportation tr [M1] ≤ 1 or tr [M−1]. If this observable M is separable, then the following inequality holds: In this section, we discuss an application to quantum teleportation. Quantum teleportation is a pro- −2 ≤ D ≤ 2. (9) tocol which transfers quantum states using local operations and classical communication [5]. Alice pre- This inequality is called the dual version of Bell– pares an unknown pure state and Alice and Bob CHSH inequality. The proof is shown in Ref. [3] in share a maximally entangled state. Alice performs general case. The condition tr [M ] = 1 or tr [M ] 1 −1 a measurement of an entangled observable M for her is necessary to the proof of the theorem but can states. Alice tells Bob the measurement result by be removed by using postprocessing. The detailed classical communication. Bob performs a unitary procedure is shown in Ref. [3] operation U to his state according to Alice’s result. In the next section, we check the violation of our i After this protocol Bob outputs a state ρ. In the inequality by an entangled effect. by LOCC is an open problem. Our inequality de- pends on the dimension of the Hilbert spaces. This dependency should be got rid of also from the view- point of the device independent entanglement detection.

Acknowledgements

Figure 2: The quantum circuit of quantum telepor- We acknowledge use of the IBM Quantum Expe- tation. The two upper qubits are Alice’s ones and rience for this work. The views expressed are those the other is Bob’s one. Alice has an unknown state of the authors and do not reflect the official policy or position of IBM or the IBM Quantum Experience |ψi and Alice and Bob have an entangled state Φ+. team. The author acknowledges support by Grant- Alice makes a Bell measurement M and tells Bob the measurement result. Bob performs a unitary in-Aid for JSPS Research Fellow (JP18J10310). gate according to Alice’s result. Bob’s output is a References state ρ. [1] J. S. Bell, “On the Einstein Podolsky Rosen Paradox*”, Physics 1, 195 (1964). ideal case, an output state is equal to an input state. This protocol is shown in Figure 2. [2] J. F. Clauser, M. A. Horne, A. Shimony, and Hereinafter, we assume that the shared entan- R. A. Holt, “Proposed Experiment to Test Lo- gled state is complete, but the Bell measurement cal Hidden-Variable Theories”, Phys. Rev. Lett. is incomplete. We assume an unbiased condition 23, 880 (1969). tr [Mi] = 1. For example, one rank projective mea- [3] I. Hamamura, “Separability criterion for quan- surement satisfies this condition. This condition is tum effects”, in press in PLA (2018) https:// necessary to use the violation of the dual Bell–CHSH doi.org/10.1016/j.physleta.2018.06.043. inequality. If each POVM element violates the dual [4] I. Hamamura, Entangled measurement, (2018) Bell–CHSH inequality, then the maximum value of the average fidelity between an input state and an https : / / github . com / Qiskit / qiskit - output state tutorial/blob/master/appendix/more_qis/ entangled_measurement.ipynb. Z [5] C. H. Bennett, G. Brassard, C. Crépeau, R. Fmax = max dM(φ) hφ | ρφi , (14) { Ui }i S Jozsa, A. Peres, and W. K. Wootters, “Tele- porting an unknown quantum state via dual is greater than 2 . Thus, the entangled observable 3 classical and einstein-podolsky-rosen channels”, which violates our inequality is “useful” for quantum Phys. Rev. Lett. 70, 1895 (1993). teleportation.

5 Conclusion We discuss the definition of the entanglement of observables and the separability criterion for effects (binary observables). The Bell measurement violates the dual version of the Bell–CHSH inequality and this violation is experimentally checked. An observable whose each POVM element violates the dual Bell–CHSH inequality is useful for quantum teleportation in some sense. Some future works remain. The most important and fundamental future work is giving an operational meaning to separable observables. For separable states, local operations and classical communication (LOCC) gives a clear operational meaning. However, whether all separable effects can be made