We consider the simultaneous coupling of a qubit system to two qubit probes, designed to measure noncommuting qubit observables when working in isolation. While the single meter model usually corresponds to an unsharp measurement of a qubit operator, the double meter model usually describes an informationally complete measurement. The double meter positive operator measurement can be characterized by three vectors associated with the space of Hermitian tracesless observables of the system. The dimension of the linear space generated by these vectors (Bloch rank) is equal to the number of independent operators (or the number of independent Bloch-vector components) which can be estimated. $\mathbb{S}$ , the subspace of parameters associated with informationally incomplete measurements (IICs), is characterized either analytically or numerically, and their Bloch rank is used to classify the IICs. Bloch rank of the ``simultaneous measurement'' of two Pauli matrices is shown to be 2 if the isolated measurements are projective and 3 if they are weak; however, two-meter weak measurements are expected to converge more slowly than their single-meter counterparts.We consider the simultaneous coupling of a qubit system to two qubit probes, designed to measure noncommuting qubit observables when working in isolation. While the single meter model usually corresponds to an unsharp measurement of a qubit operator, the double meter model usually describes an informationally complete measurement. The double meter positive operator measurement can be characterized by three vectors associated with the space of Hermitian tracesless observables of the system. The dimension of the linear space generated by these vectors (Bloch rank) is equal to the number of independent operators (or the number of independent Bloch-vector components) which can be estimated. $\mathbb{S}$, the subspace of parameters associated with informationally incomplete measurements (IICs), is characterized either analytically or numerically, and their Bloch rank is used to classify the IICs. Bloch rank of the ``simultaneous measurement'' of two Pauli matrices is shown to be 2 if the isolated measurements are projective and 3 if they are weak; however, two-meter weak measurements are expected to converge more slowly than their single-meter counterparts.

Physical Review AApr 30, 20194 citations

Extended validity of weak measurement*

Jiangdong Qiu, Changliang Ren, Zhaoxue Li +4 more

We introduce a modified weak value that is related to the mean value of input meter variable. With the help of the modified weak value, the validity conditions for various modified versions of weak value formalism are investigated, in terms of the dependence of the pointer shift on the mean value of the input meter. The weak value formalism, often used to represent the pointer shift, with the modified weak value is of great use in simplifying calculations and giving guidance of practical experiments whenever the mean value of the input meter variable is nonzero. The simulation in a qubit system is presented and coincident well with our theoretical result.We introduce a modified weak value that is related to the mean value of input meter variable. With the help of the modified weak value, the validity conditions for various modified versions of weak value formalism are investigated, in terms of the dependence of the pointer shift on the mean value of the input meter. The weak value formalism, often used to represent the pointer shift, with the modified weak value is of great use in simplifying calculations and giving guidance of practical experiments whenever the mean value of the input meter variable is nonzero. The simulation in a qubit system is presented and coincident well with our theoretical result.

Chinese Physics BApr 3, 20203 citations

Qubits, Operators and Measurement

Jack D. Hidary

Springer eBooksJan 1, 20210 citations

The power of a control qubit in weak measurements

Raul Coto, Víctor Montenegro, Vitalie Eremeev +2 more

In the late 80 s, a curious effect suggested by Aharanov et al. was found to lead to an anomalous amplification based on quantum measurements of weakly coupled systems. In this paper, we investigate the quantum control of the weak value amplification of a qubit system coupled to a meter. For the qubit system, the coupling occurs via a second non-interacting qubit, initially quantum correlated with the first one. We show that for weak measurements, the control can be remotely realized via the post-selected state of the second qubit or the degree of squeezing of the meter. In our study of the quantum control of the amplification, we can easily manipulate the degree of quantum correlations between the initially correlated qubits. We find that the degree of entanglement has no effect on the quantum control of the amplification. However, we also found a clear connection between the amplification and quantum discord like measurements as well as classical correlations between the qubits. Finally, we suggest an application of the amplification control on the enhancement of the quantum measurement accuracy, e.g. measuring the relative phase of the post-selected control qubit in a more precise way, as opposed to the non-amplified case.In the late 80 s, a curious effect suggested by Aharanov et al. was found to lead to an anomalous amplification based on quantum measurements of weakly coupled systems. In this paper, we investigate the quantum control of the weak value amplification of a qubit system coupled to a meter. For the qubit system, the coupling occurs via a second non-interacting qubit, initially quantum correlated with the first one. We show that for weak measurements, the control can be remotely realized via the post-selected state of the second qubit or the degree of squeezing of the meter. In our study of the quantum control of the amplification, we can easily manipulate the degree of quantum correlations between the initially correlated qubits. We find that the degree of entanglement has no effect on the quantum control of the amplification. However, we also found a clear connection between the amplification and quantum discord like measurements as well as classical correlations between the qubits. Finally, we suggest an application of the amplification control on the enhancement of the quantum measurement accuracy, e.g. measuring the relative phase of the post-selected control qubit in a more precise way, as opposed to the non-amplified case.

arXiv:1609.08936Scientific ReportsJul 19, 20178 citations

Transition between weak and strong measurements in the presence of post-selection

Shogo Hanashiro, Holger F. Hofmann

In the weak measurement regime, post-selection can result in the observation of anomalous weak values, seemingly contradicting the eigenvalue statistics observed when the measurement interaction is strong. Here, we investigate the dependence of meter statistics on measurement strength in a post-selected measurement. We find that the meter statistics in the intermediate regime between weak and strong measurements is nearly independent of measurement strength and show that, in this regime, the system performs a measurement of momentum on the meter. Our results show that post-selection is extremely sensitive to the action of the meter on the system, highlighting the difference between the weak measurement regime where this back-action is negligible and the average meter position is a valid measurement result, and the intermediate regime where the meter statistics should not be misinterpreted as a measurement outcome. The transition between weak values and eigenvalues is facilitated by a reversal of the roles of the system and the meter, where the post-selection acts as a readout of information about the meter. The intermediate measurement regime thus reveals that the apparent contradictions between anomalous weak values and eigenvalues originate from the non-classical effects of back-action on the post-selected meter statistics.In the weak measurement regime, post-selection can result in the observation of anomalous weak values, seemingly contradicting the eigenvalue statistics observed when the measurement interaction is strong. Here, we investigate the dependence of meter statistics on measurement strength in a post-selected measurement. We find that the meter statistics in the intermediate regime between weak and strong measurements is nearly independent of measurement strength and show that, in this regime, the system performs a measurement of momentum on the meter. Our results show that post-selection is extremely sensitive to the action of the meter on the system, highlighting the difference between the weak measurement regime where this back-action is negligible and the average meter position is a valid measurement result, and the intermediate regime where the meter statistics should not be misinterpreted as a measurement outcome. The transition between weak values and eigenvalues is facilitated by a reversal of the roles of the system and the meter, where the post-selection acts as a readout of information about the meter. The intermediate measurement regime thus reveals that the apparent contradictions between anomalous weak values and eigenvalues originate from the non-classical effects of back-action on the post-selected meter statistics.

quant-pharXiv:2609.04812Sep 7, 20260 citations

\mathbb{S}

, the subspace of parameters associated with informationally incomplete measurements (IICs), is characterized either analytically or numerically, and their Bloch rank is used to classify the IICs. Bloch rank of the ``simultaneous measurement'' of two Pauli matrices is shown to be 2 if the isolated measurements are projective and 3 if they are weak; however, two-meter weak measurements are expected to converge more slowly than their single-meter counterparts.