Direct measurement of density-matrix elements using a phase-shifting technique

A direct measurement protocol allows reconstructing specific elements of the density matrix of a quantum state without using quantum state tomography. However, the direct measurement protocols to date are primarily based on weak or strong measurements with an ancillary pointer, which interacts with the investigated system to extract information about the specified elements. Here, we present a direct measurement scheme based on phase-shifting operations which do not need ancillary pointers. In this method, estimates of at most six expectation values of projective observables suffice to determine any specific element of an unknown quantum density matrix. A concrete quantum circuit to implement this direct measurement protocol for multiqubit states is provided, which is composed of just single-qubit gates and two multiqubit controlled-phase gates. This scheme is also extended for the direct measurement of the density matrix of continuous-variable quantum states. Our method can be used in quantum information applications where only partial information about the quantum state needs to be extracted, for example, problems such as entanglement witnessing, fidelity estimation of quantum systems, and quantum coherence estimation.

Figure 1

Direct measurement of the density matrix of a general quantum state using weak measurement on a pointer. The arrow denotes the information flow from the system to the pointer. One may directly reconstruct the nondiagonal term ${\rho }_{ij}$ from the results of the measurements on the pointer.

Figure 2

Phase-shifting measurement protocol. The general quantum circuit of the phase-shifting measurement. An unknown quantum state $\rho$ can be directly measured via a projector ${\widehat{K}}_{n,m}^{(\theta ,\phi )}$ consisting of two phase-shifting operators ${\widehat{Q}}_n\left(\theta \right)$ and ${\widehat{Q}}_m\left(\phi \right)$ along with a postselection $|+\rangle \langle +|$, where $\theta =\{0,\frac{\pi }{2},-\frac{\pi }{2}\}$ and $\phi =\{0,\pi \}$.

Figure 3

The quantum circuit of phase-shifting measurements on a two-qubit system. (a) Phase-shifting measurement of a single-photon state. One can see that each blue square represents a phase shifter, which introduces a relative phase in its corresponding mode. The phase-shifting operators ${\widehat{Q}}_n\left(\theta \right)$ and ${\widehat{Q}}_m\left(\phi \right)$ can be implemented by two phase shifters in the modes $n$ and $m$, which add relative phases $e^{i\theta }$ and $e^{i\phi }$, respectively. (b) The phase-shifting operator ${\widehat{Q}}_n\left(\theta \right)$ is decomposed to four single-qubit not gates and a two-qubit controlled-phase gate. The postselection is realized by a Hadamard gate with a measurement outcome 00 in computational basis.

Figure 4

The quantum circuit of phase-shifting measurements for multiqubit states.

Figure 5

Experimental proposal for measuring the density operator of a temporal wave function or density matrix using the sequential phase-shifting protocol. An arbitrary temporal quantum state of photons can be measured by the double phase-shifting operation by EOPM and postselected by a filter with a center frequency at ${\omega }_0$.