Weak measurement and rapid state reduction in bipartite quantum systems

In this paper we consider feedback control algorithms for the rapid purification of a bipartite state consisting of two qubits, when the observer has access to only one of the qubits. We show 1) that the algorithm that maximizes the average purification rate is not the same as that that for a single qubit, and 2) that it is always possible to construct an algorithm that generates a deterministic rate of purification for {\em both} qubits. We also reveal a key difference between projective and continuous measurements with regard to state-purification.

Although quantum measurements are often treated as instantaneous, in practice measurement timescales can be significant when compared to other relevant timescales. Recent experimental advances have meant that it is now possible to observe continuous measurements of individual quantum systems. The measurement of an observable is not instantaneous, but takes place over a period of time [1,2,3,4]. This type of continuous measurement can be modeled by considering a series of projective measurements on an auxiliary system that is weakly coupled to the system of interest. The auxiliary system, an environmental degree of freedom, is then averaged out. This produces a continuous measurement record which contains information about the evolution of the quantum system of interest. The measurement record is then used to construct a best estimate of the underlying evolution -which is referred to as an 'unraveling' of the master equation for the system [7,5,6,8,9].
When continuous weak measurement is applied, it is possible to modify the evolution via Hamiltonian feedback, where the Hamiltonian applied to the system depends on the measurement record [7,8,9,10,11,12,13,14,15]. Hamiltonian feedback during measurement not only affects the final state of the system, but it can also also affect the rate of state reduction. In a protocol described by Jacobs, the average rate of state reduction (as measured by the purity of an initially mixed state) for a single qubit is maximized [16]. This process is known as rapid state reduction [15], or as rapid purification [16]. Jacobs' protocol is deterministic, but other protocols exist which are stochastic and minimize the average time for a single qubit to reach a given purity [17].
In this paper we consider the analogous situation of performing rapid state reduction in a two qubit system (shown in Fig. 1). There are two parties, identified as Alice and Bob, who may be separated and are not required to communicate. One observer (Alice) has access to one qubit, which she can measure and manipulate using local Hamiltonian feedback. She does not have access to the second qubit, which is controlled by Bob. This corresponds to a physically realistic situation in which Bob's qubit is either spatially separated from Alice, or -for architectural reasons -it is not possible to measure Bob's qubit directly. Two qubit systems are important because they form the basis for many current applications in quantum information processing, and are the simplest system which exhibits entanglement.
In this paper we highlight three aspects of rapid state reduction for bipartite qubit states: (1) We highlight a key difference between projective measurements and weak measurement. (2) We show that the measurement which provides the maximum rate of state reduction for the unprobed qubit in a two qubit system is not necessarily the same as that for either of the known optimal single qubit protocols [16,17]. (3) We show that it is always possible to purify both qubits deterministically at the same time.
The bipartite system which Alice and Bob share is described by the density matrix ρ. Without loss of generality, Alice's system undergoes a constant weak measurement, giving a measurement record, r(t). It is natural to expand ρ in the two-qubit Pauli The bipartite system they share is described by the density matrix ρ. Alice can make weak measurements on her system, but not Bob's.

basis,
where r ij are real (since ρ is Hermitian) and lie between −1 and 1, σ j are each of the three Pauli matrices and the identity. Each r ij can be found, r ij = Tr ρσ i ⊗ σ j . The stochastic master equation (SME) which governs the evolution of the density operator ρ in the presence of a weak measurement of a Hermitian observable, y, is given by Here k is the measurement strength. The first term in this equation describes the familiar drift towards the measurement axis. The second term in the equation is weighted by dW , a Wiener increment with dW 2 = dt. This term describes the update of knowledge of the density matrix conditioned on the measurement record [14]. Throughout this paper we will consider measurements on Alice's qubit alone. A measurement of Alice's qubit along a given axisn means that y is given by Here, X, Y and Z are the Pauli matrices, and I is the identity. The tensor product is implied. The measurement direction,n may change at each timestep. This is equivalent to the application of single qubit Hamiltonian feedback to Alice's qubit, except that the measurement axis rotates, rather than Alice's Bloch vector. Consider the evolution according to the SME as Alice's qubit undergoes weak measurement along then axis. The SME (Eqn. (2)) can be expressed in terms of the real Pauli coefficients of the density matrix. Ifn,m 1 andm 2 are mutually orthogonal unit vectors, then the corresponding stochastic master equation is given by for m = {m 1 ,m 2 } and j = {X, Y, Z, I}. Here, r nj is given by and similarly form. This is a system of 16 stochastic differential equations.
In this paper, we will be particularly concerned with the evolution of both Alice and Bob's reduced density matrices. Alice's reduced density matrix is given by Bob's reduced density matrix can be described by a similar equation with the indices swapped.
The purity of a quantum state, ρ can be quantified by the purity, P (ρ) = Tr ρ 2 . Purity has a minimum value of 1/d where d is the dimension of ρ, and a maximum value of 1. The purity of Alice's reduced density matrix is given by where r A is the Bloch vector of Alice's reduced density matrix. A similar expression can be given for Bob's reduced density matrix. Consider the rate of state reduction of Alice's qubit as she measures on her own system. The change in average purity of Alice's reduced density matrix given in Eqn. (7) under the evolution of the SME, is given by This expression does not depend on the state of Bob's qubit, and not surprisingly, it is identical to the one qubit case. Hamiltonian feedback can be used to implement either of the known single qubit Hamiltonian protocols without modification [16,17]. Now we consider the opposite situation, when Alice would like to find out the state of Bob's qubit. Bob's qubit is not being directly measured. It is only through the correlations in the initial system, ρ, that Alice can learn the state of Bob's system. For the most effective purification, Alice and Bob's share an known initial entangled state.
The average change in purity (according to Alice) of Bob's qubit when measured along then-axis is given by which can be written much more easily by identifying ∆R n = r nj − r nI r Ij .
∆R n is a vector with three components, one for each of j = {X, Y, Z}. Then the change in average purity of Bob's system We can write the nonlocal correlations as a matrix: for each i, j = {X, Y, Z}. To retrieve ∆R n from C we multiply by the corresponding unit vector,n. That is, The magnitude square of R n gives the rate of state reduction of Bob's qubit, as seen by Alice, when she makes a weak measurement along then direction. We now consider how to maximize this rate, in a direct analogy to the optimizing strategy used by Jacobs for a single qubit [16]. In particular we find the largest average increase in purity of Bob's reduced density matrix. Whilst there are some circumstances in which this strategy will not give a globally optimal solution, numerical simulations verify its use in this application. The average increase in purity of Bob's reduced density matrix is proportional to |∆R n | 2 . Expressing this in terms of the matrix, C, we wish to find The maximum value is given by the largest singular value, σ 2 1 , of C, therefore giving a maximum rate increase in purity of 4kσ 2 1 . The value ofn which corresponds to this maximum rate of state reduction is given by the first right singular vector, v 1 , of C.
Point (1) of this paper contrasts weak measurement and projective measurement, using a specific example. The measurement which gives the greatest increase in purity of Bob's qubit for a projective measurement is not the same as the measurement which gives the greatest rate of increase in purity of Bob's qubit for a weak measurement.
Consider the state, where −1 ≤ β ≤ 1. |ψ is the coherent superposition, with both qubits aligned in the z-basis. If β = 0 this is a maximally entangled state, and is entangled unless β = ±1. If the system of interest undergoes a dephasing process, as is common in many quantum systems, then the off diagonal terms of the density matrix decay. If the amount of dephasing is characterized by δ then the density matrix becomes where γ = √ 1 − β 2 − δ. The maximum purity from a projective measurement is when Alice measures along the z-axis. Any projective measurement by Alice along the z-axis will project Bob's state into a pure state (either |0 or |1 ), with purity, P B = 1. As Alice rotates the measurement axis, the purity of Bob's state reduces. The matrix C is  Figure 2. Purification rate of ρ T , with β = 0.5, due to weak measurement of Alice's qubit for every orientation of the measurement axis after a dephasing of δ = 0.01. φ is the zenith angle, and θ is the azimuthal angle.
This matrix is diagonal. For small values of dephasing, δ, the maximum rate of state reduction occurs in the xy-plane, as is shown in Fig. 2. Therefore the projective measurement which gives greatest purity is different from the weak measurement which gives the greatest rate of increase in purity.
Point (2) of this paper contrasts weak measurement for bipartite systems, and existing single qubit protocols. Naively, one might expect that the best way for Alice to purify Bob's system is to purify her own system fastest by applying known single qubit protocols to her own qubit. We show, by giving a specific counter-example, that this is not the case.
Consider the state, This state is perfectly aligned for Jacobs' one qubit feedback protocol; Alice's reduced density matrix lies in the xy-plane [16]. However, for the state ρ t , the matrix C is given by which has a single non-zero singular value of σ 1 = 1/ √ 5, and a corresponding vector of n 1 = 1 √ 2 (x +ẑ). We therefore expect the fastest rate of state reduction for Bob's qubit occurs when Alice measures her own system at 45 • to the z-axis and to the x-axis, as shown in Fig. 3. The numerical values for this plot were k = 0.1, dt = 0.1 and N = 10, 000 repetitions. This measurement does not correspond to either the Jacobs' single qubit scheme [16], or the scheme proposed by Wiseman and Ralph [17]. Purifying the unprobed qubit is dependant on the correlations and the nature of those correlations. As we show here, it is purified fastest by choosing a measurement axis to make use of those correlations, and not by simply using a one qubit protocol.
Point (3) of this paper is that it is possible to make bipartite state reduction deterministic. Increasing the purity of systems deterministically is a desirable property. From a theoretical standpoint it makes the equations easier, but the main advantages are practical. If the evolution of purity is deterministic, then each qubit is guaranteed to reach a set target purity in a given time. When weak measurement is being used for state preparation in a multiple qubit system, they will all reach the target purity together.
The condition required for the purity of Alice's reduced density matrix to evolve deterministically is that That is, the measurement should be taken in the plane orthogonal to the direction described by Alice's reduced density matrix. The condition for Bob's system to be deterministic is clear from Eqn. (11). It is given by where r ′ B = C T r E . Therefore for the evolution of the purity of Bob's reduced density matrix to be deterministic, the measurement axis must be chosen in a plane orthogonal to r ′ B . This plane is spanned byp 1 andp 2 . Deterministic state reduction of Bob's qubit is desirable, but we would still like to find the maximum rate of state reduction. Although the measurement axis,n can be chosen anywhere in the plane, not all orientations ofn will purify Bob's reduced density matrix at the same rate. The maximum deterministic rate of state reduction of Bob's qubit is found by taking the singular value decomposition of CP p , where P p is given by The fastest rate of average deterministic purification of Bob's reduced density matrix is given by 4kσ 2 1 where σ 1 is the largest singular value of the product CP p , and the axis of measurement is given byn det = v 1 , the first right singular vector of CP p .
It is possible to choose a weak measurement which Alice can make on her qubit so that the purity of both her reduced density matrix, and also Bob's reduced density matrix both evolve deterministically. If the measurement axis is chosen to be then the evolution of the purity of both Alice and Bob's reduced density matrices is deterministic.
In this paper we investigated the effect of weak measurements on a bipartite system consisting of two qubits -Alice's qubit and Bob's qubit. We allowed measurement on Alice's qubit alone. We gave expressions for the rate of state reduction of both systems, based on the measurement record. We have shown how to maximize the average rate of state reduction of either system, and how to achieve deterministic evolution of the purity. We have demonstrated that weak measurement of two qubits can be very different from both projective measurements and the weak measurement of a single qubit. Many interesting effects occur in bipartite quantum systems under measurement.