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Qubit POVMs by destructive weak measurements

Many quantum measurements, such as photodetection, can be destructive. In photodetection, when the detector clicks a photon has been absorbed and destroyed. Yet the lack of a click also gives information about the presence or absence of a photon. In monitoring the emission of photons from a source, one decomposes the strong measurement into a series of weak measurements, which describe the evolution of the state during the measurement process. Motivated by this example of destructive photon detection, a simple model of destructive weak measurements using qubits was studied in [1]. It has shown that the model can achieve any positive-operator valued measurement (POVM) with commuting POVM elements including projective measurements. In this paper, we use a different approach for decomposing any POVM into a series of weak measurements. The process involves three steps: randomly choose, with certain probabilities, a set of linearly independent POVM elements; perform that POVM by a series of destructive weak measurements; output the result of the series of weak measurements with certain probabilities. The probabilities of the outcomes from this process agree with those from the original POVM, and hence this model of destructive weak measurements can perform any qubit POVM.


I. INTRODUCTION
Quantum trajectory theory has been widely discussed and studied for years. Many essential aspects of its physics, such as the stochastic evolution equations arising from the interaction with the continuously measured environment leading to diffusive or jump-like behavior for the system, are well-illustrated in [2,3]. A feature worth pointing out in continuous measurements is the use of an adaptive procedure that updates the current status for the next step of measurement. In quantum optics, for example, the phase measurement of a single mode can be improved by feedback loops during the measurement process [4]. In earlier work describing a quantum measurement as a series of weak measurements [5,6], the measurement operator at each time step is updated by the next step of the weak measurement process. It is also well-known that any generalized measurement for a system is equivalent to letting the system interact with an ancilla followed by a projective measurement on the ancilla. In [7,8], it is shown that what classes of generalized measurements can be achieved by tuning the probe state or the interaction Hamiltonian during the continuous measurement process.
Since a generalized measurement is represented by a set of operators M k 's such that k M † k M k = I, each M k is naturally decomposed into a unitary multiplying by a positive semi-definite operator by the polar decomposition, i.e., M k = U k √ E k . The unitary operators U k do not affect the probabilities of getting the outcomes, but give an evolution conditional on the measurement outcome. The positive semi-definite operators E k extract the information of the POVM associated with this generalized measurement. In the light of this observation, we look at the case of "destructive" measurements where the system is projected onto a certain state after the measurement, while the probabilities of the outcomes reflect a certain POVM. A simple example of this type of measurements is the two-outcome qubit measurement consisting of M 1 = |0 +| and M 2 = |0 −|. It projects the system onto the |0 state for both outcomes while performing a POVM in the x-basis. A physical example of a destructive measurement is photodetection. In photodetection, the observer waits for photons to be absorbed by the detector. After a long monitoring time, all photons are consumed and the source system is projected onto the vacuum state. If we continuously monitor this detection process, the outcomes give a stochastic evolution for the process of photodetection.
Motivated by this example, we developed a model of destructive weak measurements for qubits in [1]. It consists of a weak unitary swap between the system and the ancilla qubit, in the initial state |0 , followed by a projective measurement on the ancilla qubit. The weak swap between the system and the |0 state is the same for each step and the cumulative effect of the swap leads to a projection onto |0 for the system at long times. Although the model is destructive, it can achieve any POVM with commuting POVM elements, including all projective measurements on a qubit. However, general POVMs can not be done by this method. In this paper, we develop another method, the random walk in a simplex, to achieve an arbitrary POVM with any number of outcomes.
The paper is structured as follows. In Sec. II, we briefly introduce the model of destructive weak measurements in [1]. In Sec. III, we present some general properties for a POVM. They are essential to prove the POVM achievability by the model of destructive weak measurement. In Sec. IV, we describe the method of random walks in arXiv:1812.00425v1 [quant-ph] 2 Dec 2018 a simplex and show how it fits our model of destructive weak measurements. In Sec. V, we show that the probabilities of getting the outcomes approach the probabilities from the POVM, as expected.

II. THE MODEL OF DESTRUCTIVE WEAK MEASUREMENTS
Recall the model of destructive weak measurements in [1]: we couple a qubit system |ψ ψ| to an ancilla in state |0 by a weak swap defined as where S is the swap operator that switches the system and the ancilla qubit, i.e., S|ψ ⊗ |0 = |0 ⊗ |ψ . The parameter φ is assumed to be a small constant. The joint system becomes after the unitary operation. We perform a POVM on the ancilla with elements of the form where The outcomeP k from the ancilla gives a measurement operator on the system, and k M † k M k = I.

III. GENERAL PROPERTIES OF POVMS
Definition 1 (LIPOVM). We say that a POVM, where E i 's are positive-semidefinite operators, and n i=1 c i E i = 0 has no nontrivial solutions for the c i 's. We denote such a POVM as a Linearly Independent POVM (LIPOVM).
A POVM, in general, can have any number of outcomes. However, it can be decomposed into a linear combination of LIPOVMs in the following sense. Proposition 1. Performing a POVM is equivalent to randomly choosing, with certain probabilities, to perform one LIPOVM from a collection of LIPOVMs.
We provide the following decomposing process to illustrate Proposition 1.
Consider a POVM, {E i } n i=1 , with linearly dependent POVM elements. The E i 's are positive-semidefinite operators with n i=1 E i = I, and n i=1 c i E i = 0 has a nontrivial solution for the c i 's. We use a labeling convention such that c 1 ≥ c 2 ≥ · · · ≥ c n , where c 1 , · · · , c m ≥ 0 and c m+1 , · · · , c n < 0. Note that c 1 > 0 and c n < 0 are always true, since otherwise they contradict the positivity of the E i 's. We can construct two POVMs, One can easily check that n j=2 E A j = I and n−1 j=1 E B j = I. We then perform {E A j } n j=2 with probability P A = c 1 /(c 1 −c n ) and perform {E B j } n−1 j=1 with probability P B = −c n /(c 1 −c n ). The probability of obtaining each outcome can be checked to agree with the original value Tr E i ρ for a given state ρ. For outcome n, (8) For outcome 1, (9) For the outcomes i, 2 ≤ i ≤ n − 1, This process reduces an n-outcome POVM into two (n − 1)-outcome POVMs with pre-processing. If either of {E A j } n j=2 and {E B j } n−1 j=1 is still linearly dependent, one can repeat the same process to reduce the number of outcomes until all POVMs are linearly independent. Hence, an n-outcome linearly dependent POVM is equivalent to randomly choosing from a set of LIPOVMs with certain probabilities. Since a qubit POVM has elements that are 2 by 2 matrices, any qubit POVM with more than 4 outcomes can always be decomposed into 4-or-feweroutcome LIPOVMs by pre-processing.
Definition 2 (PPOVM). We call a POVM with each element proportional to a projector a PPOVM, i.e., all where the a i 's are positive.
Proposition 2. Performing any qubit POVM is equivalent to performing a PPOVM followed by outputting an outcome with a probability that depends on the result of the PPOVM.
Proof. Given a qubit POVM {E i } n i=1 , each E i can be written as where a i , b i and |a i , |b i are the eigenvalues and eigenvectors of E i . Let a i ≥ b i for all i. We can rewrite each E i as We can rewrite E i in terms of theP i 's; i.e., where δ ik is the Kronecker delta and One can quickly check that n i=1 p(i|k) = 1 for any outcome k. Hence, the function p(i|k) can be interpreted as a conditional probability, in the sense that we first perform the PPOVM {P i } n i=1 and then output the final outcome i with probability p(i|k) when we got resultP k from the PPOVM. Therefore, performing a qubit POVM followed by post-processing as defined in Eq. (16).

Remark. If the qubit POVM
. This is because the transformation in Eq. (15) has an inverse. That isP where is linearly independent, then the PPOVM {P i } n i=1 will be linearly independent as well.
Combining the above properties, we have the following result: any qubit POVM is equivalent to pre-and postprocessing steps with a set of PPOVMs which are also LIPOVMs. That is, for a given qubit POVM, one uses a random number generator to choose, with probabilities given in Proposition 1, a LIPOVM, and uses Proposition 2 to find and perform the PPOVM corresponding to that LIPOVM; finally, one outputs the final outcome with probability p(·|·), defined in Eq. (16). The probability of outputting the final outcome i will be equal to Tr E i ρ, which is the probability of getting outcome i from the original POVM.
Note that the only actual measurement made in this process is the linearly independent PPOVM (LIP-POVM). Hence, for a qubit system, if one can perform any LIPPOVM then all POVMs can be accomplished. We claim that the model of lossy weak measurements [1] can achieve any qubit LIPPOVM and hence any qubit POVM. The following sections will verify the assertion.

A. POVM random walk in a simplex
In earlier work [6], it was shown that a generalized measurement can be decomposed into a series of stochastic processes such that the trajectory corresponds to a random walk in a simplex. Here we use a similar approach to parametrize the measurement operator during the sequence of weak measurements. The intuition is to map the current POVM element at each step of a sequence of weak measurements into a convex combination of the original POVM elements. We will show that, for a qubit LIPOVM, the conditions to perform a measurement where each outcome corresponds to a vector in the simplex can be satisfied.
To decompose a LIPOVM into stochastic processes of weak measurements, we use a family of measurement operators M ( x) of the form where U ( x) is a unitary operator, f ( x) is a normalization factor, and the (n−1)-simplex, n−1 . Note that for a given position x in the walk, This form characterizes how the POVM element of the weak measurement evolves as a convex combination of the original POVM elements. It will be shown in Sec. V that x will approach one of the vertices of n−1 after many measurements: x → (· · · , 0, 1, 0, · · · ). The vertex corresponds to the outcome E j of this POVM {E i } n i=1 . An example of performing a 3-outcome POVM is shown in Fig. 1.
Before making any measurement, x is at the middle of n−1 , so the random walk starts from and M ( x 0 ) = U ( x 0 ) = I. After each weak measurement is made, the new position x remains in n−1 if the weak measurement, with measurement operators satisfies for each outcome k such that Note that x k is a new position in the simplex n−1 after outcome k, and δ x k ≡ x k − x is expected to be a small quantity given that we are performing weak measurements. Using Eq. (18) for M ( x), we can rewrite Eq.
(23) as Define a new set of POVM elements by It can be easily checked that We can expand all the operators as linear combinations of Pauli operators and the identity; i.e., for each i, where σ is the vector of Pauli matrices and q i , v i are the expansion coefficients. By assumption, all q i 's are strictly positive. For convenience, the indication of x dependence for q i and v i have been dropped. We have where Using the property shown in Appendix A, the inverse square root of Eq. (29) is Similarly, we can write where and define δ r k ≡ r k − r. Inserting Eqs. (31) and (33) to Eq. (27), the condition becomes The transformation from Eq. (35) to Eq. (36) is by the properties of Pauli matrices and 1 − 2b + b 2 | r| 2 = 0 from Eq. (32). Hence, where the c k 's are positive constants to be determined. The condition to be a measurement, requires It is sufficient to solve the following conditions, Note that | r| cannot be 1 because | r| = 1 happens only if the E i 's are all proportional to the same projector, but this contradicts the assumption that the E i 's are linearly independent.
The question now becomes whether there exists a set of c k 's (> 0) and x k 's (∈ n−1 ) such that Eq. (40) is satisfied. First of all, n i=1 E i ( x) = I implies that q = (q 1 , · · · , q n ) ∈ n−1 and The v i 's and r are 3-dimensional Bloch vectors and x is an n-dimensional vector in n−1 , where n can only be 2, 3 or 4 in our case. It is shown in Appendix B that there is a one-to-one map between n−1 and the "allowed" space of r, denoted as r , For n = 2, 1 is a line and r is a line spanned by a pair of parallel and opposite-directioned vectors v 1,2 ; for n = 3, 2 is a regular triangle and r is a 2-D triangle stretched by v 1,2,3 ; for n = 4, 3 is a regular tetrahedron and r is a tetrahedron stretched by v 1,2,3,4 . An example of the n = 3 case is illustrated in Fig. 2 where the δ r k = r k − r are vectors lying in r . For example, when n = 3, the three vectors δ r k 's lie in a triangle and add up to zero vector. The three δ r k 's are given by three r k 's, which correspond to three x k 's in n−1 . In fact, we can solve for the set of x k 's if and only if we can solve for the set of δ r k 's.

B. Constraints by destructive weak measurements
So far we have shown that it is possible to perform a weak measurement while staying in a simplex. However, it was assumed that we can choose arbitrary sets of x k 's, which correspond to arbitrary weak measurements. We now show that even using a more restricted class of weak measurements-destructive weak measurement-Eq. (40) still can be satisfied. Therefore we can perform a series of destructive weak measurements such that it corresponds to a random walk in a simplex. However, the choice of δ r k 's in Eq. (40) are more restricted in this case.
In Sec. II, we reintroduced the model of destructive weak measurements in [1]. It consists of coupling the system with an ancilla qubit in state |0 by a weak swap operation, followed by a PPOVM on the ancilla. The outcomeP k = s k |e k e k | of this PPOVM corresponds to a measurement operator on the system. From the other direction, as shown in [1], for a set of eigenvectors and there exists a corresponding vector |e k ofP k on the ancilla that will give us those eigenvectors. With this |e k , the eigenvalues of M † k M k take the form On the other hand, one can check that if n k=1 M † k M k = I then the correspondingP k 's on the ancilla form a PPOVM.
This result can be translated into the Bloch sphere picture for M † k M k : where the unit vectorn k iŝ The explicit expression for M † k M k is Note that in Eq. (50) the vectorn k can be any unit vector, and the overall positive constant, can be any positive number because s k can be freely chosen. The only constraint imposed by Eq. (50) is that the Bloch vector's length, is determined by the z-component ofn k . On the other hand, the requirement that the POVM stay in the simplex means that the measurement operators must satisfy Eq. (37), which can be rewritten as wherer, δr k are the unit vectors of r, δ r k , and The term (r ⊥ · δr k )r ⊥ is just the orthogonal component of δr k in the directionr ⊥ . (We use hatsv to denote unit vectors.) The constraint is that the length of the Bloch vector of Eq. (53) must equal Eq. (52), where cos θ k ≡r · δr k and (56) Note that the right hand side of Eq. (55) depends only on the direction of δ r k (which is δr k ), and it is a number between 0 and sin φ. We can solve Eq. (55) by first determining the direction of δ r k and then solving for the length of δ r k . For a given current position r in the simplex r , we first choose the directions of δ r k 's by pointing from r toward the vertices of r ; i.e., for each outcome k, the direction of δ r k is chosen to be parallel to v k − r. Therefore, Inserting these directions back into Eq. (55), the left hand side becomes an increasing function of |δ r k |: it increases from 0 to 1 as |δ r k | goes from 0 to | v k − r|. For the directions δr k chosen above, we can aways find lengths |δ r k | to satisfy Eq. (55). For such δ r k 's, the c k 's can always be chosen to satisfy Eq. (40) because the δ r k 's are coplanar. Therefore, we have a set of δ r k 's (and hence x k 's) and c k 's that satisfy the constraints of the model of destructive weak measurement and the requirements to perform a weak measurement that corresponds to a random walk in a simplex.

V. THE PROBABILITY TO OBTAIN AN OUTCOME
So far we have shown that one can perform, by the model of destructive weak measurements, a series of weak measurements which keep the evolution of the corresponding positive operator in a simplex. In this section, we show that, at long times, the walk in the simplex will approach one of the vertices. Each vertex corresponds to an outcome of this series of weak measurements. The probability of obtaining an outcome k from the series of weak measurements approaches the probability of obtaining E k if we performed the original POVM, {E i } n i=1 . Note that we only have to consider the case of LIP-POVMs as discussed at the end of Sec. III. The following subsection will assume the POVM is a LIPPOVM.

A. Approaching a vertex
From the form of the measurement operator in Eq. (44), it is evident that after performing N steps, the measurement operators accumulate as where κ is a number depending on the string of outcomes k N · · · k 1 and The POVM element of this string of outcomes is Thus for large N , the POVM element approaches a projector, |ξ ξ|, at a rate cos 2N φ. On the other hand, because we only consider LI-PPOVMs, the simplex, r , has vertices on the surface of the Bloch sphere. Those vertices represent the LIPPOVM's elements which are proportional to projectors. The one-to-one correspondence between r and n−1 implies that their vertices are mapped to each other. Recall from Eq. (20) that when x approaches one of the vertices, the walk approaches a projector, which is one of the elements of the LIPPOVM. Therefore, in the large N limit, the random walk on the simplex given by the destructive weak measurements must approach a vertex.

B. The probability for each outcome
After N steps, we denote the position in the simplex as x (i) k N ···k1 if it is close to the vertex E i . The index k ν indicates the outcome of the νth weak measurement, and the νth measurement depends on all previous outcomes k ν−1 · · · k 1 . Recall from Eq. (20), that if the walk approaches the vertex E 1 , for example, after N steps, the POVM element becomes and C( x (1) k N ···k1 ) is a constant depending on the string of outcomes. The probability of obtaining the outcome "1" for a state ρ is where the summation is over all strings of outcomes k N · · · k 1 such that the walk approaches the vertex E 1 .
To show that p 1 indeed approaches the probability given by the usual Born's rule, Tr [ρE 1 ], it is equivalent to show thatp We first claim that the sum of the constants over the strings of outcomes, is bounded. Indeed, we can find a unit vector |ζ such that for all i ∈ {1, · · · , n}, ζ|E i |ζ = 0. Such a |ζ exists because {E i } n i=1 is a qubit LIPPOVM. From Eqs. (62) and (64), we have min{ ζ|E i |ζ } × k N ···k1∈"1" Hence, it is bounded by 1/min{ ζ|E i |ζ }. In the large N limit, for each i ∈ {2, · · · , n}, ( x (1) k N ···k1 ) i → 0 for all strings of outcomes k N · · · k 1 . Therefore, we have where max{( x (1) k N ···k1 ) i } is the maximum of ( x (1) k N ···k1 ) i among all strings of outcomes.
The coefficients, k N ···k1 ) i , approach zero for i = 1. For the same reason, other outcomes won't contribute to "1" either. Because are linearly independent, we have This verifies that p i does approach Tr [ρE i ], and the POVM element finally evolves close to a vertex E i with probability approaching Tr [ρE i ].

VI. CONCLUSION
The model of destructive weak measurement in [1] captures the behavior of a destructive strong measurement that leaves the system in a fixed final state after the measurement regardless of the outcome. The only freedom in this model is the ability to choose any PPOVM on the ancilla in an adaptive way. Surprisingly, [1] shows that, in spite of its limitations, this model can achieve any projective measurement for qubits, and the result can be generalized to any POVM with commuting POVM elements. However, the method used in [1] does not work for POVMs with non-commuting elements.
In this paper we have shown that any qubit POVM can be achieved by this model. The improvement from POVMs with commuting elements to all POVMs requires a more complicated generalization of the previous method. The approach combines classical preand post-processing with a continuous measurement that corresponds to a random walk in a simplex. The pre-processing randomly chooses a linearly independent POVM with elements proportional to projectors from the original POVM. The LIPPOVM is decomposed into a sequence of destructive weak measurements by mapping the evolution of the positive operator to a random walk in a simplex. The vertex the walk approaches corresponds to the final outcome of this succession of weak measurements. Finally, the post-processing chooses which result to output, conditioned on the outcome of the quantum measurement process. The probability of each result agrees with the probability of that result in the original POVM.
We have shown that a limited model of destructive weak measurements can achieve an arbitrary POVM for qubits. In higher dimensional systems, what measurements can be achieved by such a restricted class of weak measurements is an open question. The scheme of a system interacting with a stream of qubit probes was shown to be restricted to provide measurements with two distinct singular values [7]. Generalizing the ancilla to higher dimensions gives more degrees of freedom, and that may be a possible route to achieve POVMs for higher dimensional systems. This is the subject of ongoing research.
The inverse square root is Appendix B: Correspondence between r and n−1 The picture relating the position indicator x ∈ n−1 and its 3-D image r in Bloch sphere is presented in this section. The n can only be 2, 3 or 4 in our cases.
Recall from Eqs. (28) and (29), the definition of r is where x, q ∈ n−1 , q i > 0 for all i, and the v i 's are the Bloch vectors corresponding to a set of linearly independent POVM elements. We define a bijection T q : n−1 → n−1 for any q ∈ n−1 , where q i > 0 for all i = 1, · · · , n. The inverse map is given by We can define a vector x q ≡ T q ( x) ∈ n−1 associated with the original x. It is equivalent to analyze with x and with x q because of the bijection T q , i.e., we have a solution for x q if and only if we have a solution for x. We choose to work with x q since it is linearly related to r: is a 3 by n matrix. The space r lives in is which is a subspace spanned by { v 1 , · · · , v n }. We claim that rank(V ) = n − 1 for n = 2, 3, 4. The proof is as follows. Since n i=1 q i v i = 0 by Eq. (41), we have q ∈ Ker(V ). And by the linear independence of the set of POVM elements, we have that for any vector c = (c 1 , · · · , c n ) such that then c = 0. This implies that there is no nontrivial solution c satisfying both c · q = 0 and We can decompose the n-dimensional vector space into Span{ q} ⊕ Span{ q} ⊥ . Suppose a nontrivial vector y ∈ Span{ q} ⊥ is also in Ker(V ), i.e., y · q = 0 and , and hence no vector in Span{ q} ⊥ is also in Ker(V ). We have Span{ q} = Ker(V ) and therefore dim(Ker(V )) = 1. By the ranknullity theorem, we have rank(V ) = n − 1.
Finally, to show that the map V : n−1 → r is oneto-one, we use the following argument. If x 1,2 ∈ n−1 such that V x 1 = V x 2 , then x 1 − x 2 = a q ∈ Ker(V ) for a number a. By the fact that x 1,2 and q are in n−1 , a must be zero, and hence x 1 = x 2 . The map V : n−1 → r is onto, by the definition of r . Hence n−1 and r are isomorphic. We have dim( r ) = dim( n−1 ) = n − 1, and for n = 2, 3, 4, they are a line, a triangle, a tetrahedron.