We propose and analyze a simple framework for estimating the amplitudes of a given $n$ -qubit quantum state $\ketψ = \sum_{i=0}^{2^n-1} a_i \ket{i}$ in computational basis, utilizing a single-qubit measurement only. Previously, it was a common procedure that one could measure all qubits in order to collect measurement outcomes, from which one can estimate amplitudes of given quantum state. Here, we show that if restricting to single-qubit measurement, and one can perform measurement on arbitrary basis, then the measurement outcomes can be used to assist the finding of amplitudes in the usual computational, or Z basis. More concretely, such outcomes are capable of constructing a system of nonlinear algebraic equations, and by classically solving them, we obtain $\Tilde{a}_i$, which is the approximation to the corresponding amplitudes $a_i$ , including both real and imaginary component. We then discuss our framework from a broader perspective. First, we show that estimating all (norms of) amplitudes to additive accuracy $δ$ , i.e., $| |\Tilde{a}_i - |a_i| | \leq δ$ for all $i$ , $\mathcal{O}(4^n/δ^4)$ single-qubit measurements is sufficient. Second, we show that to achieve total variation $\sum_{i=0}^{2^n-1} | |\Tilde{a}_i|^2 - |a_i|^2| \leq δ$, $\mathcal{O}(6^n/δ^4)$ a single bit measurement is required. Finally, in order to achieve an average $L_1$ norm error $ \sum_{i=0}^{2^n-1} | |\Tilde{a}_i| - |a_i| |/2^n \leq δ$, a single bit measurement $\mathcal{O}(2^n/ δ^4)$ is needed.We propose and analyze a simple framework for estimating the amplitudes of a given $n$-qubit quantum state $\ketψ = \sum_{i=0}^{2^n-1} a_i \ket{i}$ in computational basis, utilizing a single-qubit measurement only. Previously, it was a common procedure that one could measure all qubits in order to collect measurement outcomes, from which one can estimate amplitudes of given quantum state. Here, we show that if restricting to single-qubit measurement, and one can perform measurement on arbitrary basis, then the measurement outcomes can be used to assist the finding of amplitudes in the usual computational, or Z basis. More concretely, such outcomes are capable of constructing a system of nonlinear algebraic equations, and by classically solving them, we obtain $\Tilde{a}_i$, which is the approximation to the corresponding amplitudes $a_i$, including both real and imaginary component. We then discuss our framework from a broader perspective. First, we show that estimating all (norms of) amplitudes to additive accuracy $δ$, i.e., $| |\Tilde{a}_i - |a_i| | \leq δ$ for all $i$, $\mathcal{O}(4^n/δ^4)$ single-qubit measurements is sufficient. Second, we show that to achieve total variation $\sum_{i=0}^{2^n-1} | |\Tilde{a}_i|^2 - |a_i|^2| \leq δ$, $\mathcal{O}(6^n/δ^4)$ a single bit measurement is required. Finally, in order to achieve an average $L_1$ norm error $ \sum_{i=0}^{2^n-1} | |\Tilde{a}_i| - |a_i| |/2^n \leq δ$, a single bit measurement $\mathcal{O}(2^n/ δ^4)$ is needed.

arXiv:2412.07123Dec 10, 20240 citations

Transition amplitudes

Elliot Leader

Cambridge University Press eBooksJul 13, 20230 citations

A theory of amplitudes

Stanley Gudder

A stochastic framework for quantum mechanics is presented in which the elements are measurements and amplitudes. The resulting structure provides three stochastic levels. At the bottom level are the sample points. These may be unobservable, in general. At the next level are the outcomes of measurements. The amplitude of an outcome is computed by summing the amplitudes of the sample points resulting in that outcome upon a measurement. At the highest level are the events of a measurement. The probability of an event is computed by summing the moduli squared of the amplitudes for the outcomes comprising that event. Using these guidelines, various probabilities, conditional probabilities, and expectations are defined. Superpositions of amplitude functions are investigated and superselection sectors are shown to occur in a natural way. The framework is illustrated in various mathematical models such as spin- (1)/(2) models, the two-slit experiment, and phase space quantum mechanics. Finally, a theory of discrete Feynman amplitudes is presentedA stochastic framework for quantum mechanics is presented in which the elements are measurements and amplitudes. The resulting structure provides three stochastic levels. At the bottom level are the sample points. These may be unobservable, in general. At the next level are the outcomes of measurements. The amplitude of an outcome is computed by summing the amplitudes of the sample points resulting in that outcome upon a measurement. At the highest level are the events of a measurement. The probability of an event is computed by summing the moduli squared of the amplitudes for the outcomes comprising that event. Using these guidelines, various probabilities, conditional probabilities, and expectations are defined. Superpositions of amplitude functions are investigated and superselection sectors are shown to occur in a natural way. The framework is illustrated in various mathematical models such as spin- (1)/(2) models, the two-slit experiment, and phase space quantum mechanics. Finally, a theory of discrete Feynman amplitudes is presented

Journal of Mathematical PhysicsSep 1, 198822 citations

Amplitudes

The MIT Press eBooksOct 20, 20200 citations

The geometric measure of entanglement of pure states with nonnegative amplitudes and the spectral theory of nonnegative tensors

Shenglong Hu, Liqun Qi, Guofeng Zhang

The geometric measure of entanglement for a symmetric pure state with nonnegative amplitudes has attracted much attention. On the other hand, the spectral theory of nonnegative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of nonnegative tensors can be applied to the study of the geometric measure of entanglement for a pure state with nonnegative amplitudes. Especially, an elimination method for computing the geometric measure of entanglement for symmetric pure multipartite qubit or qutrit states with nonnegative amplitudes is given. For symmetric pure multipartite qudit states with nonnegative amplitudes, a numerical algorithm with randomization is presented and proven to be convergent. We show that for the geometric measure of entanglement for pure states with nonnegative amplitudes, the nonsymmetric ones can be converted to the symmetric ones.The geometric measure of entanglement for a symmetric pure state with nonnegative amplitudes has attracted much attention. On the other hand, the spectral theory of nonnegative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of nonnegative tensors can be applied to the study of the geometric measure of entanglement for a pure state with nonnegative amplitudes. Especially, an elimination method for computing the geometric measure of entanglement for symmetric pure multipartite qubit or qutrit states with nonnegative amplitudes is given. For symmetric pure multipartite qudit states with nonnegative amplitudes, a numerical algorithm with randomization is presented and proven to be convergent. We show that for the geometric measure of entanglement for pure states with nonnegative amplitudes, the nonsymmetric ones can be converted to the symmetric ones.

arXiv:1203.3675Mar 16, 201215 citations

We propose and analyze a simple framework for estimating the amplitudes of a given

n

-qubit quantum state

\ketψ = \sum_{i=0}^{2^n-1} a_i \ket{i}

in computational basis, utilizing a single-qubit measurement only. Previously, it was a common procedure that one could measure all qubits in order to collect measurement outcomes, from which one can estimate amplitudes of given quantum state. Here, we show that if restricting to single-qubit measurement, and one can perform measurement on arbitrary basis, then the measurement outcomes can be used to assist the finding of amplitudes in the usual computational, or Z basis. More concretely, such outcomes are capable of constructing a system of nonlinear algebraic equations, and by classically solving them, we obtain $\Tilde{a}_i$, which is the approximation to the corresponding amplitudes

a_i

, including both real and imaginary component. We then discuss our framework from a broader perspective. First, we show that estimating all (norms of) amplitudes to additive accuracy

δ

, i.e., $| |\Tilde{a}_i - |a_i| | \leq δ$ for all

i

\mathcal{O}(4^n/δ^4)

single-qubit measurements is sufficient. Second, we show that to achieve total variation $\sum_{i=0}^{2^n-1} | |\Tilde{a}_i|^2 - |a_i|^2| \leq δ$,

\mathcal{O}(6^n/δ^4)

a single bit measurement is required. Finally, in order to achieve an average

L_1

norm error $ \sum_{i=0}^{2^n-1} | |\Tilde{a}_i| - |a_i| |/2^n \leq δ$, a single bit measurement

\mathcal{O}(2^n/ δ^4)

is needed.