Введение
Объекты, подобные распределениям вероятностей, нарушающие сигма-аддитивность; полезны в вычислительной физике.
Распределение квазивероятностей – это математический объект, схожий с распределением вероятностей, но ослабляющий некоторые аксиомы теории вероятностей Колмогорова. Квазивероятности обладают рядом общих свойств с обычными вероятностями, важнейшим из которых является возможность вычисления математических ожиданий относительно весов распределения. Однако они могут нарушать аксиому сигма-аддитивности: интегрирование по ним не обязательно дает вероятности взаимоисключающих состояний. Более того, распределения квазивероятностей также имеют области отрицательной плотности вероятности, что, контринтуитивно, противоречит первой аксиоме. Распределения квазивероятностей естественно возникают при изучении квантовой механики в фазовом пространстве, что обычно используется в квантовой оптике, временном и частотном анализе и других областях.
Objects like probability distributions that violate σ additivity; useful in computational physics
A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, the ability to yield expectation values with respect to the weights of the distribution. However, they can violate the σ additivity axiom: integrating over them does not necessarily yield probabilities of mutually exclusive states. Indeed, quasiprobability distributions also have regions of negative probability density, counterintuitively, contradicting the first axiom. Quasiprobability distributions arise naturally in the study of quantum mechanics when treated in phase space formulation, commonly used in quantum optics, time frequency analysis, and elsewhere.
A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, the ability to yield expectation values with respect to the weights of the distribution. However, they can violate the σ additivity axiom: integrating over them does not necessarily yield probabilities of mutually exclusive states. Indeed, quasiprobability distributions also have regions of negative probability density, counterintuitively, contradicting the first axiom. Quasiprobability distributions arise naturally in the study of quantum mechanics when treated in phase space formulation, commonly used in quantum optics, time frequency analysis, and elsewhere.
Введение
В наиболее общей форме динамика квантово-механической системы определяется мастер-уравнением в гильбертовом пространстве: уравнением движения для оператора плотности (обычно записываемого как ρ) системы. Оператор плотности определяется относительно полной ортонормальной базы. Хотя возможно напрямую интегрировать это уравнение для очень малых систем (т. е., систем с небольшим числом частиц или степеней свободы), это быстро становится неразрешимым для больших систем. Однако можно доказать, что оператор плотности всегда может быть записан в диагональной форме, при условии, что он относится к переполненной (сверхполной) основе. Когда оператор плотности представлен в такой переполненной основе, он может быть записан таким образом, который больше напоминает обычную функцию, ценой того, что функция обладает свойствами квазивероятностного распределения. Эволюция системы затем полностью определяется эволюцией функции квазивероятностного распределения. Когерентные состояния, т. е. собственные правые состояния оператора уничтожения a, служат переполненной основой в описанной выше конструкции. По определению, когерентные состояния обладают следующим свойством, а также некоторыми другими интересными свойствами. Например, никакие два когерентных состояния не являются ортогональными. Действительно, если |α⟩ и |β⟩ – пара когерентных состояний, то
In the most general form, the dynamics of a quantum mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written ) of the system. The density operator is defined with respect to a complete orthonormal basis. Although it is possible to directly integrate this equation for very small systems (i. e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove that the density operator can always be written in a diagonal form, provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in a manner more resembling of an ordinary function, at the expense that the function has the features of a quasiprobability distribution. The evolution of the system is then completely determined by the evolution of the quasiprobability distribution function. The coherent states, i. e. right eigenstates of the annihilation operator serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property,
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. Proof of the overcompleteness of the coherent statesIntegration over the complex plane can be written in terms of polar coordinates with Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
Clearly, one can span the Hilbert space by writing a state as
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete. In the coherent states basis, however, it is always possible which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. Proof of the overcompleteness of the coherent statesIntegration over the complex plane can be written in terms of polar coordinates with Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
Clearly, one can span the Hilbert space by writing a state as
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete. In the coherent states basis, however, it is always possible which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
Отметим, что эти состояния, однако, правильно нормированы: ⟨α|α⟩ = 1. В силу полноты базиса состояний Фока, выбор базиса когерентных состояний должен быть переполненным (сверхполным). Нажмите, чтобы увидеть неформальное доказательство. Доказательство переполненности когерентных состояний. Интегрирование по комплексной плоскости может быть записано в полярных координатах, где обмен суммой и интегралом допустим, что приводит к простому интегральному выражению для гамма-функции:
In the most general form, the dynamics of a quantum mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written ) of the system. The density operator is defined with respect to a complete orthonormal basis. Although it is possible to directly integrate this equation for very small systems (i. e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove that the density operator can always be written in a diagonal form, provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in a manner more resembling of an ordinary function, at the expense that the function has the features of a quasiprobability distribution. The evolution of the system is then completely determined by the evolution of the quasiprobability distribution function. The coherent states, i. e. right eigenstates of the annihilation operator serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property,
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. Proof of the overcompleteness of the coherent statesIntegration over the complex plane can be written in terms of polar coordinates with Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
Clearly, one can span the Hilbert space by writing a state as
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete. In the coherent states basis, however, it is always possible which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. Proof of the overcompleteness of the coherent statesIntegration over the complex plane can be written in terms of polar coordinates with Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
Clearly, one can span the Hilbert space by writing a state as
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete. In the coherent states basis, however, it is always possible which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
Очевидно, что можно охватить гильбертово пространство, записав состояние как
In the most general form, the dynamics of a quantum mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written ) of the system. The density operator is defined with respect to a complete orthonormal basis. Although it is possible to directly integrate this equation for very small systems (i. e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove that the density operator can always be written in a diagonal form, provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in a manner more resembling of an ordinary function, at the expense that the function has the features of a quasiprobability distribution. The evolution of the system is then completely determined by the evolution of the quasiprobability distribution function. The coherent states, i. e. right eigenstates of the annihilation operator serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property,
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. Proof of the overcompleteness of the coherent statesIntegration over the complex plane can be written in terms of polar coordinates with Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
Clearly, one can span the Hilbert space by writing a state as
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete. In the coherent states basis, however, it is always possible which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. Proof of the overcompleteness of the coherent statesIntegration over the complex plane can be written in terms of polar coordinates with Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
Clearly, one can span the Hilbert space by writing a state as
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete. In the coherent states basis, however, it is always possible which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
С другой стороны, несмотря на правильную нормализацию состояний, фактор π > 1 доказывает, что эта база является переполненной. Однако в базисе когерентных состояний всегда возможно выражение, связанное с симметричным упорядочением операторов. В квантовой оптике, в частности, операторы, представляющие интерес, особенно оператор числа частиц, часто естественно выражаются в нормальном порядке. В этом случае соответствующее представление фазового пространства является представлением Глаубера — Сударшана P. Квазивероятностная природа этих распределений фазового пространства лучше всего проявляется в представлении P благодаря следующему ключевому утверждению:
In the most general form, the dynamics of a quantum mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written ) of the system. The density operator is defined with respect to a complete orthonormal basis. Although it is possible to directly integrate this equation for very small systems (i. e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove that the density operator can always be written in a diagonal form, provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in a manner more resembling of an ordinary function, at the expense that the function has the features of a quasiprobability distribution. The evolution of the system is then completely determined by the evolution of the quasiprobability distribution function. The coherent states, i. e. right eigenstates of the annihilation operator serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property,
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. Proof of the overcompleteness of the coherent statesIntegration over the complex plane can be written in terms of polar coordinates with Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
Clearly, one can span the Hilbert space by writing a state as
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete. In the coherent states basis, however, it is always possible which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. Proof of the overcompleteness of the coherent statesIntegration over the complex plane can be written in terms of polar coordinates with Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
Clearly, one can span the Hilbert space by writing a state as
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete. In the coherent states basis, however, it is always possible which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
Это широкое утверждение не работает в других представлениях. Например, функция Вигнера состояния ЭПР положительно определена, но не имеет классического аналога. Помимо представлений, определенных выше, существует множество других квазивероятностных распределений, возникающих в альтернативных представлениях фазового пространства. Другое популярное представление – представление Хусими Q, которое полезно, когда операторы находятся в антинормальном порядке. В последнее время положительное представление P и более широкий класс обобщенных представлений P использовались для решения сложных задач в квантовой оптике. Все они эквивалентны и могут быть преобразованы друг в друга, а именно, функция распределения классов Коэна.
In the most general form, the dynamics of a quantum mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written ) of the system. The density operator is defined with respect to a complete orthonormal basis. Although it is possible to directly integrate this equation for very small systems (i. e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove that the density operator can always be written in a diagonal form, provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in a manner more resembling of an ordinary function, at the expense that the function has the features of a quasiprobability distribution. The evolution of the system is then completely determined by the evolution of the quasiprobability distribution function. The coherent states, i. e. right eigenstates of the annihilation operator serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property,
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. Proof of the overcompleteness of the coherent statesIntegration over the complex plane can be written in terms of polar coordinates with Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
Clearly, one can span the Hilbert space by writing a state as
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete. In the coherent states basis, however, it is always possible which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
They also have some further interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. Proof of the overcompleteness of the coherent statesIntegration over the complex plane can be written in terms of polar coordinates with Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function:
Clearly, one can span the Hilbert space by writing a state as
On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete. In the coherent states basis, however, it is always possible which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.