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Indeed, I agree with you, standard notation is, in my personal view, already sufficiently clear and bra-ket notation should be used when it is really useful. A typical case in QM is when a state vector is determined by a set of quantum numbers like this $$\left|l m s \right\rangle$$ Another case concerns the use of the so-called occupation numbers $$\left|n_{k_1} n_{k_2}\right\rangle$$ in QFT. Also q-bits notation for states $\left|0\right\rangle$, $\left|1\right\rangle$ in quantum information theory is meaningful... Finally the use of bra ket notation permits one to denote orthogonal projectors onto subspaces in a very effective manner $$\sum_{|m|\leq l}\left|l m \right\rangle \left\langle l m\right|\:.$$

A reason for its, in my view, nowadays not completely justified use is historical and due to the famous P.A.M. Dirac's textbook. In the 1930s, mathematical objects like Hilbert spaces and dual spaces, self-adjoint operators, were not very familiar mathematical tools to physicists. (The modern notion of Hilbert space was invented in 1932 by J. von Neumann in his less famous textbook on mathematical foundations of QM.) Dirac proposed a very nice notation which embodied a fundamental part of the formalism. However it also includes some drawbacks. In particular, manipulating non-self adjoint operators, e.g., symmetries, turns out to be very cumbersome within bra-ket formalism. If $A$ is self-adjoint, in $\left\langle \psi\right| A\left| \phi\right\rangle$ the operator can be viewed, indifferently, as acting on the left or on the right. If the operator is not self-adjoint this is false.

I think bra-ket notation is a very useful tool, but should be used "cum grano salis" in QM. In my view $\left|\psi\right\rangle$ where $\psi$ is a qunatum mechanics wavefunction, may be a dangerous notation, especially for students, as it generates misleading questions like this, $A\left|\psi\right\rangle = \left|A\psi \right\rangle$?

ADDENDUM. I understand that I interpreted the question into a broader view, regarding the use of bra-ket notation in QM rather than the restricted field of quantum information theory.

Indeed, I agree with you, standard notation is, in my personal view, already sufficiently clear and bra-ket notation should be used when it is really useful. A typical case in QM is when a state vector is determined by a set of quantum numbers like this $$\left|l m s \right\rangle$$ Another case concerns the use of the so-called occupation numbers $$\left|n_{k_1} n_{k_2}\right\rangle$$ in QFT. Also q-bits notation for states $\left|0\right\rangle$, $\left|1\right\rangle$ in quantum information theory is meaningful... Finally the use of bra ket notation permits one to denote orthogonal projectors onto subspaces in a very effective manner $$\sum_{|m|\leq l}\left|l m \right\rangle \left\langle l m\right|\:.$$

A reason for its, in my view, nowadays not completely justified use is historical and due to the famous P.A.M. Dirac's textbook. In the 1930s, mathematical objects like Hilbert spaces and dual spaces, self-adjoint operators, were not very familiar mathematical tools to physicists. (The modern notion of Hilbert space was invented in 1932 by J. von Neumann in his less famous textbook on mathematical foundations of QM.) Dirac proposed a very nice notation which embodied a fundamental part of the formalism. However it also includes some drawbacks. In particular, manipulating non-self adjoint operators, e.g., symmetries, turns out to be very cumbersome within bra-ket formalism. If $A$ is self-adjoint, in $\left\langle \psi\right| A\left| \phi\right\rangle$ the operator can be viewed, indifferently, as acting on the left or on the right preserving the final result. If the operator is not self-adjoint this is false.

I think bra-ket notation is a very useful tool, but should be used "cum grano salis" in QM. In my view $\left|\psi\right\rangle$ where $\psi$ is a qunatum mechanics wavefunction, may be a dangerous notation, especially for students, as it generates misleading questions like this, $A\left|\psi\right\rangle = \left|A\psi \right\rangle$?

ADDENDUM. I understand that I interpreted the question into a broader view, regarding the use of bra-ket notation in QM rather than the restricted field of quantum information theory.

Indeed, I agree with you, standard notation is, in my personal view, already sufficiently clear and bra-ket notation should be used when it is really useful. A typical case in QM is when a state vector is determined by a set of quantum numbers like this $$\left|l m s \right\rangle$$ Another case concerns the use of the so-called occupation numbers $$\left|n_{k_1} n_{k_2}\right\rangle$$ in QFT. Also q-bits notation for states $\left|0\right\rangle$, $\left|1\right\rangle$ in quantum information theory is meaningful... Finally the use of bra ket notation permits one to denote orthogonal projectors onto subspaces in a very effective manner $$\sum_{|m|\leq l}\left|l m \right\rangle \left\langle l m\right|\:.$$

A reason for its, in my view, nowadays not completely justified use is historical and due to the famous P.A.M. Dirac's textbook. In the 30ths of past century, mathematical objects like Hilbert spaces and dual spaces, self-adjoint operators, were not very familiar mathematical tools to physicists. (The modern notion of Hilbert space was invented in 1932 by J. von Neumann in his less famous textbook on mathematical foundations of QM.) Dirac proposed a very nice notation which embodied a fundamental part of the formalism. However it also includes some drawbacks. In particular, manipulating non-self adjoint operators, e.g., symmetries, turns out to be very cumbersome within bra-ket formalism. If $A$ is self-adjoint, in $\left\langle \psi\right| A\left| \phi\right\rangle$ the operator can be viewed, indifferently, as acting on the left or on the right. If the operator is not self-adjoint this is false.

I think bra-ket notation is a very useful tool, but should be used "cum grano salis" in QM. In my view $\left|\psi\right\rangle$ where $\psi$ is a qunatum mechanics wavefunction, may be a dangerous notation, especially for students, as it generates misleading questions like this, $A\left|\psi\right\rangle = \left|A\psi \right\rangle$?

ADDENDUM. I understand that I interpreted the question into a broader view, regarding the use of bra-ket notation in QM rather than the restricted field of quantum information theory.

Indeed, I agree with you, standard notation is, in my personal view, already sufficiently clear and bra-ket notation should be used when it is really useful. A typical case in QM is when a state vector is determined by a set of quantum numbers like this $$\left|l m s \right\rangle$$ Another case concerns the use of the so-called occupation numbers $$\left|n_{k_1} n_{k_2}\right\rangle$$ in QFT. Also q-bits notation for states $\left|0\right\rangle$, $\left|1\right\rangle$ in quantum information theory is meaningful... Finally the use of bra ket notation permits one to denote orthogonal projectors onto subspaces in a very effective manner $$\sum_{|m|\leq l}\left|l m \right\rangle \left\langle l m\right|\:.$$

A reason for its, in my view, nowadays not completely justified use is historical and due to the famous P.A.M. Dirac's textbook. In the 1930s, mathematical objects like Hilbert spaces and dual spaces, self-adjoint operators, were not very familiar mathematical tools to physicists. (The modern notion of Hilbert space was invented in 1932 by J. von Neumann in his less famous textbook on mathematical foundations of QM.) Dirac proposed a very nice notation which embodied a fundamental part of the formalism. However it also includes some drawbacks. In particular, manipulating non-self adjoint operators, e.g., symmetries, turns out to be very cumbersome within bra-ket formalism. If $A$ is self-adjoint, in $\left\langle \psi\right| A\left| \phi\right\rangle$ the operator can be viewed, indifferently, as acting on the left or on the right. If the operator is not self-adjoint this is false.

I think bra-ket notation is a very useful tool, but should be used "cum grano salis" in QM. In my view $\left|\psi\right\rangle$ where $\psi$ is a qunatum mechanics wavefunction, may be a dangerous notation, especially for students, as it generates misleading questions like this, $A\left|\psi\right\rangle = \left|A\psi \right\rangle$?

ADDENDUM. I understand that I interpreted the question into a broader view, regarding the use of bra-ket notation in QM rather than the restricted field of quantum information theory.

Indeed, I agree with you, standard notation is, in my personal view, already sufficiently clear and bra-ket notation should be used when it is really useful. A typical case in QM is when a state vector is determined by a set of quantum numbers like this $$|l m s \rangle$$ Another case concerns the use of the so-called occupation numbers $$|n_{k_1} n_{k_2}\rangle$$ in QFT. Also q-bits notation for states $|0\rangle$, $|1\rangle$ in quantum information theory is meaningful... Finally the use of bra ket notation permits one to denote orthogonal projectors onto subspaces in a very effective manner $$\sum_{|m|\leq l}|l m \rangle \langle l m|\:.$$

A reason for its, in my view, nowadays not completely justified use is historical and due to the famous P.A.M. Dirac's textbook. In the 30ths of past century, mathematical objects like Hilbert spaces and dual spaces, self-adjoint operators, were not very familiar mathematical tools to physicists. (The modern notion of Hilbert space was invented in 1932 by J. von Neumann in his less famous textbook on mathematical foundations of QM.) Dirac proposed a very nice notation which embodied a fundamental part of the formalism. However it also includes some drawbacks. In particular, manipulating non-self adjoint operators, e.g., symmetries, turns out to be very cumbersome within bra-ket formalism. If $A$ is self-adjoint, in $\langle \psi| A| \phi\rangle$ the operator can be viewed, indifferently, as acting on the left or on the right. If the operator is not self-adjoint this is false.

I think bra-ket notation is a very useful tool, but should be used "cum grano salis" in QM. In my view $|\psi\rangle$ where $\psi$ is a qunatum mechanics wavefunction, may be a dangerous notation, especially for students, as it generates misleading questions like this, $A|\psi\rangle = |A\psi \rangle$?

ADDENDUM. I understand that I interpreted the question into a broader view, regarding the use of bra-ket notation in QM rather than the restricted field of quantum information theory.

Indeed, I agree with you, standard notation is, in my personal view, already sufficiently clear and bra-ket notation should be used when it is really useful. A typical case in QM is when a state vector is determined by a set of quantum numbers like this $$\left|l m s \right\rangle$$ Another case concerns the use of the so-called occupation numbers $$\left|n_{k_1} n_{k_2}\right\rangle$$ in QFT. Also q-bits notation for states $\left|0\right\rangle$, $\left|1\right\rangle$ in quantum information theory is meaningful... Finally the use of bra ket notation permits one to denote orthogonal projectors onto subspaces in a very effective manner $$\sum_{|m|\leq l}\left|l m \right\rangle \left\langle l m\right|\:.$$

A reason for its, in my view, nowadays not completely justified use is historical and due to the famous P.A.M. Dirac's textbook. In the 30ths of past century, mathematical objects like Hilbert spaces and dual spaces, self-adjoint operators, were not very familiar mathematical tools to physicists. (The modern notion of Hilbert space was invented in 1932 by J. von Neumann in his less famous textbook on mathematical foundations of QM.) Dirac proposed a very nice notation which embodied a fundamental part of the formalism. However it also includes some drawbacks. In particular, manipulating non-self adjoint operators, e.g., symmetries, turns out to be very cumbersome within bra-ket formalism. If $A$ is self-adjoint, in $\left\langle \psi\right| A\left| \phi\right\rangle$ the operator can be viewed, indifferently, as acting on the left or on the right. If the operator is not self-adjoint this is false.

I think bra-ket notation is a very useful tool, but should be used "cum grano salis" in QM. In my view $\left|\psi\right\rangle$ where $\psi$ is a qunatum mechanics wavefunction, may be a dangerous notation, especially for students, as it generates misleading questions like this, $A\left|\psi\right\rangle = \left|A\psi \right\rangle$?

ADDENDUM. I understand that I interpreted the question into a broader view, regarding the use of bra-ket notation in QM rather than the restricted field of quantum information theory.