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4. Consider a quantum system Q described by a Hilbert space H. (a) Suppose we are given a subspace Ho of H and a linear map from kets in Ho to others in h. That is, Vo: e) ), in a linear way. This map preserves inner products of vectors in Ho, so that (l) (vlo). Show that there exists a unitary operator V on H such that ) = Vl) for all |) e Ho. (Hint: Use the fact that any orthonormal set of vectors may be extended to a complete basis.) This allows us to build up a "unitary-like" map on a subspace into a full unitary operator, and it can be used for the following: (b) Start with a map E on density operators for Q with Kraus representation E(p) = E AupA;. Append a system E in state 10) and consider the following linear map on kets: lw) e 10) (Ale)) le4). where {les)} is a fixed orthonormal basis of HE). Show that there exists a unitary UQE) such that E) = Tr)(UQE(p 8 |0)(0)(UQ)t) In other words, every & on Q that has a Kraus representation can be realized as the result of unitary evolution on a larger system QE.

4. Consider a quantum system Q described by a Hilbert space H. (a) Suppose we are given a subspace Ho of H and a linear map from kets in Ho to others in h. That is, Vo: e) ), in a linear way. This map preserves inner products of vectors in Ho, so that (l) (vlo). Show that there exists a unitary operator V on H such that ) = Vl) for all |) e Ho. (Hint: Use the fact that any orthonormal set of vectors may be extended to a complete basis.) This allows us to build up a "unitary-like" map on a subspace into a full unitary operator, and it can be used for the following: (b) Start with a map E on density operators for Q with Kraus representation E(p) = E AupA;. Append a system E in state 10) and consider the following linear map on kets: lw) e 10) (Ale)) le4). where {les)} is a fixed orthonormal basis of HE). Show that there exists a unitary UQE) such that E) = Tr)(UQE(p 8 |0)(0)(UQ)t) In other words, every & on Q that has a Kraus representation can be realized as the result of unitary evolution on a larger system QE.

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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No SIM 10:04 AM 70% Thursday Edit 9:32 AM 4. Consider a quantum system Q described by a Hilbert space H. (a) Suppose we are given a subspace Ho of H and a linear map from kets in Ho to others in h. That is, Vo : |) + |), in a linear way. This map preserves inner products of vectors in Ho, so that (o) = (wlø). Show that there exists a unitary operator V on H such that ) = V|) for all |) e Ho. (Hint: Use the fact that any orthonormal set of vectors may be extended to a complete basis.) This allows us to build up a "unitary-like" map on a subspace into a full unitary operator, and it can be used for the following: (b) Start with a map E on density operators for Q with Kraus representation E(p) = AnpA;, £AAk = 1. Append a system E in state |0) and consider the following linear map on kets: where {Je)} is a fixed orthonormal basis of HE). Show that there exists a unitary UQE) such that E(p) = Tr)(U(QE)(p 8 |0)(0|)(U(QE)t) In other words, every E on Q that has a Kraus representation can be realized as the result of unitary evolution on a larger system QE.
Transcribed Image Text:No SIM 10:04 AM 70% Thursday Edit 9:32 AM 4. Consider a quantum system Q described by a Hilbert space H. (a) Suppose we are given a subspace Ho of H and a linear map from kets in Ho to others in h. That is, Vo : |) + |), in a linear way. This map preserves inner products of vectors in Ho, so that (o) = (wlø). Show that there exists a unitary operator V on H such that ) = V|) for all |) e Ho. (Hint: Use the fact that any orthonormal set of vectors may be extended to a complete basis.) This allows us to build up a "unitary-like" map on a subspace into a full unitary operator, and it can be used for the following: (b) Start with a map E on density operators for Q with Kraus representation E(p) = AnpA;, £AAk = 1. Append a system E in state |0) and consider the following linear map on kets: where {Je)} is a fixed orthonormal basis of HE). Show that there exists a unitary UQE) such that E(p) = Tr)(U(QE)(p 8 |0)(0|)(U(QE)t) In other words, every E on Q that has a Kraus representation can be realized as the result of unitary evolution on a larger system QE.
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