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Introduction to Quantum Information Processing Lecture 4 Michele Mosca

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Overview l Von Neumann measurements l General measurements l Traces and density matrices and partial traces

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“Von Neumann measurement in the computational basis” l Suppose we have a universal set of quantum gates, and the ability to measure each qubit in the basis l If we measure we get with probability

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In section 2.2.5, this is described as follows l We have the projection operators and satisfying l We consider the projection operator or “observable” l Note that 0 and 1 are the eigenvalues l When we measure this observable M, the probability of getting the eigenvalue is and we are in that case left with the state

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“Expected value” of an observable If we associate with outcome the eigenvalue then the expected outcome is

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“Von Neumann measurement in the computational basis” l Suppose we have a universal set of quantum gates, and the ability to measure each qubit in the basis l Say we have the state l If we measure all n qubits, then we obtain with probability l Notice that this means that probability of measuring a in the first qubit equals

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Partial measurements l (This is similar to Bayes Theorem) l If we only measure the first qubit and leave the rest alone, then we still get with probability l The remaining n-1 qubits are then in the renormalized state

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In section 2.2.5 l This partial measurement corresponds to measuring the observable

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Von Neumann Measurements l A Von Neumann measurement is a type of projective measurement. Given an orthonormal basis, if we perform a Von Neumann measurement with respect to of the state then we measure with probability

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Von Neumann Measurements l E.x. Consider Von Neumann measurement of the state with respect to the orthonormal basis l Note that l We therefore getwith probability

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Von Neumann Measurements l Note that

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How do we implement Von Neumann measurements? l If we have access to a universal set of gates and bit-wise measurements in the computational basis, we can implement Von Neumann measurements with respect to an arbitrary orthonormal basisas follows.

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How do we implement Von Neumann measurements? l Construct a quantum network that implements the unitary transformation l Then “conjugate” the measurement operation with the operation

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Another approach

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Ex. Bell basis change l Consider the orthonormal basis consisting of the “Bell” states l Note that

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Bell measurement l We can “destructively” measure l Or non-destructively project

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Most general measurement

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Trace of a matrix The trace of a matrix is the sum of its diagonal elements e.g. Some properties: Orthonormal basis { }

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Density Matrices Notice that 0 = 0| , and 1 = 1| . So the probability of getting 0 when measuring | is: where = | | is called the density matrix for the state |

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Mixture of pure states A state described by a state vector | is called a pure state. What if we have a qubit which is known to be in the pure state | 1 with probability p 1, and in | 2 with probability p 2 ? More generally, consider probabilistic mixtures of pure states (called mixed states):

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Density matrix of a mixed state …then the probability of measuring 0 is given by conditional probability: whereis the density matrix for the mixed state Density matrices contain all the useful information about an arbitrary quantum state.

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Density Matrix If we apply the unitary operation U to the resulting state is with density matrix

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Density Matrix If we apply the unitary operation U to the resulting state is with density matrix

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Density Matrix If we perform a Von Neumann measurement of the state wrt a basis containing, the probability of obtaining is

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Density Matrix If we perform a Von Neumann measurement of the state wrt a basis containing the probability of obtaining is

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Density Matrix In other words, the density matrix contains all the information necessary to compute the probability of any outcome in any future measurement.

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Spectral decomposition l Often it is convenient to rewrite the density matrix as a mixture of its eigenvectors l Recall that eigenvectors with distinct eigenvalues are orthogonal; for the subspace of eigenvectors with a common eigenvalue (“degeneracies”), we can select an orthonormal basis

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Spectral decomposition l In other words, we can always “diagonalize” a density matrix so that it is written as where is an eigenvector with eigenvalue and forms an orthonormal basis

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Partial Trace l How can we compute probabilities for a partial system? l E.g.

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Partial Trace l If the 2 nd system is taken away and never again (directly or indirectly) interacts with the 1 st system, then we can treat the first system as the following mixture l E.g.

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Partial Trace

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Why? l the probability of measuring e.g. in the first register depends only on

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Partial Trace l Notice that it doesn’t matter in which orthonormal basis we “trace out” the 2 nd system, e.g. l In a different basis

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Partial Trace

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Distant transformations don’t change the local density matrix l Notice that the previous observation implies that a unitary transformation on the system that is traced out does not affect the result of the partial trace l I.e.

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Distant transformations don’t change the local density matrix l In fact, any legal quantum transformation on the traced out system, including measurement (without communicating back the answer) does not affect the partial trace l I.e.

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Why?? l Operations on the 2 nd system should not affect the statistics of any outcomes of measurements on the first system l Otherwise a party in control of the 2 nd system could instantaneously communicate information to a party controlling the 1 st system.

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Principle of implicit measurement l If some qubits in a computation are never used again, you can assume (if you like) that they have been measured (and the result ignored) l The “reduced density matrix” of the remaining qubits is the same

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Partial Trace l This is a linear map that takes bipartite states to single system states. l We can also trace out the first system l We can compute the partial trace directly from the density matrix description

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Partial Trace using matrices l Tracing out the 2 nd system

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Most general measurement

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