Problem set 7

... Quantum Mechanics 3, Spring 2012 CMI Problem set 7 Due by beginning of class on Monday Mar 5, 2012 BCH formula for x and p , SHO 1. Consider the function f (t) = etA Be−tA where A, B are a pair of operators (e.g. position and momentum or creation and annihilation operators etc.). t is a parameter wh ...

... Quantum Mechanics 3, Spring 2012 CMI Problem set 7 Due by beginning of class on Monday Mar 5, 2012 BCH formula for x and p , SHO 1. Consider the function f (t) = etA Be−tA where A, B are a pair of operators (e.g. position and momentum or creation and annihilation operators etc.). t is a parameter wh ...

Aalborg Universitet The Landauer-Büttiker formula and resonant quantum transport

... dot levels across the fixed Fermi level of the system (recall that the latter is entirely controlled by the semi-infinite leads). Otherwise stated, the eigenvalues of H S (Vg ) equal the ones of H S (Vg = 0) (we denote them by {Ei }), up to a global shift Vg . Using the Landauer-Büttiker formula (8 ...

... dot levels across the fixed Fermi level of the system (recall that the latter is entirely controlled by the semi-infinite leads). Otherwise stated, the eigenvalues of H S (Vg ) equal the ones of H S (Vg = 0) (we denote them by {Ei }), up to a global shift Vg . Using the Landauer-Büttiker formula (8 ...

Explicit solution of the continuous Baker-Campbell

... term in the expansion of Q. Lower order terms of this expansion have been calculated in the literature before by iterative methods. Recently, Wilcox (5) carried out this calculation up to n = 4. It can be generally proved, that in every order of perturbation theory D is a linear combination of integ ...

... term in the expansion of Q. Lower order terms of this expansion have been calculated in the literature before by iterative methods. Recently, Wilcox (5) carried out this calculation up to n = 4. It can be generally proved, that in every order of perturbation theory D is a linear combination of integ ...

Entanglement measure for rank-2 mixed states

... to study the entanglement for a wide class of complex quantum systems. For example, consider a pure state 兩典 of an 共n + 1兲-partite quantum system AB1B2 ¯ Bn, where A is a qubit and B j are arbitrary quantum systems. Let be the state found by tracing out the qubit A. The I-tangle formula Eq. 共13兲 ...

... to study the entanglement for a wide class of complex quantum systems. For example, consider a pure state 兩典 of an 共n + 1兲-partite quantum system AB1B2 ¯ Bn, where A is a qubit and B j are arbitrary quantum systems. Let be the state found by tracing out the qubit A. The I-tangle formula Eq. 共13兲 ...

Properties of higher-order Trotter formulas

... Higher-order Trotter formulas provide a very general and systematic way to derive discrete approximations of path integrals for quantum mechanical or quantum statistical systems. These discretizations converge more rapidly while being only slightly more complicated than the usual low-order formulati ...

... Higher-order Trotter formulas provide a very general and systematic way to derive discrete approximations of path integrals for quantum mechanical or quantum statistical systems. These discretizations converge more rapidly while being only slightly more complicated than the usual low-order formulati ...

Time Evolution of States for Open Quantum

... and the state ρ̂S (0) of the system is pure i.e is an orthogonal projector on a unit vector ψ of HS . A non pure state will be called a mixed state. A density matrix ρ̂S is a mixed state if and only if ρ̂S has an eigenvalue λ, such that 0 < λ < 1. If ρ̂ is a pure state then ρ̂ = Πψ , ψ ∈ H, kψk = 1 ...

... and the state ρ̂S (0) of the system is pure i.e is an orthogonal projector on a unit vector ψ of HS . A non pure state will be called a mixed state. A density matrix ρ̂S is a mixed state if and only if ρ̂S has an eigenvalue λ, such that 0 < λ < 1. If ρ̂ is a pure state then ρ̂ = Πψ , ψ ∈ H, kψk = 1 ...

tions processing as well as in quantum information processing. In anal

... Information is quantized in classical digital informations processing as well as in quantum information processing. In analogy to the classical bit, the elementary quantum of information in quantum information processing is called a qubit. In the first part of this chapter we will learn how qubits c ...

... Information is quantized in classical digital informations processing as well as in quantum information processing. In analogy to the classical bit, the elementary quantum of information in quantum information processing is called a qubit. In the first part of this chapter we will learn how qubits c ...

Aalborg Universitet

... connected to ideal leads where the carriers are quasi free fermions, is completely characterized by a one particle scattering matrix. Many people have since contributed to the justification of this formalism, starting from the first principles of non-equilibrium quantum statistical mechanics. In thi ...

... connected to ideal leads where the carriers are quasi free fermions, is completely characterized by a one particle scattering matrix. Many people have since contributed to the justification of this formalism, starting from the first principles of non-equilibrium quantum statistical mechanics. In thi ...

Quantum Error Correction and Orthogonal Geometry

... A quantum error-correcting code is a way of encoding quantum states into qubits (two-state quantum systems) so that error or decoherence in a small number of individual qubits has little or no effect on the encoded data. The existence of quantum error-correcting codes was discovered only recently [1 ...

... A quantum error-correcting code is a way of encoding quantum states into qubits (two-state quantum systems) so that error or decoherence in a small number of individual qubits has little or no effect on the encoded data. The existence of quantum error-correcting codes was discovered only recently [1 ...

A DIRECT PROOF OF THE QUANTUM VERSION OF MONK`S

... Ωw (F• ) = {V• ∈ F`(E) | dim(Vp ∩ Fq ) ≥ p − rw (p, n − q) ∀p, q} where rw (p, q) = #{i ≤ p | w(i) ≤ q}. The codimension of this variety is equal to the length `(w) of w. Notice that the rank conditions on Vp are redundant unless w has a descent at position p, i.e. w(p) > w(p + 1). Given a sequence ...

... Ωw (F• ) = {V• ∈ F`(E) | dim(Vp ∩ Fq ) ≥ p − rw (p, n − q) ∀p, q} where rw (p, q) = #{i ≤ p | w(i) ≤ q}. The codimension of this variety is equal to the length `(w) of w. Notice that the rank conditions on Vp are redundant unless w has a descent at position p, i.e. w(p) > w(p + 1). Given a sequence ...

On a Quantum Version of Pieri`s Formula

... substitution is well defined, due to Lemma 2.1. Our main result can be stated as follows: Theorem 3.1 (Quantum Pieri’s formula) Let I be a subset in {1, 2, . . . , n}, and let J = {1, 2, . . . , n} \ I. Then, for k ≥ 1, we have in the algebra Enp : X Ek (θI ; p) = τa1 b1 τa2 b2 · · · τak bk , ...

... substitution is well defined, due to Lemma 2.1. Our main result can be stated as follows: Theorem 3.1 (Quantum Pieri’s formula) Let I be a subset in {1, 2, . . . , n}, and let J = {1, 2, . . . , n} \ I. Then, for k ≥ 1, we have in the algebra Enp : X Ek (θI ; p) = τa1 b1 τa2 b2 · · · τak bk , ...

PPT - Institute of Physics, Bhubaneswar

... • If two parties don’t share a reference frame, rotationally invariant states are required to communicate. Bartlett et al., Rev. Mod. Phys. 79, 555 (2007) • If we encode in a DFS, universal quantum computing can be performed on a subspace even though it is not possible on the whole Hilbert space. Ke ...

... • If two parties don’t share a reference frame, rotationally invariant states are required to communicate. Bartlett et al., Rev. Mod. Phys. 79, 555 (2007) • If we encode in a DFS, universal quantum computing can be performed on a subspace even though it is not possible on the whole Hilbert space. Ke ...

High-fidelity Z-measurement error encoding of optical qubits

... in Fig. 4. The average fidelity for all of these states 共excluding , = 0° and 90°兲 is 0.96± 0.03, which is the same, to within error, as for the reconstructed states shown in Fig. 3. The behavior observed in Fig. 4 can be explained in terms of the classical and nonclassical interference requireme ...

... in Fig. 4. The average fidelity for all of these states 共excluding , = 0° and 90°兲 is 0.96± 0.03, which is the same, to within error, as for the reconstructed states shown in Fig. 3. The behavior observed in Fig. 4 can be explained in terms of the classical and nonclassical interference requireme ...

Quantum State Reconstruction From Incomplete Data

... prepared in the state , which is arbitrary but same for all qubits. The state of reservoir is described by the density matrix ...

... prepared in the state , which is arbitrary but same for all qubits. The state of reservoir is described by the density matrix ...

The Quantum Error Correcting Criteria

... where Ckl is a hermitian matrix. This equation is called the quantum error correcting criteria. It tells us when our encoding into a subspace can protect us from quantum errors Ek . As such it is a very important criteria for the theory of quantum error correction. Let’s show that this is a necessar ...

... where Ckl is a hermitian matrix. This equation is called the quantum error correcting criteria. It tells us when our encoding into a subspace can protect us from quantum errors Ek . As such it is a very important criteria for the theory of quantum error correction. Let’s show that this is a necessar ...

Coherent and incoherent evolution of qubits in

... waves and double defects • Spin qubits at defects and the use of control spins ...

... waves and double defects • Spin qubits at defects and the use of control spins ...

Unitary time evolution

... This simple result has many profound consequences. For one, the state |ψi of a system is not an observable. Given a quantum system, there is no way to tell in what state |ψi it was prepared. If the state |ψi is known, the state can be “copied” by preparing another system. But it is impossible to co ...

... This simple result has many profound consequences. For one, the state |ψi of a system is not an observable. Given a quantum system, there is no way to tell in what state |ψi it was prepared. If the state |ψi is known, the state can be “copied” by preparing another system. But it is impossible to co ...

superconducting qubits solid state qubits

... “Charge qubits” and “spin qubits” The qubits levels can be formed by either the energy levels of an electron in a potential well (such as a quantum dot or an impurity ion) or by the spin states of the electron (or the nucleus). The former are examples of charge qubits. The charge qubits have high e ...

... “Charge qubits” and “spin qubits” The qubits levels can be formed by either the energy levels of an electron in a potential well (such as a quantum dot or an impurity ion) or by the spin states of the electron (or the nucleus). The former are examples of charge qubits. The charge qubits have high e ...

Lecture 4: Some Properties of Qubits Introduction A Brief Recap

... • So we can deduce a0 and a1 by making lots of measurements and just counting n(0) and n(1), right? • No: remember that after the measurement, the state of the system collapses onto the result of the measurement • We could prepare a large number of qubits in the same state and measure them all: this ...

... • So we can deduce a0 and a1 by making lots of measurements and just counting n(0) and n(1), right? • No: remember that after the measurement, the state of the system collapses onto the result of the measurement • We could prepare a large number of qubits in the same state and measure them all: this ...

QUANTUM ERROR CORRECTING CODES FROM THE

... formalism" that may lead to such a general method. In particular, we cast the general problem of finding correctable codes for quantum channels into a matrix analysis framework. We then utilize a new tool recently introduced in [15]---called the "higher-rank numerical range"--the study of which was ...

... formalism" that may lead to such a general method. In particular, we cast the general problem of finding correctable codes for quantum channels into a matrix analysis framework. We then utilize a new tool recently introduced in [15]---called the "higher-rank numerical range"--the study of which was ...

QUANTUM ERROR CORRECTING CODES FROM THE

... primarily to mitigate the effects of such operators on quantum information encoded in evolving systems. By “active quantum error correction”, we mean protocols that involve active intervention into the system to correct errors. The basic method for active quantum error correction [1–4] identifies qu ...

... primarily to mitigate the effects of such operators on quantum information encoded in evolving systems. By “active quantum error correction”, we mean protocols that involve active intervention into the system to correct errors. The basic method for active quantum error correction [1–4] identifies qu ...