Phase-space picture

An N-particle system can be represented in non-relativistic quantum mechanics by a wave function ${\displaystyle \psi (x_{1},x_{2},\dots ,x_{N})}$, where each x_i is a point in 3-dimensional space. This has analogies with the classical phase space. A classical phase space contains a real-valued function in 6N dimensions (each particle contributes 3 spatial coordinates and 3 momenta). In this case a "quantum" phase space, on the other hand, involves a complex-valued function on a 3N-dimensional space. The position and momenta are represented by operators that do not commute, and ${\displaystyle \psi }$ lives in the mathematical structure of a Hilbert space. Aside from these differences, however, the rough analogy holds.

Different previously isolated, non-interacting systems occupy different phase spaces. Alternatively we can say that they occupy different lower-dimensional subspaces in the phase space of the joint system. The effective dimensionality of a system's phase space is the number of degrees of freedom present, which—in non-relativistic models—is 6 times the number of a system's free particles. For a macroscopic system this will be a very large dimensionality. When two systems (the environment being one system) start to interact, though, their associated state vectors are no longer constrained to the subspaces. Instead the combined state vector time-evolves a path through the "larger volume", whose dimensionality is the sum of the dimensions of the two subspaces. The extent to which two vectors interfere with each other is a measure of how "close" they are to each other (formally, their overlap or Hilbert space multiplies together) in the phase space. When a system couples to an external environment, the dimensionality of, and hence "volume" available to, the joint state vector increases enormously. Each environmental degree of freedom contributes an extra dimension.

The original system's wave function can be expanded in many different ways as a sum of elements in a quantum superposition. Each expansion corresponds to a projection of the wave vector onto a basis. The basis can be chosen at will. Choosing an expansion where the resulting basis elements interact with the environment in an element-specific way, such elements will—with overwhelming probability—be rapidly separated from each other by their natural unitary time evolution along their own independent paths. After a very short interaction, there is almost no chance of further interference. The process is effectively irreversible. The different elements effectively become "lost" from each other in the expanded phase space created by coupling with the environment. In phase space, this decoupling is monitored through the Wigner quasi-probability distribution. The original elements are said to have decohered. The environment has effectively selected out those expansions or decompositions of the original state vector that decohere (or lose phase coherence) with each other. This is called "environmentally-induced superselection", or einselection.^[6] The decohered elements of the system no longer exhibit quantum interference between each other, as in a double-slit experiment. Any elements that decohere from each other via environmental interactions are said to be quantum-entangled with the environment. The converse is not true: not all entangled states are decohered from each other.

Any measuring device or apparatus acts as an environment, since at some stage along the measuring chain, it has to be large enough to be read by humans. It must possess a very large number of hidden degrees of freedom. In effect, the interactions may be considered to be quantum measurements. As a result of an interaction, the wave functions of the system and the measuring device become entangled with each other. Decoherence happens when different portions of the system's wave function become entangled in different ways with the measuring device. For two einselected elements of the entangled system's state to interfere, both the original system and the measuring in both elements device must significantly overlap, in the scalar product sense. If the measuring device has many degrees of freedom, it is very unlikely for this to happen.

As a consequence, the system behaves as a classical statistical ensemble of the different elements rather than as a single coherent quantum superposition of them. From the perspective of each ensemble member's measuring device, the system appears to have irreversibly collapsed onto a state with a precise value for the measured attributes, relative to that element. This provides one explanation of how the Born rule coefficients effectively act as probabilities as per the measurement postulate constituting a solution to the quantum measurement problem.

Dirac notation

Using Dirac notation, let the system initially be in the state

${\displaystyle |\psi \rangle =\sum _{i}|i\rangle \langle i|\psi \rangle ,}$

where the ${\displaystyle |i\rangle }$s form an einselected basis (environmentally induced selected eigenbasis^[6]), and let the environment initially be in the state ${\displaystyle |\epsilon \rangle }$. The vector basis of the combination of the system and the environment consists of the tensor products of the basis vectors of the two subsystems. Thus, before any interaction between the two subsystems, the joint state can be written as

${\displaystyle |{\text{before}}\rangle =\sum _{i}|i\rangle |\epsilon \rangle \langle i|\psi \rangle ,}$

where ${\displaystyle |i\rangle |\epsilon \rangle }$ is shorthand for the tensor product ${\displaystyle |i\rangle \otimes |\epsilon \rangle }$. There are two extremes in the way the system can interact with its environment: either (1) the system loses its distinct identity and merges with the environment (e.g. photons in a cold, dark cavity get converted into molecular excitations within the cavity walls), or (2) the system is not disturbed at all, even though the environment is disturbed (e.g. the idealized non-disturbing measurement). In general, an interaction is a mixture of these two extremes that we examine.

Loss of interference and the transition from quantum to classical probabilities

The utility of decoherence lies in its application to the analysis of probabilities, before and after environmental interaction, and in particular to the vanishing of quantum interference terms after decoherence has occurred. If we ask what is the probability of observing the system making a transition from ${\displaystyle \psi }$ to ${\displaystyle \phi }$ before ${\displaystyle \psi }$ has interacted with its environment, then application of the Born probability rule states that the transition probability is the squared modulus of the scalar product of the two states:

${\displaystyle \operatorname {prob} _{\text{before}}(\psi \to \phi )=\left|\langle \psi |\phi \rangle \right|^{2}=\left|\sum _{i}\psi _{i}^{*}\phi _{i}\right|^{2}=\sum _{i}|\psi _{i}^{*}\phi _{i}|^{2}+\sum _{ij;i\neq j}\psi _{i}^{*}\psi _{j}\phi _{j}^{*}\phi _{i},}$

where ${\displaystyle \psi _{i}=\langle i|\psi \rangle }$, ${\displaystyle \psi _{i}^{*}=\langle \psi |i\rangle }$, and ${\displaystyle \phi _{i}=\langle i|\phi \rangle }$ etc.

The above expansion of the transition probability has terms that involve ${\displaystyle i\neq j}$; these can be thought of as representing interference between the different basis elements or quantum alternatives. This is a purely quantum effect and represents the non-additivity of the probabilities of quantum alternatives.

To calculate the probability of observing the system making a quantum leap from ${\displaystyle \psi }$ to ${\displaystyle \phi }$ after ${\displaystyle \psi }$ has interacted with its environment, then application of the Born probability rule states that we must sum over all the relevant possible states ${\displaystyle |\epsilon _{i}\rangle }$ of the environment before squaring the modulus:

${\displaystyle \operatorname {prob} _{\text{after}}(\psi \to \phi )=\sum _{j}\,\left|\langle {\text{after}}\right|\phi ,\epsilon _{j}\rangle |^{2}=\sum _{j}\,\left|\sum _{i}\psi _{i}^{*}\langle i,\epsilon _{i}|\phi ,\epsilon _{j}\rangle \right|^{2}=\sum _{j}\left|\sum _{i}\psi _{i}^{*}\phi _{i}\langle \epsilon _{i}|\epsilon _{j}\rangle \right|^{2}.}$

The internal summation vanishes when we apply the decoherence/einselection condition ${\displaystyle \langle \epsilon _{i}|\epsilon _{j}\rangle \approx \delta _{ij}}$, and the formula simplifies to

${\displaystyle \operatorname {prob} _{\text{after}}(\psi \to \phi )\approx \sum _{j}|\psi _{j}^{*}\phi _{j}|^{2}=\sum _{i}|\psi _{i}^{*}\phi _{i}|^{2}.}$

If we compare this with the formula we derived before the environment introduced decoherence, we can see that the effect of decoherence has been to move the summation sign ${\displaystyle \textstyle \sum _{i}}$ from inside of the modulus sign to outside. As a result, all the cross- or quantum interference-terms

${\displaystyle \sum _{ij;i\neq j}\psi _{i}^{*}\psi _{j}\phi _{j}^{*}\phi _{i}}$

have vanished from the transition-probability calculation. The decoherence has irreversibly converted quantum behaviour (additive probability amplitudes) to classical behaviour (additive probabilities).^[6][7][8] However, Ballentine^[9] shows that the significant impact of decoherence to reduce interference need not have significance for the transition of quantum systems to classical limits.

In terms of density matrices, the loss of interference effects corresponds to the diagonalization of the "environmentally traced-over" density matrix.^[6]

Density-matrix approach

The effect of decoherence on density matrices is essentially the decay or rapid vanishing of the off-diagonal elements of the partial trace of the joint system's density matrix, i.e. the trace, with respect to any environmental basis, of the density matrix of the combined system and its environment. The decoherence irreversibly converts the "averaged" or "environmentally traced-over"^[6] density matrix from a pure state to a reduced mixture; it is this that gives the appearance of wave-function collapse. Again, this is called "environmentally induced superselection", or einselection.^[6] The advantage of taking the partial trace is that this procedure is indifferent to the environmental basis chosen.

Initially, the density matrix of the combined system can be denoted as

${\displaystyle \rho =|{\text{before}}\rangle \langle {\text{before}}|=|\psi \rangle \langle \psi |\otimes |\epsilon \rangle \langle \epsilon |,}$

where ${\displaystyle |\epsilon \rangle }$ is the state of the environment. Then if the transition happens before any interaction takes place between the system and the environment, the environment subsystem has no part and can be traced out, leaving the reduced density matrix for the system:

${\displaystyle \rho _{\text{sys}}=\operatorname {Tr} _{\textrm {env}}(\rho )=|\psi \rangle \langle \psi |\langle \epsilon |\epsilon \rangle =|\psi \rangle \langle \psi |.}$

Now the transition probability will be given as

${\displaystyle \operatorname {prob} _{\text{before}}(\psi \to \phi )=\langle \phi |\rho _{\text{sys}}|\phi \rangle =\langle \phi |\psi \rangle \langle \psi |\phi \rangle ={\big |}\langle \psi |\phi \rangle {\big |}^{2}=\sum _{i}|\psi _{i}^{*}\phi _{i}|^{2}+\sum _{ij;i\neq j}\psi _{i}^{*}\psi _{j}\phi _{j}^{*}\phi _{i},}$

where ${\displaystyle \psi _{i}=\langle i|\psi \rangle }$, ${\displaystyle \psi _{i}^{*}=\langle \psi |i\rangle }$, and ${\displaystyle \phi _{i}=\langle i|\phi \rangle }$ etc.

Now the case when transition takes place after the interaction of the system with the environment. The combined density matrix will be

${\displaystyle \rho =|{\text{after}}\rangle \langle {\text{after}}|=\sum _{i,j}\psi _{i}\psi _{j}^{*}|i,\epsilon _{i}\rangle \langle j,\epsilon _{j}|=\sum _{i,j}\psi _{i}\psi _{j}^{*}|i\rangle \langle j|\otimes |\epsilon _{i}\rangle \langle \epsilon _{j}|.}$

To get the reduced density matrix of the system, we trace out the environment and employ the decoherence/einselection condition and see that the off-diagonal terms vanish (a result obtained by Erich Joos and H. D. Zeh in 1985):^[10]

${\displaystyle \rho _{\text{sys}}=\operatorname {Tr} _{\text{env}}{\Big (}\sum _{i,j}\psi _{i}\psi _{j}^{*}|i\rangle \langle j|\otimes |\epsilon _{i}\rangle \langle \epsilon _{j}|{\Big )}=\sum _{i,j}\psi _{i}\psi _{j}^{*}|i\rangle \langle j|\langle \epsilon _{j}|\epsilon _{i}\rangle =\sum _{i,j}\psi _{i}\psi _{j}^{*}|i\rangle \langle j|\delta _{ij}=\sum _{i}|\psi _{i}|^{2}|i\rangle \langle i|.}$

Similarly, the final reduced density matrix after the transition will be

${\displaystyle \sum _{j}|\phi _{j}|^{2}|j\rangle \langle j|.}$

The transition probability will then be given as

${\displaystyle \operatorname {prob} _{\text{after}}(\psi \to \phi )=\sum _{i,j}|\psi _{i}|^{2}|\phi _{j}|^{2}\langle j|i\rangle \langle i|j\rangle =\sum _{i}|\psi _{i}^{*}\phi _{i}|^{2},}$

which has no contribution from the interference terms

${\displaystyle \sum _{ij;i\neq j}\psi _{i}^{*}\psi _{j}\phi _{j}^{*}\phi _{i}.}$

The density-matrix approach has been combined with the Bohmian approach to yield a reduced-trajectory approach, taking into account the system reduced density matrix and the influence of the environment.^[11]

Operator-sum representation

Consider a system S and environment (bath) B, which are closed and can be treated quantum-mechanically. Let ${\displaystyle {\mathcal {H}}_{S}}$ and ${\displaystyle {\mathcal {H}}_{B}}$ be the system's and bath's Hilbert spaces respectively. Then the Hamiltonian for the combined system is

${\displaystyle {\hat {H}}={\hat {H}}_{S}\otimes {\hat {I}}_{B}+{\hat {I}}_{S}\otimes {\hat {H}}_{B}+{\hat {H}}_{I},}$

where ${\displaystyle {\hat {H}}_{S},{\hat {H}}_{B}}$ are the system and bath Hamiltonians respectively, ${\displaystyle {\hat {H}}_{I}}$ is the interaction Hamiltonian between the system and bath, and ${\displaystyle {\hat {I}}_{S},{\hat {I}}_{B}}$ are the identity operators on the system and bath Hilbert spaces respectively. The time-evolution of the density operator of this closed system is unitary and, as such, is given by

${\displaystyle \rho _{SB}(t)={\hat {U}}(t)\rho _{SB}(0){\hat {U}}^{\dagger }(t),}$

where the unitary operator is ${\displaystyle {\hat {U}}=e^{-i{\hat {H}}t/\hbar }}$. If the system and bath are not entangled initially, then we can write ${\displaystyle \rho _{SB}=\rho _{S}\otimes \rho _{B}}$. Therefore, the evolution of the system becomes

${\displaystyle \rho _{SB}(t)={\hat {U}}(t)[\rho _{S}(0)\otimes \rho _{B}(0)]{\hat {U}}^{\dagger }(t).}$

The system–bath interaction Hamiltonian can be written in a general form as

${\displaystyle {\hat {H}}_{I}=\sum _{i}{\hat {S}}_{i}\otimes {\hat {B}}_{i},}$

where ${\displaystyle {\hat {S}}_{i}\otimes {\hat {B}}_{i}}$ is the operator acting on the combined system–bath Hilbert space, and ${\displaystyle {\hat {S}}_{i},{\hat {B}}_{i}}$ are the operators that act on the system and bath respectively. This coupling of the system and bath is the cause of decoherence in the system alone. To see this, a partial trace is performed over the bath to give a description of the system alone:

${\displaystyle \rho _{S}(t)=\operatorname {Tr} _{B}{\big [}{\hat {U}}(t)[\rho _{S}(0)\otimes \rho _{B}(0)]{\hat {U}}^{\dagger }(t){\big ]}.}$

${\displaystyle \rho _{S}(t)}$ is called the reduced density matrix and gives information about the system only. If the bath is written in terms of its set of orthogonal basis kets, that is, if it has been initially diagonalized, then ${\displaystyle \textstyle \rho _{B}(0)=\sum _{j}a_{j}|j\rangle \langle j|}$. Computing the partial trace with respect to this (computational) basis gives

${\displaystyle \rho _{S}(t)=\sum _{l}{\hat {A}}_{l}\rho _{S}(0){\hat {A}}_{l}^{\dagger },}$

where ${\displaystyle {\hat {A}}_{l},{\hat {A}}_{l}^{\dagger }}$ are defined as the Kraus operators and are represented as (the index ${\displaystyle l}$ combines indices ${\displaystyle k}$ and ${\displaystyle j}$):

${\displaystyle {\hat {A}}_{l}={\sqrt {a_{j}}}\langle k|{\hat {U}}|j\rangle .}$

This is known as the operator-sum representation (OSR). A condition on the Kraus operators can be obtained by using the fact that ${\displaystyle \operatorname {Tr} [\rho _{S}(t)]=1}$; this then gives

${\displaystyle \sum _{l}{\hat {A}}_{l}^{\dagger }{\hat {A}}_{l}={\hat {I}}_{S}.}$

This restriction determines whether decoherence will occur or not in the OSR. In particular, when there is more than one term present in the sum for ${\displaystyle \rho _{S}(t)}$, then the dynamics of the system will be non-unitary, and hence decoherence will take place.

Semigroup approach

A more general consideration for the existence of decoherence in a quantum system is given by the master equation, which determines how the density matrix of the system alone evolves in time (see also the Belavkin equation^[12][13][14] for the evolution under continuous measurement). This uses the Schrödinger picture, where evolution of the state (represented by its density matrix) is considered. The master equation is

${\displaystyle \rho '_{S}(t)={\frac {-i}{\hbar }}{\big [}{\tilde {H}}_{S},\rho _{S}(t){\big ]}+L_{D}{\big [}\rho _{S}(t){\big ]},}$

where ${\displaystyle {\tilde {H}}_{S}=H_{S}+\Delta }$ is the system Hamiltonian ${\displaystyle H_{S}}$ along with a (possible) unitary contribution ${\displaystyle \Delta }$ from the bath, and ${\displaystyle L_{D}}$ is the Lindblad decohering term.^[5] The Lindblad decohering term is represented as

${\displaystyle L_{D}{\big [}\rho _{S}(t){\big ]}={\frac {1}{2}}\sum _{\alpha ,\beta =1}^{M}b_{\alpha \beta }{\Big (}{\big [}\mathbf {F} _{\alpha },\rho _{S}(t)\mathbf {F} _{\beta }^{\dagger }{\big ]}+{\big [}\mathbf {F} _{\alpha }\rho _{S}(t),\mathbf {F} _{\beta }^{\dagger }{\big ]}{\Big )}.}$

The ${\displaystyle \{\mathbf {F} _{\alpha }\}_{\alpha =1}^{M}}$ are basis operators for the M-dimensional space of bounded operators that act on the system Hilbert space ${\displaystyle {\mathcal {H}}_{S}}$ and are the error generators.^[15] The matrix elements ${\displaystyle b_{\alpha \beta }}$ represent the elements of a positive semi-definite Hermitian matrix; they characterize the decohering processes and, as such, are called the noise parameters.^[15] The semigroup approach is particularly nice, because it distinguishes between the unitary and decohering (non-unitary) processes, which is not the case with the OSR. In particular, the non-unitary dynamics are represented by ${\displaystyle L_{D}}$, whereas the unitary dynamics of the state are represented by the usual Heisenberg commutator. Note that when ${\displaystyle L_{D}{\big [}\rho _{S}(t){\big ]}=0}$, the dynamical evolution of the system is unitary. The conditions for the evolution of the system density matrix to be described by the master equation are:^[5]

1. the evolution of the system density matrix is determined by a one-parameter semigroup
2. the evolution is "completely positive" (i.e. probabilities are preserved)
3. the system and bath density matrices are initially decoupled

Rotational decoherence

Consider a system of N qubits that is coupled to a bath symmetrically. Suppose this system of N qubits undergoes a rotation around the ${\displaystyle |{\uparrow }\rangle \langle {\uparrow }|,|{\downarrow }\rangle \langle {\downarrow }|}$ ${\displaystyle {\big (}|0\rangle \langle 0|,|1\rangle \langle 1|{\big )}}$ eigenstates of ${\displaystyle {\hat {J_{z}}}}$. Then under such a rotation, a random phase ${\displaystyle \phi }$ will be created between the eigenstates ${\displaystyle |0\rangle }$, ${\displaystyle |1\rangle }$ of ${\displaystyle {\hat {J_{z}}}}$. Thus these basis qubits ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$ will transform in the following way:

${\displaystyle |0\rangle \to |0\rangle ,\quad |1\rangle \to e^{i\phi }|1\rangle .}$

This transformation is performed by the rotation operator

${\displaystyle R_{z}(\phi )={\begin{pmatrix}1&0\\0&e^{i\phi }\end{pmatrix}}.}$

Since any qubit in this space can be expressed in terms of the basis qubits, then all such qubits will be transformed under this rotation. Consider the ${\displaystyle j}$th qubit in a pure state ${\displaystyle \vert \psi _{j}\rangle \langle \psi _{j}\vert }$ where ${\displaystyle |\psi _{j}\rangle =a|0\rangle +b|1\rangle }$. Before application of the rotation this state is:

${\displaystyle \rho _{j,{\text{init}}}={\begin{pmatrix}|a|^{2}&ab^{\ast }\\a^{\ast }b&|b|^{2}\end{pmatrix}}}$.

This state will decohere, since it is not ‘encoded’ with (dependent upon) the dephasing factor ${\displaystyle e^{i\phi }}$. This can be seen by examining the density matrix averaged over the random phase ${\displaystyle \phi }$:

${\displaystyle \rho _{j}=\mathbb {E} [R_{z}(\phi )\vert \psi _{j}\rangle \langle \psi _{j}\vert R_{z}^{\dagger }(\phi )]=\int \limits _{-\infty }^{\infty }R_{z}(\phi )|\psi _{j}\rangle \langle \psi _{j}|R_{z}^{\dagger }(\phi )\;P({\text{d}}\phi )}$,

where ${\displaystyle P(\cdot )}$ is a probability measure of the random phase, ${\displaystyle \phi }$. Although not entirely necessary, let us assume for simplicity that this is given by the Gaussian distribution, i.e. ${\displaystyle P({\text{d}}\phi )={\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {1}{2}}({\frac {\phi }{\sigma }})^{2}}\,{\text{d}}\phi }$, where ${\displaystyle \sigma }$ represents the spread of the random phase. Then the density matrix computed as above is

${\displaystyle \rho _{j}={\begin{pmatrix}|a|^{2}&ab^{\ast }\,e^{-{\frac {1}{2}}\sigma ^{2}}\\a^{\ast }b\,e^{-{\frac {1}{2}}\sigma ^{2}}&|b|^{2}\end{pmatrix}}}$.

Observe that the off-diagonal elements—the coherence terms—decay as the spread of the random phase, ${\displaystyle \sigma }$, increases over time (which is a realistic expectation). Thus the density matrices for each qubit of the system become indistinguishable over time. This means that no measurement can distinguish between the qubits, thus creating decoherence between the various qubit states. In particular, this dephasing process causes the qubits to collapse to one of the pure states in ${\displaystyle \{\vert 0\rangle \langle 0\vert ,\vert 1\rangle \langle 1\vert \}}$. This is why this type of decoherence process is called collective dephasing, because the mutual phases between all qubits of the N-qubit system are destroyed.

Depolarizing

Depolarizing is a non-unitary transformation on a quantum system which maps pure states to mixed states. This is a non-unitary process because any transformation that reverses this process will map states out of their respective Hilbert space thus not preserving positivity (i.e. the original probabilities are mapped to negative probabilities, which is not allowed). The 2-dimensional case of such a transformation would consist of mapping pure states on the surface of the Bloch sphere to mixed states within the Bloch sphere. This would contract the Bloch sphere by some finite amount and the reverse process would expand the Bloch sphere, which cannot happen.

Dissipation

Dissipation is a decohering process by which the populations of quantum states are changed due to entanglement with a bath. An example of this would be a quantum system that can exchange its energy with a bath through the interaction Hamiltonian. If the system is not in its ground state and the bath is at a temperature lower than that of the system's, then the system will give off energy to the bath, and thus higher-energy eigenstates of the system Hamiltonian will decohere to the ground state after cooling and, as such, will all be non-degenerate. Since the states are no longer degenerate, they are not distinguishable, and thus this process is irreversible (non-unitary).

Quantitative measurement

The decoherence rate depends on a number of factors, including temperature or uncertainty in position, and many experiments have tried to measure it depending on the external environment.^[18]

The process of a quantum superposition gradually obliterated by decoherence was quantitatively measured for the first time by Serge Haroche and his co-workers at the École Normale Supérieure in Paris in 1996.^[19] Their approach involved sending individual rubidium atoms, each in a superposition of two states, through a microwave-filled cavity. The two quantum states both cause shifts in the phase of the microwave field, but by different amounts, so that the field itself is also put into a superposition of two states. Due to photon scattering on cavity-mirror imperfection, the cavity field loses phase coherence to the environment.

Haroche and his colleagues measured the resulting decoherence via correlations between the states of pairs of atoms sent through the cavity with various time delays between the atoms.

Reducing environmental decoherence

In July 2011, researchers from University of British Columbia and University of California, Santa Barbara were able to reduce environmental decoherence rate "to levels far below the threshold necessary for quantum information processing" by applying high magnetic fields in their experiment.^[20][21][22]

In August 2020 scientists reported that ionizing radiation from environmental radioactive materials and cosmic rays may substantially limit the coherence times of qubits if they aren't shielded adequately which may be critical for realizing fault-tolerant superconducting quantum computers in the future.^[23][24][25]