diff --git a/.github/workflows/main.yml b/.github/workflows/main.yml
index 31f9fc848..b6cc81115 100644
--- a/.github/workflows/main.yml
+++ b/.github/workflows/main.yml
@@ -37,6 +37,32 @@ jobs:
           QISKIT_IN_PARALLEL: TRUE
         run: tox -e py
         if: runner.os == 'macOS'
+  tests-no-opt:
+    name: tests-python3.8-no-optional-dependencies
+    runs-on: ${{ matrix.os }}
+    strategy:
+      matrix:
+        python-version: [3.8]
+        os: ["ubuntu-latest"]
+    steps:
+      - uses: actions/checkout@v2
+      - name: Set up Python ${{ matrix.python-version }}
+        uses: actions/setup-python@v2
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Pip cache
+        uses: actions/cache@v2
+        with:
+          path: ~/.cache/pip
+          key: ${{ runner.os }}-${{ matrix.python-version }}-pip-tests-${{ hashFiles('setup.py','requirements-dev.txt','constraints.txt') }}
+          restore-keys: |
+            ${{ runner.os }}-${{ matrix.python-version }}-pip-tests-
+            ${{ runner.os }}-${{ matrix.python-version }}-pip-
+            ${{ runner.os }}-${{ matrix.python-version }}
+      - name: Install Deps
+        run: python -m pip install -U tox setuptools virtualenv wheel
+      - name: Install and Run Tests
+        run: tox -e no-opt
   windows-tests:
     name: tests-python${{ matrix.python-version }}-windows
     runs-on: windows-latest
diff --git a/.pylintrc b/.pylintrc
index 7626bcf45..e8956278b 100644
--- a/.pylintrc
+++ b/.pylintrc
@@ -295,7 +295,7 @@ ignore-mixin-members=yes
 # (useful for modules/projects where namespaces are manipulated during runtime
 # and thus existing member attributes cannot be deduced by static analysis. It
 # supports qualified module names, as well as Unix pattern matching.
-ignored-modules=matplotlib.cm,numpy.random,retworkx
+ignored-modules=matplotlib.cm,numpy.random,retworkx,numba
 
 # List of class names for which member attributes should not be checked (useful
 # for classes with dynamically set attributes). This supports the use of
diff --git a/constraints.txt b/constraints.txt
index 180ca0f9a..2344507f4 100644
--- a/constraints.txt
+++ b/constraints.txt
@@ -2,3 +2,4 @@ pylint==2.4.4
 astroid==2.3.3
 pywin32==225
 setuptools==49.6.0
+pyfakefs==4.1.0
diff --git a/docs/conf.py b/docs/conf.py
index f097808ae..7a542a1b1 100644
--- a/docs/conf.py
+++ b/docs/conf.py
@@ -46,7 +46,7 @@
 # The short X.Y version
 version = ''
 # The full version, including alpha/beta/rc tags
-release = '0.5.0'
+release = '0.5.1'
 
 # -- General configuration ---------------------------------------------------
 
diff --git a/qiskit/ignis/VERSION.txt b/qiskit/ignis/VERSION.txt
index 8f0916f76..4b9fcbec1 100644
--- a/qiskit/ignis/VERSION.txt
+++ b/qiskit/ignis/VERSION.txt
@@ -1 +1 @@
-0.5.0
+0.5.1
diff --git a/qiskit/ignis/mitigation/expval/base_meas_mitigator.py b/qiskit/ignis/mitigation/expval/base_meas_mitigator.py
index c6b445b3f..bed5b6387 100644
--- a/qiskit/ignis/mitigation/expval/base_meas_mitigator.py
+++ b/qiskit/ignis/mitigation/expval/base_meas_mitigator.py
@@ -166,13 +166,13 @@ def stddev_upper_bound(self, shots: int = 1, qubits: Optional[List[int]] = None)
         return gamma / np.sqrt(shots)
 
     def plot_assignment_matrix(self,
-                               qubits: Optional[List[int]] = None,
-                               ax: Optional[plt.axes] = None) -> plt.axes:
+                               qubits=None,
+                               ax=None):
         """Matrix plot of the readout error assignment matrix.
 
         Args:
-            qubits: Optional, qubits being measured for operator expval.
-            ax: Optional. Axes object to add plot to.
+            qubits (list(int)): Optional, qubits being measured for operator expval.
+            ax (axes): Optional. Axes object to add plot to.
 
         Returns:
             plt.axes: the figure axes object.
@@ -195,13 +195,13 @@ def plot_assignment_matrix(self,
         return ax
 
     def plot_mitigation_matrix(self,
-                               qubits: Optional[List[int]] = None,
-                               ax: Optional[plt.axes] = None) -> plt.axes:
+                               qubits=None,
+                               ax=None):
         """Matrix plot of the readout error mitigation matrix.
 
         Args:
-            qubits: Optional, qubits being measured for operator expval.
-            ax: Optional. Axes object to add plot to.
+            qubits (list(int)): Optional, qubits being measured for operator expval.
+            ax (plt.axes): Optional. Axes object to add plot to.
 
         Returns:
             plt.axes: the figure axes object.
@@ -247,12 +247,12 @@ def _int_to_bitstring(i, num_qubits=None):
         return label
 
     @staticmethod
-    def _plot_axis(mat: np.ndarray, ax: plt.axes) -> plt.axes:
+    def _plot_axis(mat, ax):
         """Helper function for setting up axes for plots.
 
         Args:
-            mat: the N-qubit matrix to plot.
-            ax: Optional. Axes object to add plot to.
+            mat (np.ndarray): the N-qubit matrix to plot.
+            ax (plt.axes): Optional. Axes object to add plot to.
 
         Returns:
             plt.axes: the figure object and axes object.
diff --git a/qiskit/ignis/verification/tomography/fitters/base_fitter.py b/qiskit/ignis/verification/tomography/fitters/base_fitter.py
index 00322cba7..d31531d08 100644
--- a/qiskit/ignis/verification/tomography/fitters/base_fitter.py
+++ b/qiskit/ignis/verification/tomography/fitters/base_fitter.py
@@ -40,6 +40,7 @@ class TomographyFitter:
     """Base maximum-likelihood estimate tomography fitter class"""
 
     _HAS_SDP_SOLVER = None
+    _HAS_SDP_SOLVER_NOT_SCS = False
 
     def __init__(self,
                  result: Union[Result, List[Result]],
@@ -120,12 +121,14 @@ def fit(self,
             **kwargs) -> np.array:
         r"""Reconstruct a quantum state using CVXPY convex optimization.
 
-                **Fitter method**
+        **Fitter method**
 
-        The ``cvx`` fitter method used CVXPY convex optimization package.
-        The ``lstsq`` method uses least-squares fitting (linear inversion).
-        The ``auto`` method will use 'cvx' if the CVXPY package is found on
-        the system, otherwise it will default to 'lstsq'.
+        The ``'cvx'`` fitter method uses the CVXPY convex optimization package
+        with a SDP solver.
+        The ``'lstsq'`` method uses least-squares fitting.
+        The ``'auto'`` method will use ``'cvx'`` if the both the CVXPY and a suitable
+        SDP solver packages are found on the system, otherwise it will default
+        to ``'lstsq'``.
 
         **Objective function**
 
@@ -165,9 +168,14 @@ def fit(self,
         **CVXPY Solvers:**
 
         Various solvers can be called in CVXPY using the `solver` keyword
-        argument. See the `CVXPY documentation
+        argument. If ``psd=True`` an SDP solver is required other an SOCP
+        solver is required. See the `CVXPY documentation
         <https://www.cvxpy.org/tutorial/advanced/index.html#solve-method-options>`_
         for more information on solvers.
+        Note that the default SDP solver ('SCS') distributed
+        with CVXPY will not be used for the ``'auto'`` method due its reduced
+        accuracy compared to other solvers. When using the ``'cvx'`` method we
+        strongly recommend installing one of the other supported SDP solvers.
 
         References:
 
@@ -200,7 +208,11 @@ def fit(self,
         # Choose automatic method
         if method == 'auto':
             self._check_for_sdp_solver()
-            if self._HAS_SDP_SOLVER:
+            if self._HAS_SDP_SOLVER_NOT_SCS:
+                # We don't use the SCS solver for automatic method as it has
+                # lower accuracy than the other supported SDP solvers which
+                # typically results in the returned matrix not being
+                # completely positive.
                 method = 'cvx'
             else:
                 method = 'lstsq'
@@ -516,7 +528,9 @@ def _check_for_sdp_solver(cls):
                 # pylint:disable=import-error
                 import cvxpy
                 solvers = cvxpy.installed_solvers()
-                if 'CVXOPT' in solvers:
+                # Check for other SDP solvers cvxpy supports
+                if 'CVXOPT' in solvers or 'MOSEK' in solvers:
+                    cls._HAS_SDP_SOLVER_NOT_SCS = True
                     cls._HAS_SDP_SOLVER = True
                     return
                 if 'SCS' in solvers:
@@ -524,7 +538,7 @@ def _check_for_sdp_solver(cls):
                     # SCS solver cannot solver larger than 2x2 matrix
                     # problems without BLAS
                     try:
-                        var = cvxpy.Variable((4, 4), PSD=True)
+                        var = cvxpy.Variable((5, 5), PSD=True)
                         obj = cvxpy.Minimize(cvxpy.norm(var))
                         cvxpy.Problem(obj).solve(solver='SCS')
                         cls._HAS_SDP_SOLVER = True
diff --git a/qiskit/ignis/verification/tomography/fitters/cvx_fit.py b/qiskit/ignis/verification/tomography/fitters/cvx_fit.py
index 7c2e9a9a5..9f7aeee08 100644
--- a/qiskit/ignis/verification/tomography/fitters/cvx_fit.py
+++ b/qiskit/ignis/verification/tomography/fitters/cvx_fit.py
@@ -198,6 +198,8 @@ def cvx_fit(data: np.array,
     if 'solver' not in kwargs:
         if 'CVXOPT' in cvxpy.installed_solvers():
             kwargs['solver'] = 'CVXOPT'
+        elif 'MOSEK' in cvxpy.installed_solvers():
+            kwargs['solver'] = 'MOSEK'
 
     problem_solved = False
     while not problem_solved:
diff --git a/releasenotes/notes/no-auto-scs-b82ebef53508fc7b.yaml b/releasenotes/notes/no-auto-scs-b82ebef53508fc7b.yaml
new file mode 100644
index 000000000..71ab7a4d6
--- /dev/null
+++ b/releasenotes/notes/no-auto-scs-b82ebef53508fc7b.yaml
@@ -0,0 +1,13 @@
+---
+fixes:
+  - |
+    Fix the ``"auto"`` method of the
+    :class:`~qiskit.ignis.verification.tomography.TomographyFitter`,
+    :class:`~qiskit.ignis.verification.tomography.StateTomographyFitter`, and
+    :class:`~qiskit.ignis.verification.tomography.ProcessTomographyFitter` to
+    only use ``"cvx"`` if CVXPY is installed *and* a third-party SDP solver
+    other than SCS is available. This is because the SCS solver has lower
+    accuracy than other solver methods and often returns a density matrix or
+    Choi-matrix that is not completely-positive and fails validation when used
+    with the :func:`qiskit.quantum_info.state_fidelity` or
+    :func:`qiskit.quantum_info.process_fidelity` functions.